Asymptotic Behavior of Solutions to a Doubly Weighted Quasi-linear Equation

Abstract

We establish pointwise asymptotic estimates at infinity for solutions to the doubly weighted quasi-linear equation: ${\begin{aligned} - div (w_{1} | \nabla u |^{p - 2} \nabla u) & = w_{2} | u |^{q - 2} u & in Ω, \\ u \in D^{1, p, w_{1}} & (Ω), \end{aligned}$ where $w_{1}$ and $w_{2}$ are compatible weights and $q > p > 1$ is a critical exponent in the sense of Sobolev.

Keywords

weighted quasilinear equations decay at infinity local boundedness Harnack inequality

1. Introduction

In this article, we study qualitative and quantitative properties of weak solutions to the following equation:

{\begin{aligned} - div (w_{1} {| \nabla u |}^{p - 2} \nabla u) & = w_{2} {| u |}^{q - 2} u & in Ω, \\ u & \in D^{1, p, w_{1}} (Ω), \end{aligned}

(1)

for weights

w_{1}, w_{2}

and

q > p > 1

critical for the weighted Sobolev embedding from

D^{1, p, w_{1}} (Ω)

(defined before Theorem 4) into

L^{q, w_{2}} (Ω)

. In particular, we are interested in the point-wise asymptotic behavior of a solution

u

to (1).

The main motivation behind studying this problem comes from the results in Castro (2021), where the existence of extremals to a Sobolev inequality with monomial weights was analyzed (see also Cabré & Ros-Oton, 2013; Castro, 2017). It is known that extremals to a weighted Sobolev inequality can be viewed as positive solutions to (1) for appropriate weights $w_{1}, w_{2}$ , and our goal is to obtain as much information as possible regarding said extremals and, in general, of solutions to (1).

The functions $w_{1}, w_{2}$ will be weight functions, meaning locally Lebesgue integrable nonnegative functions over $Ω \subseteq R^{N}$ satisfying at least the following two conditions: if we abuse the notation and we also write $w$ as the measure induced by $w$ , that is, $w (B) = \int_{B} w d x$ , we require that $w$ is a doubling measure in $Ω$ , that is, there exists a doubling constant $γ > 0$ such that

w (2 B) \leq γ w (B)

(2)

holds for every (open) ball such that

2 B \subset Ω

, where

ρ B

denotes the ball with the same center as

B

, but with its radius multiplied by

ρ > 0

. The smallest possible

γ > 0

for which (2) holds for every ball will be denoted by

γ_{w} > 0

from now on. Additionally, we will suppose that

0 < w < \infty λ - almost everywhere,

(3)

where

λ

denotes the

N

-dimensional Lebesgue measure. Observe that these two conditions ensure mutual absolute continuity between the measures

w

and

λ

In addition to (2) and (3), we will suppose that the weight $w_{1}$ satisfies the following local $(1, p)$ Poincaré inequality: if we write $⨏_{B} f w d x = [1 / w (B)] \int f w d x$ then (PI)

Local weighted $(1, p)$ -Poincaré inequality: There exists $ρ \geq 1$ such that if $u \in C^{1} (Ω)$ then for all balls $B \subset Ω$ of radius $l (B)$ one has

⨏_{B} | u - u_{B, w_{1}} | w_{1} d x \leq C_{1} l (B) {(⨏_{ρ B} {| \nabla u |}^{p} w_{1} d x)}^{\frac{1}{p}},

(4)

where

u_{B, w} = ⨏_{B} u w d x

is the weighted average of

u

over

B

As it can be seen in Heinonen et al. (2006, Chapter 20), when a weight function $w$ satisfies (2), (3), and (4), then $w_{1}$ is $p$ -admissible, that is, it also satisfies the following properties: (PII)

Uniqueness of the gradient: If $(u_{n})_{n \in N} \subseteq C^{1} (Ω)$ satisfy

\int_{Ω} {| u_{n} |}^{p} w_{1} d x \to_{n \to \infty}^{0} and \int_{Ω} {| \nabla u_{n} - v |}^{p} w_{1} d x \to_{n \to \infty}^{0}

for some

v : Ω \to R^{N}

, then

v = 0

(PIII)

Local Poincaré–Sobolev inequality: There exist constants $C_{3} > 0$ and $χ_{1} > 1$ such that for all balls $B \subset Ω$ one has

{(⨏_{B} {| u - u_{B, w_{1}} |}^{χ_{1} p} w_{1} d x)}^{\frac{1}{χ_{1} p}} \leq C_{2} l (B) {(⨏_{B} {| \nabla u |}^{p} w_{1} d x)}^{\frac{1}{p}}

(5)

for bounded

u \in C^{1} (B)

(PIV)

Local Sobolev inequality: There exist constants $C_{2} > 0$ and $χ_{1} > 1$ (same as above) such that for all balls $B \subset Ω$ one has

{(⨏_{B} {| u |}^{χ_{1} p} w_{1} d x)}^{\frac{1}{χ_{1} p}} \leq C_{2} l (B) {(⨏_{B} {| \nabla u |}^{p} w_{1} d x)}^{\frac{1}{p}}

(6)

for

u \in C_{c}^{1} (B)

Remark 1

The value of $χ_{1}$ is a dimensional constant associated with the weight $w_{1}$ , namely, it can be seen that if $w$ is a doubling weight then

\frac{w (B_{R} (y))}{w (B_{r} (x))} \leq C {(\frac{R}{r})}^{D_{w}}, for all 0 < r \leq R < \infty with B_{r} (x) \subseteq B_{R} (y) \subseteq Ω,

(7)

for

D_{w} = \log_{2} γ_{w}

, and if we denote

D_{1} := \log_{2} γ_{w_{1}}

then we can take

χ_{1} = D_{1} / (D_{1} - p)

in (5) and (6).

Remark 2

An important class of $p$ -admissible weights is the class of Muckenhoupt weights $A_{p}$ . A weight $w$ belongs to $A_{p}$ for $p > 1$ if there exists a constant $C > 0$ such that for any ball $B$

(\int_{B} w) {(\int_{B} w^{\frac{1}{1 - p}})}^{p - 1} \leq C λ (B),

where as before

λ

denotes the Lebesgue measure in

R^{N}

. See Heinonen et al. (2006, Chapter 15) for more details.

Regarding the weight $w_{2}$ , in addition to satisfy (2) and (3) (in particular, $w_{2}$ also satisfies (7) for $D_{2} := \log_{2} γ_{w_{2}}$ ), we require that the following compatibility condition with the weight $w_{1}$ is met: there exists $q > p$ such that

\frac{r}{R} {(\frac{w_{2} (B_{r})}{w_{2} (B_{R})})}^{\frac{1}{q}} \leq C {(\frac{w_{1} (B_{r})}{w_{1} (B_{R})})}^{\frac{1}{p}},

(8)

holds for all balls

B_{r} \subset B_{R} \subset Ω

. From Franchi et al. (1994), see also Björn (2001, Theorem 7), we know that if

1 \leq p < q < \infty

w_{1}

p

-admissible,

w_{2}

is doubling, and (8) is satisfied, then the pair of weights

(w_{1}, w_{2})

satisfy the

(q, p)

-local Poincaré–Sobolev inequality

{(⨏_{B_{R}} {| u - u_{B, w_{2}} |}^{q} w_{2} d x)}^{\frac{1}{q}} \leq C R {(⨏_{B_{R}} {| \nabla u |}^{p} w_{1} d x)}^{\frac{1}{p}}, \forall u \in C^{1} (B_{R}),

(9)

and the

(q, p)

-local Sobolev inequality

{(⨏_{B_{R}} {| u |}^{q} w_{2} d x)}^{\frac{1}{q}} \leq C R {(⨏_{B_{R}} {| \nabla u |}^{p} w_{1} d x)}^{\frac{1}{p}}, \forall u \in C_{c}^{1} (B_{R}) .

(10)

Remark 3

As it will be useful later we write $D = q p / (q - p)$ and $χ_{2} = D / (D - p) = q / p$ . Notice that this $D$ comes from (8) and in general it has nothing to do with $D_{2} = \log_{2} γ_{w_{2}}$ , the dimensional constant associated with the doubling weight $w_{2}$ mentioned before.

Remark 4

In fact, it can be seen that either (9) or (10) implies (8). The idea is to apply (9) (respectively, (10)) to $u (x) = | x - x_{0} | η (x)$ for suitable $η \in C_{c}^{1} (B_{R} (x_{0}))$ , see Chanillo and Wheeden (1985) and Franchi et al. (1994, Remark 4.10) for the details.

In order to establish the main results of this work, we recall some definitions regarding weighted spaces. For an admissible weight $w$ , we consider the weighted Lebesgue space

L^{p, w} (Ω) = {u : Ω \to R measurable : \int_{Ω} {| u |}^{p} w d x < \infty}

equipped with the norm

{‖ u ‖}_{p, w}^{p} = \int_{Ω} {| u |}^{p} w d x .

In what follows, we will omit writing

d x

when there is no confusion about the variable of integration.

The $p$ -admissibility of $w_{1}$ is useful to have a proper definition for weighted Sobolev spaces: for an open set $Ω \subseteq R^{N}$ , we define the weighted Sobolev space $H^{1, p, w_{1}} (Ω)$

H^{1, p, w_{1}} (Ω) = the completion of {u \in C^{1} (Ω) : u, \frac{\partial u}{\partial x_{i}} \in L^{p, w_{1}} (Ω) for all i}

equipped with the norm

{‖ u ‖}_{1, p, w_{1}}^{p} = {‖ u ‖}_{p, w_{1}}^{p} + \sum_{i = 1}^{N} {‖ \frac{\partial u}{\partial x_{i}} ‖}_{p, w_{1}}^{p} .

As we mentioned before, the goal of this work is to obtain point-wise estimates at infinity for solutions to (1). To do so, we first study the local regularity of weak solutions in the following quasi-linear problem:

{\begin{aligned} div A (x, u, \nabla u) & = B (x, u, \nabla u), & in Ω \subseteq R^{N}, \\ u & \in H_{loc}^{1, p, w_{1}} (Ω), \end{aligned}

(11)

where

A : Ω \times R \times R^{N} \to R^{N}

and

B : Ω \times R \times R^{N} \to R

are functions verifying the Serrin-like conditions

\begin{aligned} A (x, u, z) \cdot z \geq w_{1} (x) (a^{- 1} {| z |}^{p} - d_{1} {| u |}^{p} - g), \end{aligned}

(H1)

\begin{aligned} | A (x, u, z) | \leq w_{1} (x) (a {| z |}^{p - 1} + b {| u |}^{p - 1} + e), \end{aligned}

(H2)

\begin{aligned} | B (x, u, z) | \leq w_{2} (x) (c {| z |}^{p - 1} + d_{2} {| u |}^{p - 1} + f), \end{aligned}

(H3)

for a constant

a > 0

and measurable functions

b, c, d_{1}, d_{2}, e, f, g : Ω \to R^{+} \cup {0}

satisfying

\begin{aligned} b, e \in L^{\frac{D_{1}}{p - 1}, w_{1}} (B_{2}), & c {(\frac{w_{2}}{w_{1}})}^{1 - \frac{1}{p}} \in L^{\frac{D_{1}}{1 - ε}, w_{2}} (B_{2}), \\ d_{1}, g \in L^{\frac{D_{1}}{p - ε}, w_{1}} (B_{2}), & d_{2}, f \in L^{\frac{D}{p - ε}, w_{2}} (B_{2}), \end{aligned}

(H_ε)

for some

0 \leq ε < 1

Equations of the type (11) have been recently studied in the context of double-phase energies (see, e.g., He & Ji, 2025; Papageorgiou et al., 2023). In particular, Papageorgiou et al. (2023) examine the doubly weighted Dirichlet eigenvalue problem

- div (w_{1} {| \nabla u |}^{p - 2} \nabla u) = λ w_{2} {| u |}^{p - 2} u in Ω \subseteq R^{N},

where

w_{1} \in C^{0, 1} (\bar{Ω}) \cap A_{p}

and

w_{2} = m \cdot w_{1}

for some essentially bounded, almost everywhere positive function

m \in L^{\infty} (Ω)

. One of their results shows that every eigenfunction associated with the first eigenvalue is bounded in

Ω

and has a constant sign.

We are now ready to state the main results of this work. From this point onward, the functions $w_{1}, w_{2}$ will be nonnegative locally integrable weight functions satisfying (2) and (3), $w_{1}$ will satisfy the local weighted $(1, p)$ -Poincaré inequality (4) and the pair $(w_{1}, w_{2})$ will verify the compatibility condition (8). We will also suppose that $1 < p < min {D_{1}, D}$ .

The first result of this work shows that weak solutions to (11) are locally bounded.

Theorem 1

Suppose that there exists $0 < ε < 1$ such that (H $_{ε}$ ) is satisfied, then there exists a constant $C > 0$ depending on the norms of $a, b, c, d_{1}, d_{2}$ such that for any weak solution to (11) in $B_{2} (x_{0})$ , we have

{‖ u ‖}_{L^{\infty} (B_{1})} \leq C ([u]_{p, B_{2}} + k),

where

\begin{aligned} k = {[{(⨏_{B_{2}} {| e |}^{\frac{D_{1}}{p - 1}} w_{1})}^{\frac{p - 1}{D_{1}}} + {(⨏_{B_{2}} {| f |}^{\frac{D}{p - ε}} w_{2})}^{\frac{p - ε}{D}}]}^{\frac{1}{p - 1}} + {[{(⨏_{B_{2}} {| g |}^{\frac{D_{1}}{p - ε}} w_{1})}^{\frac{p - ε}{D_{1}}}]}^{\frac{1}{p}} \end{aligned}

(12)

and for

s > 1

and

B \subseteq Ω

, we write

[u]_{s, B} = {(⨏_{B} {| u |}^{s} w_{1})}^{\frac{1}{s}} + {(⨏_{B} {| u |}^{s} w_{2})}^{\frac{1}{s}} .

(13)

Remark 5

We have chosen to exhibit the local regularity results only for the case $B_{1} \subset B_{2} \subset Ω$ as the general case $B_{R} \subseteq B_{2 R} \subseteq Ω$ can be easily obtained by a suitable scaling argument.

Next, we consider the case $ε = 0$ and we show that weak solutions are in $L^{s, w_{i}} (B_{1})$ for every $s > p$ .

Theorem 2

Suppose that (H $_{ε}$ ) is satisfied for $ε = 0$ , then there exists a constant $C > 0$ depending on the norms of $a, b, c, d_{1}, d_{2}$ such that for any weak solution to (11) in $B_{2} (x_{0})$ satisfies

[u]_{s, B_{1}} \leq C_{s} ([u]_{p, B_{2}} + k)

for every

s > p

and

k

as in (12).

Finally, we show that the Harnack inequality holds for nonnegative weak solutions to (11).

Theorem 3 Harnack inequality

Under the same hypotheses of Theorem 1 with the additional assumption that $u$ is a nonnegative weak solution of $div A = B$ in $B_{3} (x_{0})$ , then

max_{B_{1}} u \leq C (min_{B_{1}} u + k),

where

C

and

k

are as in Theorem 1.

Remark 6
It is worth emphasizing that, while the results in Theorems 1 to 3 may be anticipated in light of the foundational works of Serrin (1964) and Kenig–Fabes–Serapioni (Fabes et al., 1982), to the best of our knowledge, they have not been explicitly established in the literature at this level of generality.

With the aid of the above theorems, we are able to study (1) and to obtain a general result regarding the behavior at infinity of solutions. To do that, we will suppose that in addition to the above conditions, both weights $w_{1}, w_{2}$ verify global Sobolev inequalities, that is, there exists a constant $C > 0$ such that
${(\int_{Ω} {| u |}^{q_{1}} w_{1})}^{\frac{1}{q_{1}}} \leq C {(\int_{Ω} | \nabla u | w_{1})}^{\frac{1}{p}}$
(14)
for $q_{1} = χ_{1} p$ and
${(\int_{Ω} {| u |}^{q} w_{2})}^{\frac{1}{q}} \leq C {(\int_{Ω} | \nabla u | w_{1})}^{\frac{1}{p}}$
(15)
for $q$ as in (8), and all $u \in C_{c}^{1} (Ω)$ . Under these assumptions, and if we define $D^{1, p, w_{1}} (Ω)$ as the closure of $C_{c}^{\infty} (Ω)$ under the (semi) norm ${‖ \nabla u ‖}_{p, w_{1}}$ then $D^{1, p, w_{1}} (Ω)$ embeds continuously into both $L^{q_{1}, w_{1}} (Ω)$ and $L^{q, w_{2}} (Ω)$ and we are able to prove
Theorem 4 Decay

Suppose $u \in D^{1, p, w_{1}} (Ω)$ is a weak solution to (1). Then, there exists $R_{0} > 1$ , $C > 0$ , and $λ > 0$ such that

| u (x) | \leq \frac{C}{{| x |}^{\frac{p}{q_{1} - p} + λ}},

for all

| x | > R_{0}

Ω

Remark 7
It is important to mention that this decay behavior is not optimal, but it can be used as a starting point to obtain better results. This could be done with the aid of a comparison principle and the construction of a suitable barrier function depending on the weights $w_{1}, w_{2}$ .

The rest of this article is dedicated to the proofs of the above results. In Section 2, we study (11) and obtain the proofs of Theorems 1 to 3, whereas in Section 3, we turn to the proof of Theorem 4. Finally, in Section 4, we show that the above results apply to the class of monomial weights $x^{A}$ studied in Castro (2017).
2. Local Estimates

Throughout the different proofs in this section, we will use the dimensional constants of the weights, denoted $D_{i} := D_{w_{i}}$ , as well as the local Sobolev exponents $q_{1} := (D_{1} p) / (D_{1} - p)$ and $D = q p / (q - p)$ for $q$ given by (8). With these notations, we also have

χ_{1} = \frac{q_{1}}{p} = \frac{D_{1}}{D_{1} - p} and χ_{2} = \frac{q}{p} = \frac{D}{D - p} .

Following Serrin (1964), we define

F : [k, \infty) \to R

F (x) = F_{α, k, l} (x) = {\begin{cases} x^{α} & if k \leq x \leq l, \\ l^{α - 1} (α x - (α - 1) l) & if x > l, \end{cases}

which is in

C^{1} ([k, \infty))

, with

| F^{'} (x) | \leq α l^{α - 1}

. We consider

\bar{x} = | x | + k

and

G : R \to R

defined as

G (x) = G_{α, k, l} (x) = sign (x) (F (\bar{x}) {| F^{'} (\bar{x}) |}^{p - 1} - α^{p - 1} k^{β}),

where

β = 1 + p (α - 1)

. Observe that

G

is a piecewise smooth function, which is linear if

| x | > l - k

and that both

F

and

G

satisfy

\begin{aligned} | G | & \leq F (\bar{x}) {| F^{'} (\bar{x}) |}^{p - 1}, \\ \bar{x} F^{'} (\bar{x}) & \leq α F (\bar{x}), \\ F^{'} (\bar{x}) & \leq α F (\bar{x})^{1 - \frac{1}{α}}, \end{aligned}

and

G^{'} (x) = {\begin{cases} \frac{β}{α} {| F^{'} (\bar{x}) |}^{p} & if | x | < l - k, \\ {| F^{'} (\bar{x}) |}^{p} & if | x | > l - k . \end{cases}

Finally, observe that if

η \in C_{c}^{\infty} (Ω)

and if

u \in H_{l o c}^{1, p, w_{1}} (Ω)

then

φ = η^{p} G (u)

is a valid test function for the weak formulation

\int_{Ω} A (x, u, \nabla u) \nabla φ + B (x, u, \nabla u) φ = 0

thanks to the results in Heinonen et al. (2006, Chapter 1) regarding weighted Sobolev spaces for

p

-admissible weights.

We can now prove the local boundedness of weak solutions.

Proof of Theorem 1. Proof of Theorem 1.

By using (H1)–(H3), we can write

\begin{aligned} | A (x, u, z) | & \leq w_{1} (a {| z |}^{p - 1} + \bar{b} {\bar{u}}^{p - 1}), \\ A (x, u, z) \cdot z & \geq w_{1} ({| z |}^{p} - {\bar{d}}_{1} {\bar{u}}^{p}), \\ | B (x, u, z) | & \leq w_{2} (c {| z |}^{p - 1} + {\bar{d}}_{2} {\bar{u}}^{p - 1}), \end{aligned}

(16)

where

\begin{aligned} \bar{b} & = b + k^{1 - p} e, \\ {\bar{d}}_{1} & = d_{1} + k^{- p} g, \\ {\bar{d}}_{2} & = d_{2} + k^{1 - p} f, \end{aligned}

and

\bar{u} = | u | + k

for

k \geq 0

defined by¹

k = {[{(⨏_{B_{2}} {| e |}^{\frac{D_{1}}{p - 1}} w_{1})}^{\frac{p - 1}{D_{1}}} + {(⨏_{B_{2}} {| f |}^{\frac{D}{p - ε}} w_{2})}^{\frac{p - ε}{D}}]}^{\frac{1}{p - 1}} + {[{(⨏_{B_{2}} {| g |}^{\frac{D_{1}}{p - ε}} w_{1})}^{\frac{p - ε}{D_{1}}}]}^{\frac{1}{p}} .

Observe that (H

_{ε}

) implies that

⨏_{B_{2}} {| \bar{b} |}^{\frac{D_{1}}{p - 1}} w_{1} \leq C, ⨏_{B_{2}} {| {\bar{d}}_{1} |}^{\frac{D_{1}}{p - ε}} w_{1} \leq C, ⨏_{B_{2}} {| {\bar{d}}_{2} |}^{\frac{D}{p - ε}} w_{2} \leq C,

(17)

for some constant

C > 0

depending on the respective local norms of

b, d_{1}, d_{2}, e, f, g

For a local weak solution $u$ and arbitrary $η \in C_{c}^{\infty} (B_{2})$ , we use $φ = η^{p} G (u)$ and with the aid of (16) one can obtain the estimate

\begin{aligned} A \cdot \nabla φ + B φ & = η^{p} G^{'} (u) A \cdot \nabla u + p η^{p - 1} G (u) A \cdot \nabla η + η^{p} G (u) B \\ \geq η^{p} G^{'} (u) w_{1} ({| \nabla u |}^{p} - {\bar{d}}_{1} {\bar{u}}^{p}) \\ - p η^{p - 1} | \nabla η G (u) | w_{1} (a {| \nabla u |}^{p - 1} + \bar{b} {\bar{u}}^{p - 1}) \\ - η^{p} | G (u) | w_{2} (c {| \nabla u |}^{p - 1} + {\bar{d}}_{2} {\bar{u}}^{p - 1}), \end{aligned}

so that if

v = F (\bar{u})

one reaches

\begin{aligned} A \cdot \nabla φ + B φ \geq & {| η \nabla v |}^{p} w_{1} - p a | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} - p α^{p - 1} \bar{b} | v \nabla η | {| η v |}^{p - 1} w_{1} \\ - β α^{p - 1} {\bar{d}}_{1} {| η v |}^{p} w_{1} - c η v {| η \nabla v |}^{p - 1} w_{2} - α^{p - 1} {\bar{d}}_{2} {| η v |}^{p} w_{2} . \end{aligned}

(18)

We integrate over $B_{2}$ and divide by $w_{1} (B_{2})$ to obtain

\begin{aligned} ⨏_{B_{2}} {| η \nabla v |}^{p} w_{1} & \leq p a ⨏_{B_{2}} | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} + p α^{p - 1} ⨏_{B_{2}} \bar{b} | v \nabla η | {| v η |}^{p - 1} w_{1} \\ + β α^{p - 1} ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{p} w_{1} + \frac{1}{w_{1} (B_{2})} \int_{B_{2}} c v η {| η \nabla v |}^{p - 1} w_{2} \\ + \frac{α^{p - 1}}{w_{1} (B_{2})} \int_{B_{2}} {\bar{d}}_{2} {| v η |}^{p} w_{2}, \end{aligned}

but since

w_{2} (B_{2}) = C w_{1} (B_{2})

for

C = C (x_{0}, w_{1}, w_{2}) = w_{2} (B_{2}) / w_{1} (B_{2})

, we can write

\begin{aligned} ⨏_{B_{2}} {| η \nabla v |}^{p} w_{1} & \leq p a ⨏_{B_{2}} | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} + p α^{p - 1} ⨏_{B_{2}} \bar{b} | v \nabla η | {| v η |}^{p - 1} w_{1} \\ + β α^{p - 1} ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{p} w_{1} + C ⨏_{B_{2}} c v η {| η \nabla v |}^{p - 1} w_{2} \\ + C α^{p - 1} ⨏_{B_{2}} {\bar{d}}_{2} {| v η |}^{p} w_{2}, \end{aligned}

(19)

and each term on the right-hand side of the inequality can be estimated using (6), (10), and (17) as follows:

⨏_{B_{2}} | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} \leq {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}},

(20)

D_{1}

the dimensional constant associated with the weight

w_{1}

then

\begin{aligned} ⨏_{B_{2}} \bar{b} | v \nabla η | {| v η |}^{p - 1} w_{1} & \leq {(⨏_{B_{2}} {\bar{b}}^{\frac{D_{1}}{p - 1}} w_{1})}^{\frac{p - 1}{D_{1}}} {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| v η |}^{χ_{1} p} w_{1})}^{\frac{p - 1}{χ_{1} p}} \\ \leq C {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| \nabla (v η) |}^{p} w_{1})}^{1 - \frac{1}{p}}, \end{aligned}

(21)

and

\begin{aligned} ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{p} w_{1} & = ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{ε} {| v η |}^{p - ε} w_{1} \\ \leq {(⨏_{B_{2}} {\bar{d}}_{1}^{\frac{D_{1}}{p - ε}} w_{1})}^{\frac{p - ε}{D_{1}}} {(⨏_{B_{2}} {| v η |}^{p} w_{1})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| v η |}^{χ_{1} p} w_{1})}^{\frac{p - ε}{χ_{1} p}} \\ \leq C {(⨏_{B_{2}} {| v η |}^{p} w_{1})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| \nabla (v η) |}^{p} w_{1})}^{1 - \frac{ε}{p}}, \end{aligned}

(22)

whereas for

D = p q / (q - p)

and

\bar{c} = c {(w_{2} / w_{1})}^{1 - (1 / p)}

, we have

\begin{aligned} ⨏_{B_{2}} c v η {| η \nabla v |}^{p - 1} w_{2} & = ⨏_{B_{2}} \bar{c} w_{2}^{\frac{1 - ε}{D}} {| v η |}^{ε} w_{2}^{\frac{ε}{p}} {| v η |}^{1 - ε} w_{2}^{\frac{1 - ε}{q}} {| η \nabla v |}^{p - 1} w_{1}^{1 - \frac{1}{p}} \\ \leq {(⨏_{B_{2}} {| \bar{c} |}^{\frac{D}{1 - ε}} w_{2})}^{\frac{1 - ε}{D}} \\ \times {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| v η |}^{q} w_{2})}^{\frac{1 - ε}{q}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} \\ \leq C {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| \nabla (v η) |}^{p} w_{1})}^{\frac{1 - ε}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}}, \end{aligned}

(23)

and

\begin{aligned} ⨏_{B_{2}} {\bar{d}}_{2} {| v η |}^{p} w_{2} & = ⨏_{B_{2}} {\bar{d}}_{2} {| v η |}^{ε} {| v η |}^{p - ε} w_{2} \\ \leq {(⨏_{B_{2}} {\bar{d}}_{2}^{\frac{D}{p - ε}} w_{2})}^{\frac{p - ε}{D}} {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| v η |}^{q} w_{2})}^{\frac{p - ε}{q}} \\ \leq C {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} {(⨏_{B_{2}} {| \nabla (v η) |}^{p} w_{1})}^{1 - \frac{ε}{p}} . \end{aligned}

(24)

Therefore, (19), (20), (21), (22), (23), and (24) give

\begin{aligned} ⨏_{B_{2}} {| η \nabla v |}^{p} w_{1} & \leq p a {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} \\ + C p α^{p - 1} [(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1}) + {(\int_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}}] \\ + C β α^{p - 1} {(⨏_{B_{2}} {| v η |}^{p} w_{1})}^{\frac{ε}{p}} [{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{1 - \frac{ε}{p}} + {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{ε}{p}}] \\ + C {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} \\ \times [{(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} {(\int_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1 - ε}{p}} + {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{ε}{p}}] \\ + C α^{p - 1} {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{ε}{p}} [{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{1 - \frac{ε}{p}} + {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{ε}{p}}] . \end{aligned}

(25)

If one considers

z = \frac{{(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{\frac{1}{p}}}{{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}}}

and

ζ = \frac{{(⨏_{B_{2}} {| η v |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| η v |}^{p} w_{2})}^{\frac{1}{p}}}{{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}}},

then, because

α \geq 1

, (25) becomes

z^{p} \leq C (z^{p - 1} + α^{p - 1} (1 + z^{p - 1}) + ζ^{ε} (z^{p - 1} + z^{p - ε}) + (1 + β) α^{p - 1} ζ^{ε} (1 + z^{p - ε}))

for some constant

C > 0

depending on

a, b, c, d, e, f, g, w_{1}, w_{2}

and

p

. With the aid of Serrin (1964, Lemma 2), we obtain

z \leq C α^{\frac{p}{ε}} (1 + ζ),

which gives

\begin{aligned} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{\frac{1}{p}} & \leq C α^{\frac{p}{ε}} ({(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| η v |}^{p} w_{1})}^{\frac{1}{p}} \\ + {(⨏_{B_{2}} {| η v |}^{p} w_{2})}^{\frac{1}{p}}) . \end{aligned}

(26)

Now, by (6) and (10), the local Sobolev inequalities for the pairs $(w_{1}, w_{1})$ and $(w_{1}, w_{2})$ , respectively, we obtain

\begin{aligned} {(⨏_{B_{2}} {| η v |}^{χ_{i} p} w_{i})}^{\frac{1}{χ_{i} p}} \leq & C α^{\frac{p}{ε}} ({(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| η v |}^{p} w_{1})}^{\frac{1}{p}} \\ + {(⨏_{B_{2}} {| η v |}^{p} w_{2})}^{\frac{1}{p}}), \end{aligned}

(27)

where we recall that

χ_{1} = D_{1} / (D_{1} - p)

and

χ_{2} = q / p = D / (D - p)

To continue we consider a sequence of cut-off functions as follows: we take $η_{n} \in C_{c}^{\infty} (B_{h_{n}})$ such that $η_{n} \equiv 1$ in $B_{h_{n + 1}}$ and $| \nabla η_{n} | \leq C 2^{n}$ , where $h_{n} = 1 + 2^{- n}$ . If one recalls that both weights are doubling so that $w_{i} (B_{h_{n}}) \leq γ_{w_{i}} w_{i} (B_{h_{n + 1}})$ , we deduce from (27) that (after passing to the limit $l \to \infty$ )

\begin{aligned} {(⨏_{B_{h_{n + 1}}} {| \bar{u} |}^{α χ_{1} p} w_{1})}^{\frac{1}{χ_{1} p}} + {(⨏_{B_{h_{n + 1}}} {| \bar{u} |}^{α χ_{2} p} w_{2})}^{\frac{1}{χ_{2} p}} \\ \leq C 2^{n} α^{\frac{p}{ε}} [{(⨏_{B_{h_{n}}} {| \bar{u} |}^{α p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{h_{n}}} {| \bar{u} |}^{α p} w_{2})}^{\frac{1}{p}}], \end{aligned}

(28)

which is valid for all

α \geq 1

. Recall the definition of

[u]_{s, B}

given by (13), that is,

[u]_{s, B} = {(⨏_{B} {| \bar{u} |}^{s} w_{1})}^{\frac{1}{s}} + {(⨏_{B} {| \bar{u} |}^{s} w_{2})}^{\frac{1}{s}}

and observe that if

χ = min {χ_{1}, χ_{2}}

then

{(⨏_{B_{h_{n + 1}}} {| \bar{u} |}^{χ^{n + 1} p} w_{i})}^{\frac{1}{χ^{(n + 1)} p}} \leq {(⨏_{B_{h_{n + 1}}} {| \bar{u} |}^{χ^{n} χ_{i} p} w_{i})}^{\frac{1}{χ^{n} χ_{i} p}},

for

i = 1, 2

. Therefore, if we select

α_{n} = χ^{n} > 1

in (28), we are led to

[\bar{u}]_{s_{n + 1}, B_{h_{n + 1}}} \leq C^{χ^{- n}} 2^{n χ^{- n}} χ^{\frac{p}{ε} n χ^{- n}} [\bar{u}]_{s_{n}, B_{h_{n}}},

where

s_{n} = p χ^{n}

. Since

χ > 1

, then

\sum_{k = 0}^{\infty} k χ^{- k}

and

\sum_{k = 0}^{\infty} χ^{- k}

are convergent series, so we can iterate the above inequality to obtain

[\bar{u}]_{s_{n}, B_{h_{n}}} \leq C [\bar{u}]_{p, B_{2}},

for some constant

C

independent of

n

. After passing to the limit

n \to \infty

, we obtain

{‖ u ‖}_{L^{\infty} (B_{1})} \leq C [{(⨏_{B_{2}} {| u |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| u |}^{p} w_{2})}^{\frac{1}{p}} + k],

and the result follows.

Proof of Theorem 2. Proof of Theorem 2.

Thanks to the interpolation inequality in $L^{s, w_{i}}$ , it is enough to find a sequence $s_{n} \to_{n \to \infty}^{+} \infty$ for which one has

[\bar{u}]_{s_{n}, B_{1}} \leq C_{n} [\bar{u}]_{p, B_{2}},

where

\bar{u} = | u | + k

. As in the proof of Theorem 1, by using the test function

φ = η^{p} G (u)

, we arrive at the inequality

\begin{aligned} ⨏_{B_{2}} {| η \nabla v |}^{p} w_{1} & \leq a p ⨏_{B_{2}} | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} + p α^{p - 1} ⨏_{B_{2}} \bar{b} | v \nabla η | {| v η |}^{p - 1} w_{1} \\ + β α^{p - 1} ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{p} w_{1} + ⨏_{B_{2}} c v η {| η \nabla v |}^{p - 1} w_{2} \\ + α^{p - 1} ⨏_{B_{2}} {\bar{d}}_{2} {| v η |}^{p} w_{2}, \end{aligned}

however, since

ε = 0

, we cannot apply the same estimates from (20) to (22) directly. Instead, we firstly estimate the term involving

\bar{b}

as follows:

\begin{aligned} ⨏_{B_{2}} \bar{b} | v \nabla η | {| v η |}^{p - 1} w_{1} & \leq {(⨏_{B_{2}} {\bar{b}}^{\frac{D_{1}}{p - 1}} w_{1})}^{\frac{p - 1}{D_{1}}} {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| v η |}^{χ_{1} p} w_{1})}^{\frac{p - 1}{χ_{1} p}} \\ \leq C {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{1 / p} \\ \times [{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{1 - \frac{1}{p}} + {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}}] . \end{aligned}

For the terms involving

c

and

\bar{d}

, we consider

\bar{c} = c {(w_{2} / w_{1})}^{1 - (1 / p)}

, and for each

M > 0

we let

C_{M} = {\bar{c} \leq M}

and proceed as follows:

\begin{aligned} ⨏_{B_{2}} c v η {| η \nabla v |}^{p - 1} w_{2} & = \frac{1}{w_{2} (B_{2})} [\int_{B_{2} \cap C_{M}} c v η {| η \nabla v |}^{p - 1} w_{2} \\ + \int_{B_{2} \cap C_{M}^{c}} \bar{c} w_{2}^{\frac{1}{D}} | v η | w_{2}^{\frac{1}{q}} {| η \nabla v |}^{p - 1} w_{1}^{1 - \frac{1}{p}}] \\ \leq M {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{1}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} \\ + C {(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} \\ + C {(\frac{1}{w_{2} (B_{2})} \int_{B_{2} \cap C_{M}^{c}} {| \bar{c} |}^{D} w_{2})}^{\frac{1}{D}} (⨏_{B_{2}} {| η \nabla v |}^{p} w_{1}) . \end{aligned}

Similarly, if

D_{i, M} = {{\bar{d}}_{i} \leq M}

for

i = 1, 2

, we have

\begin{aligned} ⨏_{B_{2}} {\bar{d}}_{1} {| v η |}^{p} w_{1} & = \frac{1}{w_{1} (B_{2})} [\int_{B_{2} \cap D_{1, M}} {\bar{d}}_{1} {| v η |}^{p} w_{1} + \int_{B_{2} \cap D_{1, M}^{c}} {\bar{d}}_{1} {| v η |}^{p} w_{1}] \\ \leq M ⨏_{B_{2}} {| v η |}^{p} w_{1} + C (⨏_{B_{2}} {| v \nabla η |}^{p} w_{1}) \\ + {(\frac{1}{w_{1} (B_{2})} \int_{B_{2} \cap D_{1, M}^{c}} {| {\bar{d}}_{1} |}^{\frac{D_{1}}{p}} w_{1})}^{\frac{p}{D_{1}}} (⨏_{B_{2}} {| η \nabla v |}^{p} w_{1}), \end{aligned}

and

\begin{aligned} ⨏_{B_{2}} {\bar{d}}_{2} {| v η |}^{p} w_{2} & = [\frac{1}{w_{2} (B_{2})} \int_{B_{2} \cap D_{2, M}} {\bar{d}}_{2} {| v η |}^{p} w_{2} + \int_{B_{2} \cap D_{1, M}^{c}} {\bar{d}}_{2} {| v η |}^{p} w_{2}] \\ \leq M ⨏_{B_{2}} {| v η |}^{p} w_{2} + C (⨏_{B_{2}} {| v \nabla η |}^{p} w_{1}) \\ + {(\frac{1}{w_{2} (B_{2})} \int_{B_{2} \cap D_{1, M}^{c}} {| {\bar{d}}_{2} |}^{\frac{D}{p}} w_{2})}^{\frac{p}{D}} (⨏_{B_{2}} {| η \nabla v |}^{p} w_{1}) . \end{aligned}

Because $\bar{c} \in L^{D, w_{2}}$ , ${\bar{d}}_{1} \in L^{(D_{1} / p), w_{1}}$ , and ${\bar{d}}_{2} \in L^{(D / p), w_{2}}$ then for any $δ > 0$ , we can find $M > 0$ such that

\begin{aligned} {(\frac{1}{w_{2} (B_{2})} \int_{B_{2} \cap C_{M}^{c}} {| \bar{c} |}^{D} w_{2})}^{\frac{1}{D}} & + {(\frac{1}{w_{1} (B_{2})} \int_{B_{2} \cap D_{1, M}^{c}} {| {\bar{d}}_{1} |}^{\frac{D_{1}}{p}} w_{1})}^{\frac{p}{D_{1}}} \\ + {(\frac{1}{w_{2} (B_{2})} \int_{B_{2} \cap D_{2, M}^{c}} {| {\bar{d}}_{2} |}^{\frac{D}{p}} w_{2})}^{\frac{p}{D}} \leq δ, \end{aligned}

therefore, for any

α \geq 1

, we can find

δ > 0

sufficiently small and a constant

C_{α} > 0

such that

\begin{aligned} ⨏_{B_{2}} {| η \nabla v |}^{p} w_{1} & \leq C_{α} [{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| v η |}^{p} w_{2})}^{\frac{1}{p}}] {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{1 - \frac{1}{p}} \\ + C_{α} (⨏_{B_{2}} {| v \nabla η |}^{p} w_{1}) + C_{α} (⨏_{B_{2}} {| v η |}^{p} w_{1}) + C_{α} (⨏_{B_{2}} {| v η |}^{p} w_{2}) . \end{aligned}

The above inequality allows us to use Serrin (1964, Lemma 2) once again and obtain an inequality analogous to (26), namely

\begin{aligned} {(⨏_{B_{2}} {| η \nabla v |}^{p} w_{1})}^{\frac{1}{p}} \leq C_{α} [{(⨏_{B_{2}} {| v \nabla η |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| η v |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{2}} {| η v |}^{p} w_{2})}^{\frac{1}{p}}], \end{aligned}

(29)

the main difference being that the constant

C_{α}

is no longer explicit. Nonetheless, we can continue the argument from the proof of Theorem 1 by choosing appropriate cut-off functions

η

to reach

[\bar{u}]_{s_{n + 1}, B_{h_{n + 1}}} \leq C_{n} [\bar{u}]_{s_{n}, B_{h_{n}}},

where

s_{n} = p χ^{n}

h_{n} = 1 + 2^{- n}

and

[u]_{s, B}

is defined in (13). Observe that while we do not obtain a uniform estimate for

C_{n}

, we can still iterate the above to conclude that

[\bar{u}]_{s_{n}, B_{1}} \leq C_{n} [\bar{u}]_{p, B_{2}}

and the result is proved.

Proof of Theorem 3. Proof of Theorem 3.

By Theorem 1, $u$ is bounded on any compact subset of $B_{3}$ , so for any $β \in R$ and any $δ > 0$ the function $φ = η^{p} {\bar{u}}^{β}$ is a valid test function provided $\bar{u} = u + k + δ$ and $η \in C_{c}^{\infty} (B_{3})$ , where $k$ is defined as in Theorem 1.

For $β = 1 - p$ and $v = \log \bar{u}$ we obtain

\begin{aligned} (p - 1) \int_{B_{3}} {| η \nabla v |}^{p} w_{1} & \leq p a \int_{B_{3}} | \nabla η | {| η \nabla v |}^{p - 1} w_{1} + p \int_{B_{3}} \bar{b} η^{p - 1} | \nabla η | w_{1} \\ + \int_{B_{3}} c η {| η \nabla v |}^{p - 1} w_{2} + (p - 1) \int_{B_{3}} {\bar{d}}_{1} η^{p} w_{1} + \int_{B_{3}} {\bar{d}}_{2} η^{p} w_{2}, \end{aligned}

(30)

for any

η \in C_{c}^{\infty} (B_{3})

. To continue with the proof denote by

z = {(\int_{B_{3}} {| η \nabla v |}^{p} w_{1})}^{1 / p}

and observe that, with the aid of Hölder’s inequality, (30) becomes

z^{p} \leq C_{1} z^{p - 1} + C_{2},

where for

\bar{c} = c {(w_{2} / w_{1})}^{1 - (1 / p)}

, we have

\begin{aligned} C_{1} = \frac{p a}{p - 1} {(\int_{B_{3}} {| \nabla η |}^{p} w_{1})}^{\frac{1}{p}} + \frac{1}{p - 1} {(\int_{B_{3}} {| \bar{c} η |}^{p} w_{2})}^{\frac{1}{p}}, \end{aligned}

(31)

\begin{aligned} C_{2} = \frac{p}{p - 1} \int_{B_{3}} \bar{b} η^{p - 1} | \nabla η | w_{1} + \int_{B_{3}} {\bar{d}}_{1} η^{p} w_{1} + \frac{1}{p - 1} \int_{B_{3}} {\bar{d}}_{2} η^{p} w_{2} . \end{aligned}

(32)

Thanks to Young’s inequality, we deduce that

z^{p} \leq C (C_{1}^{p} + C_{2}),

for some constant

C

. To continue, we estimate

C_{1}

and

C_{2}

using appropriate

η

. For any

0 < h < 2

such that

B_{h} \subset B_{2}

(not necessarily concentric) we have that

B_{3 h / 2} \subset B_{3}

and we consider

η \in C_{c}^{\infty} (B_{3 h / 2})

such that

η \equiv 1

B_{h}

0 \leq η \leq 1

and

| \nabla η | \leq C h^{- 1}

We use such $η$ in (31) and (32) and we get the following estimates using Hölder’s inequality and the properties of $η$ :

\begin{aligned} \int_{B_{3}} {| \nabla η |}^{p} w_{1} & \leq \frac{C}{h^{p}} w_{1} (B_{\frac{3 h}{2}}), \\ \int_{B_{3}} \bar{b} η^{p - 1} | \nabla η | w_{1} & \leq \frac{C}{h} w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - 1}{D_{1}}} {(\int_{B_{3}} {| \bar{b} |}^{\frac{D_{1}}{p - 1}} w_{1})}^{\frac{p - 1}{D_{1}}}, \\ \int_{B_{3}} {| \bar{c} η |}^{p} w_{2} & \leq C w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{(1 - ε) p}{D}} {(\int_{B_{3}} {| \bar{c} |}^{\frac{D}{1 - ε}} w_{2})}^{\frac{(1 - ε) p}{D}}, \\ \int_{B_{3}} {\bar{d}}_{1} η^{p} w_{1} & \leq C w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - ε}{D_{1}}} {(\int_{B_{3}} {| {\bar{d}}_{1} |}^{\frac{D_{1}}{p - ε}} w_{1})}^{\frac{p - ε}{D_{1}}}, \\ \int_{B_{3}} {\bar{d}}_{2} η^{p} w_{2} & \leq C w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{p - ε}{D}} {(\int_{B_{3}} {| {\bar{d}}_{2} |}^{\frac{D}{p - ε}} w_{2})}^{\frac{p - ε}{D}} . \end{aligned}

Therefore, one obtains

\begin{aligned} h^{p} ⨏_{B_{h}} {| \nabla v |}^{p} w_{1} & \leq \frac{h^{p}}{w_{1} (B_{h})} \int_{B_{3}} {| η \nabla v |}^{p} w_{1} \\ \leq \frac{C h^{p}}{w_{1} (B_{h})} (C_{1}^{p} + C_{2}) \\ \leq C (\frac{w_{1} (B_{\frac{3 h}{2}})}{w_{1} (B_{h})} + h^{p - 1} \frac{w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - 1}{D_{1}}}}{w_{1} (B_{h})} + h^{p} \frac{w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - ε}{D_{1}}}}{w_{1} (B_{h})} \\ + h^{p} \frac{w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{(1 - ε) p}{D}}}{w_{1} (B_{h})} + h^{p} \frac{w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{p - ε}{D}}}{w_{1} (B_{h})}), \end{aligned}

where

C

depends on

\int_{B_{3}} {| \bar{b} |}^{D_{1} / (p - 1)} w_{1}, \int_{B_{3}} {| \bar{c} |}^{D / (1 - ε)} w_{2}

\int_{B_{3}} {| {\bar{d}}_{1} |}^{D_{1} / (p - ε)} w_{1}

, and

\int_{B_{3}} {| {\bar{d}}_{2} |}^{D / (p - ε)} w_{2}

. We claim that the right-hand side of the above inequality is bounded independently of

0 < h \leq 2

, indeed because

w_{1}

is doubling we have

\frac{w_{1} (B_{\frac{3 h}{2}})}{w_{1} (B_{h})} \leq C,

and because

B_{3 h / 2} \subset B_{3}

we deduce from (7) that

C h^{D_{1}} w_{1} (B_{3}) \leq w_{1} (B_{3 h / 2})

, hence

h^{p - 1} \frac{w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - 1}{D_{1}}}}{w_{1} (B_{h})} \leq \frac{γ_{w_{1}} h^{p - 1}}{w_{1} (B_{\frac{3 h}{2}})^{\frac{p - 1}{D_{1}}}} \leq C,

also

h^{p} \frac{w_{1} (B_{\frac{3 h}{2}})^{1 - \frac{p - ε}{D_{1}}}}{w_{1} (B_{h})} \leq \frac{γ_{w_{1}} h^{p}}{w_{1} (B_{\frac{3 h}{2}})^{\frac{p - ε}{D_{1}}}} \leq C h^{ε} .

From (8), we deduce

\begin{aligned} h^{p} \frac{w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{(1 - ε) p}{D}}}{w_{1} (B_{h})} & = h^{p} \frac{w_{2} (B_{\frac{3 h}{2}})^{\frac{p}{q} + ε (1 - \frac{p}{q})}}{w_{1} (B_{h})} \\ = h^{p} {(\frac{w_{2} (B_{\frac{3 h}{2}})^{\frac{1}{q}}}{w_{1} (B_{h})^{\frac{1}{p}}})}^{p} w_{2} (B_{\frac{3 h}{2}})^{ε (1 - \frac{p}{q})} \\ \leq γ_{w_{2}}^{\frac{p}{q}} h^{p} {(\frac{w_{2} (B_{h})^{\frac{1}{q}}}{w_{1} (B_{h})^{\frac{1}{p}}})}^{p} w_{2} (B_{3})^{ε (1 - \frac{p}{q})} \\ \leq γ_{w_{2}}^{\frac{p}{q}} h^{p} {(C (\frac{3}{h}) \frac{w_{2} (B_{3})^{\frac{1}{q}}}{w_{1} (B_{3})^{\frac{1}{p}}})}^{p} w_{2} (B_{3})^{ε (1 - \frac{p}{q})} \\ \leq C \end{aligned}

and similarly

h^{p} \frac{w_{2} (B_{\frac{3 h}{2}})^{1 - \frac{(p - ε)}{D}}}{w_{1} (B_{h})} = h^{p} {(\frac{w_{2} (B_{\frac{3 h}{2}})^{\frac{1}{q}}}{w_{1} (B_{h})^{\frac{1}{p}}})}^{p} w_{2} (B_{\frac{3 h}{2}})^{\frac{ε}{p} (1 - \frac{p}{q})} \leq C .

Hence, for any

ε \geq 0

, each term on the right-hand side is bounded independently of

0 < h \leq 2

Finally, the local Poincaré–Sobolev inequalities (5) and (9) tell us that

\begin{aligned} ⨏_{B_{h}} | v - v_{B_{h}} | w_{i} & \leq {(⨏_{B_{h}} {| v - v_{B_{h}} |}^{q_{i}} w_{i})}^{\frac{1}{q_{i}}} \\ \leq C h {(⨏_{B_{h}} {| \nabla v |}^{p} w_{1})}^{\frac{1}{p}} \\ \leq C, \end{aligned}

for any ball

B_{h} \subseteq B_{2}

and both

i = 1, 2

. We conclude that

⨏_{B_{h}} | v - v_{B_{h}} | w_{i} \leq C,

(33)

where

C > 0

is a constant independent of

h

, in other words,

v \in BMO (B_{2}, w_{i} d x)

. If we denote by

{‖ v ‖}_{BMO (B_{2}, w_{i})}

as the least possible

C > 0

in (33) then the John–Nirenberg lemma for doubling measures (Heinonen et al., 2006, Appendix II) tells us that there exist constants

p_{0, i}, C > 0

such that

⨏_{B} e^{p_{0, i} | v - v_{B} |} w_{i} \leq C

for all balls

B \subseteq B_{2}

. In particular, this gives

(⨏_{B_{2}} e^{p_{0, i} v} w_{i}) \cdot (⨏_{B_{2}} e^{- p_{0, i} v} w_{i}) \leq C^{2},

and because

v = \log \bar{u}

we have obtained

⨏_{B_{2}} {\bar{u}}^{p_{0, i}} w_{i} \leq C {(⨏_{B_{2}} {\bar{u}}^{- p_{0, i}} w_{i})}^{- 1} .

Denote by

p_{0} = min {p_{0, 1}, p_{0, 2}}

and observe that

⨏_{B_{2}} {\bar{u}}^{p_{0}} w_{i} \leq C {(⨏_{B_{2}} {\bar{u}}^{- p_{0}} w_{i})}^{- 1},

holds for both

i = 1, 2

because

p_{0} \leq p_{0, i}

and Hölder inequality. Therefore, if we denote by

Ψ (p, h) = {(⨏_{B_{h}} {\bar{u}}^{p} w_{1})}^{1 / p} + {(⨏_{B_{h}} {\bar{u}}^{p} w_{2})}^{1 / p}

then the inequality above implies

Ψ (p_{0}, 2) \leq C Ψ (- p_{0}, 2) .

(34)

The rest of the proof consists in using $φ = η^{p} {\bar{u}}^{β}$ for $β \neq 1 - p, 0$ as test function and $v = {\bar{u}}^{α}$ for $α$ given by $p β = p + α - 1$ . This gives

\begin{aligned} {| α |}^{p} (A \cdot \nabla φ + B φ) & \geq w_{1} (β {| η \nabla v |}^{p} - β {| α |}^{p} {\bar{d}}_{1} {| η v |}^{p}) \\ - w_{1} (a p | α | | \nabla η v | {| η \nabla v |}^{p - 1} + p {| α |}^{p} \bar{b} {| η v |}^{p - 1} | \nabla η v |) \\ - w_{2} (| α | c | η v | {| η \nabla v |}^{p - 1} + {| α |}^{p} {\bar{d}}_{2} {| η v |}^{p - 1}), \end{aligned}

which after integrating over

B_{3}

becomes

\begin{aligned} 0 & \geq ⨏_{B_{3}} (β {| η \nabla v |}^{p} - β {| α |}^{p} {\bar{d}}_{1} {| η v |}^{p}) w_{1} \\ - ⨏_{B_{3}} (a p | α | | \nabla η v | {| η \nabla v |}^{p - 1} + p {| α |}^{p} \bar{b} {| η v |}^{p - 1} | \nabla η v |) w_{1} \\ - C ⨏_{B_{3}} (c | α | | η v | {| η \nabla v |}^{p - 1} + {| α |}^{p} {\bar{d}}_{2} {| η v |}^{p - 1}) w_{2}, \end{aligned}

where

C = w_{2} (B_{3}) / w_{1} (B_{3})

. Depending on

β

, we have:

If $β > 0$ then

\begin{aligned} β ⨏_{B_{3}} {| η \nabla v |}^{p} w_{1} & \leq a p | α | ⨏_{B_{3}} | \nabla η v | {| η \nabla v |}^{p - 1} w_{1} + p {| α |}^{p} ⨏_{B_{3}} \bar{b} {| η v |}^{p - 1} | \nabla η v | w_{1} \\ + β {| α |}^{p} ⨏_{B_{3}} {\bar{d}}_{1} {| η v |}^{p} w_{1} + C | α | ⨏_{B_{3}} c | η v | {| η \nabla v |}^{p - 1} w_{2} \\ + C {| α |}^{p} ⨏_{B_{3}} {\bar{d}}_{2} {| η v |}^{p - 1} w_{2}, \end{aligned}

and if we proceed as in the proof of Theorem 1 to estimate each integral on the right-hand side, we obtain

\begin{aligned} {(⨏_{B_{3}} {| η \nabla v |}^{χ_{i} p} w_{i})}^{\frac{1}{χ_{i} p}} \leq C α^{\frac{p}{ε}} (1 + β^{- 1})^{\frac{1}{ε}} \\ \times [{(⨏_{B_{3}} {| η v |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{3}} {| η v |}^{p} w_{2})}^{\frac{1}{p}} + {(⨏_{B_{3}} {| \nabla η v |}^{p} w_{1})}^{\frac{1}{p}}] . \end{aligned}

η \in C_{c}^{\infty} (B_{h})

is such that

η \equiv 1

B_{h^{'}}

for

1 \leq h^{'} < h \leq 2

with

| \nabla η | \leq C (h - h^{'})^{- 1}

then

\begin{aligned} {(⨏_{B_{h^{'}}} {| v |}^{χ_{i} p} w_{i})}^{\frac{1}{χ_{i} p}} & \leq C {(\frac{w_{i} (B_{3})}{w_{i} (B_{h^{'}})})}^{\frac{1}{χ_{i} p}} \frac{α^{\frac{p}{ε}} (1 + β^{- 1})^{\frac{1}{ε}}}{h - h^{'}} \\ \times {[{(\frac{w_{1} (B_{h})}{w_{1} (B_{3})})}^{\frac{1}{p}} ⨏_{B_{h}} {| v |}^{p} w_{1} + {(\frac{w_{2} (B_{h})}{w_{2} (B_{3})})}^{\frac{1}{p}} ⨏_{B_{h}} {| v |}^{p} w_{2}]}^{\frac{1}{p}}, \end{aligned}

but since

1 \leq h^{'} < h \leq 2

, we have

\frac{w_{i} (B_{3})}{w_{i} (B_{h^{'}})} \leq \frac{w_{i} (B_{4 h^{'}})}{w_{i} (B_{h^{'}})} \leq γ_{w_{i}}^{2} and \frac{w_{i} (B_{h})}{w_{i} (B_{3})} \leq 1,

hence, for

χ = min {χ_{1}, χ_{2}}

, we have

Ψ (χ p, h^{'}) \leq C \frac{α^{\frac{p}{ε}} (1 + β^{- 1})^{\frac{1}{ε}}}{h - h^{'}} Ψ (p, h) .

(35)

Similarly, for $1 - p < β < 0$ one has

Ψ (χ p, h^{'}) \leq C \frac{(1 - β^{- 1})^{\frac{1}{ε}}}{h - h^{'}} Ψ (p, h) .

(36)

And if $β < 1 - p$ then one obtains

Ψ (χ p^{'}, h^{'}) \leq C \frac{(1 + | α |)^{\frac{p}{ε}}}{h - h^{'}} Ψ (p, h) .

(37)

If we observe that $Ψ (s, r) \to_{s \to + \infty}^{2} max_{B_{r}} \bar{u}$ and $Ψ (s, r) \to_{s \to - \infty}^{2} min_{B_{r}} \bar{u}$ , then we can repeat the iterative argument from the proof of Serrin (1964, Theorem 5) to deduce that (35) and (36) imply

max_{B_{1}} \bar{u} \leq C Ψ (p_{0}^{'}, 2)

for some

p_{0}^{'} \leq p_{0}

chosen appropriately, whereas (37) will give

min_{B_{1}} \bar{u} \geq C^{- 1} Ψ (- p_{0}, 2) .

Finally, we can use (34) to obtain a constant

C > 0

depending on the structural parameters such that

max_{B_{1}} \bar{u} \leq C min_{B_{1}} \bar{u}

and because

\bar{u} = u + k + δ

we conclude by letting

δ \to 0^{+}

3. Behavior at Infinity

In this section, we obtain a decay estimate for weak solutions to the equation

{\begin{aligned} - div (w_{1} {| \nabla u |}^{p - 2} \nabla u) & = w_{2} {| u |}^{q - 2} u & in Ω, \\ u & \in D^{1, p, w_{1}} (Ω), \end{aligned}

(38)

where the set

Ω \subseteq R^{N}

(bounded or not) is such that there exists a constant

C > 0

for which the global weighted Sobolev inequalities (14) and (15) hold. With the aid of the results regarding the equation

div A = B

, we are able to prove that weak solutions to (38) are locally bounded.

Lemma 1

Let $u \in D^{1, p, w_{1}} (Ω)$ be a weak solution of

- div (w_{1} {| \nabla u |}^{p - 2} \nabla u) = w_{2} {| u |}^{q - 2} u in Ω .

Then for every

R > 0

such that

B_{4 R} (x_{0}) \subseteq Ω

there exists

C_{R} > 0

such that

{‖ u ‖}_{L^{\infty} (B_{R} (x_{0}))} \leq C_{R} [u]_{p, B_{4 R} (x_{0})} .

Proof.

Observe that equation (38) can be written in the from $div A = B$ for $a = 1$ , $b = c = d_{1} = e = f = g = 0$ and $d_{2} = {| u |}^{q - p}$ . Firstly, from Theorem 2, we know that if $d_{2} \in L^{(D / p), w_{2}}$ then for every $s \geq 1$ and $R > 0$ the weak solution $u$ satisfies

\begin{aligned} {(⨏_{B_{2 R} (x_{0})} {| u |}^{s} w_{1})}^{\frac{1}{s}} + {(⨏_{B_{2 R} (x_{0})} {| u |}^{s} w_{2})}^{\frac{1}{s}} \\ \leq C_{R, s} [{(⨏_{B_{4 R} (x_{0})} {| u |}^{p} w_{1})}^{\frac{1}{p}} + {(⨏_{B_{4 R} (x_{0})} {| u |}^{p} w_{2})}^{\frac{1}{p}}], \end{aligned}

and

C_{R, s}

depends on

s

and on

{(⨏_{B_{4 R} (x_{0})} {| d_{2} |}^{(D / p)} w_{2})}^{p / D}

. But because

u \in D^{1, p, w_{1}} (Ω)

and the weights

w_{1}, w_{2}

verify (8) then the local Sobolev inequality (10) holds. Therefore

u \in L^{q, w_{2}} (Ω)

, hence

d \in L^{(D / p), w_{2}} (B_{4 R} (x_{0})) \Leftrightarrow q = D p / (D - p)

. In particular, this shows that

u \in L^{s, w_{2}} (B_{2 R} (x_{0}))

for every

s

and as a consequence

d_{2} = {| u |}^{q - p} \in L^{[D / (p - ε)], w_{2}} (B_{2 R} (x_{0}))

for every

0 < ε < p

. Therefore, we can now use Theorem 1 to conclude that

{‖ u ‖}_{L^{\infty} (B_{R} (x_{0}))} \leq C_{R} [u]_{p, B_{4 R} (x_{0})},

where

C_{R}

depends on

R > 0

and the norm of

u

D^{1, p, w_{1}} (Ω)

Now we estimate the decay of the $L^{q_{1}, w_{1}}$ norm of weak solutions as one leaves the set $Ω$ .

Lemma 2

Suppose $u \in D^{1, p, w_{1}} (Ω)$ is a weak solution of (38), then there exists $R_{0} > 0$ and $τ > 0$ such that for $R \geq R_{0}$ one has

{‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})} \leq {(\frac{R_{0}}{R})}^{τ} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R_{0}})} .

Proof.

Observe that for any $η \in W^{1, \infty} (R^{N})$ the function $φ = η^{p} u$ is a valid test function in

\int_{Ω} {| \nabla u |}^{p - 2} \nabla u \nabla φ w_{1} = \int_{Ω} {| u |}^{q - 2} u φ w_{2} .

On the one hand, using Young’s inequality, we can find

C_{p} > 0

such that

\begin{aligned} \int_{Ω} {| \nabla u |}^{p - 2} \nabla u \nabla φ w_{1} & = \int_{Ω} {| η \nabla u |}^{p} w_{1} + p \int_{Ω} η^{p - 1} {| \nabla u |}^{p - 2} \nabla u \cdot u \nabla η w_{1} \\ \geq \frac{1}{2} \int_{Ω} {| η \nabla u |}^{p} w_{1} - C_{p} \int_{Ω} {| u \nabla η |}^{p} w_{1} . \end{aligned}

On the other hand, since $q > p$ , we can write

\begin{aligned} \int_{Ω} {| u |}^{q - 2} u φ w_{2} & = \int_{Ω} u^{q} η^{p} w_{2} \\ = \int_{Ω} {| u |}^{q - p} {| η u |}^{p} w_{2} \\ \leq {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} {(\int_{Ω} {| η u |}^{q} w_{2})}^{\frac{p}{q}}, \end{aligned}

hence

\begin{aligned} \int_{Ω} {| \nabla (η u) |}^{p} w_{1} & = \int_{Ω} {| η \nabla u + u \nabla η |}^{p} w_{1} \\ \leq 2^{p - 1} \int_{Ω} {| η \nabla u |}^{p} w_{1} + 2^{p - 1} \int_{Ω} {| u \nabla η |}^{p} w_{1} \\ \leq 2^{p - 1} (2 \int_{Ω} {| \nabla u |}^{p - 2} \nabla u \nabla φ w_{1} + C_{p} \int_{Ω} {| u \nabla η |}^{p} w_{1}) \\ + 2^{p - 1} \int_{Ω} {| u \nabla η |}^{p} w_{1} \\ \leq C_{p} \int_{Ω} {| u \nabla η |}^{p} w_{1} + 2^{p} {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} {(\int_{Ω} {| η u |}^{q} w_{2})}^{\frac{p}{q}} . \end{aligned}

Therefore, the global Sobolev inequality (15) tells us that there exists a constant

C_{p, w_{1}, w_{2}} > 0

such that

\int_{Ω} {| \nabla (η u) |}^{p} w_{1} \leq C_{p} \int_{Ω} {| u \nabla η |}^{p} w_{1} + C_{p, w_{1}, w_{2}} {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} (\int_{Ω} {| \nabla (η u) |}^{p} w_{1}) .

(39)

We now choose $η$ . First of all, because ${‖ u ‖}_{q, w_{2}}$ is finite for any given $ε > 0$ we can find $R_{0} = R_{0} (ε) > 0$ such that

\int_{Ω ∖ B_{R}} {| u |}^{q} w_{2} \leq ε,

for any

R \geq R_{0}

. Hence, we select

R_{0} > 0

such that

C_{p, w_{1}, w_{2}} {(\int_{Ω ∖ B_{R_{0}}} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} \leq \frac{1}{2},

and we suppose that

R \geq R_{0}

from now on. We consider

η \in W^{1, \infty} (R^{N})

, such that

0 \leq η \leq 1

η (x) = 0

for

x \in B_{R}

η (x) = 1

for

x \notin B_{2 R}

, and

| \nabla η | \leq C R^{- 1}

. If we use such

η

in (39), we obtain a constant

C > 0

independent of

R

such that

\int_{Ω} {| \nabla (η u) |}^{p} w_{1} \leq C_{p} \int_{Ω} {| u \nabla η |}^{p} w_{1},

which after using (14) gives

{(\int_{Ω} {| η u |}^{q_{1}} w_{1})}^{\frac{1}{q_{1}}} \leq C {(\int_{Ω} {| u \nabla η |}^{p} w_{1})}^{\frac{1}{p}} .

(40)

By the choice of $η$ , we also have

\begin{aligned} \int_{Ω} {| u \nabla η |}^{p} w_{1} & \leq C R^{- p} \int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{p} w_{1} \\ \leq C R^{- p} {(w_{1} (Ω \cap B_{2 R}))}^{1 - \frac{1}{χ_{1}}} {(\int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1})}^{\frac{1}{χ_{1}}} \\ \leq C R^{- p} {(w_{1} (Ω \cap B_{R_{0}}) {(\frac{2 R}{R_{0}})}^{D_{1}})}^{1 - \frac{1}{χ_{1}}} {(\int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1})}^{\frac{1}{χ_{1}}} \\ = C {(\frac{w_{1} (Ω \cap B_{R_{0}})}{R_{0}^{D_{1}}})}^{1 - \frac{1}{χ_{1}}} R^{D_{1} (1 - \frac{1}{χ_{1}}) - p} {(\int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1})}^{\frac{1}{χ_{1}}} \\ \leq C R^{D_{1} (1 - \frac{1}{χ_{1}}) - p} {(\int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1})}^{\frac{1}{χ_{1}}} \\ = C {(\int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1})}^{\frac{1}{χ_{1}}}, \end{aligned}

(41)

where we have used (7) and the fact that

(1 / q_{1}) = (1 / D_{1}) - (1 / p)

. From (40) and (41), we obtain

\int_{Ω} {| η u |}^{q_{1}} w_{1} \leq C \int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1},

for some constant

C > 0

depending on

p, q_{1}, R_{0}

but independent of

R

. To continue, observe that since

η \equiv 1

B_{2 R}^{c}

, we can write

\begin{aligned} \int_{Ω ∖ B_{2 R}} {| u |}^{q_{1}} w_{1} & \leq \int_{Ω} {| η u |}^{q_{1}} w_{1} \\ \leq C \int_{Ω \cap B_{2 R} ∖ B_{R}} {| u |}^{q_{1}} w_{1} \\ = C \int_{Ω ∖ B_{R}} {| u |}^{q_{1}} w_{1} - C \int_{Ω ∖ B_{2 R}} {| u |}^{q_{1}} w_{1}, \end{aligned}

thus, if

θ = (C / C + 1) \in (0, 1)

, then we obtain

\int_{Ω ∖ B_{2 R}} {| u |}^{q_{1}} w_{1} \leq θ \int_{Ω ∖ B_{R}} {| u |}^{q_{1}} w_{1} .

If we consider $f (R) = \int_{Ω ∖ B_{R}} {| u |}^{q} w_{1}$ , then (3) tells us that

f (2 R) \leq θ f (R),

and if we take

R = R_{n} = 2^{n} R_{0}

for

n \geq 0

, the above inequality gives

f (2^{n} R_{0}) \leq θ f (2^{n - 1} R_{0})

, which after iterating it gives

f (2^{n} R_{0}) \leq θ^{n} f (R_{0}) .

Because

R \geq R_{0}

then one can find

n \geq 1

such that

2^{n - 1} R_{0} \leq R < 2^{n} R_{0}

the above can be written as

f (R) \leq f (2^{n} R_{0}) \leq θ^{n} f (R_{0}) \leq θ^{\log_{2} (R R_{0}^{- 1})} f (R_{0}) .

Hence

\int_{Ω ∖ B_{R}} {| u |}^{q} w_{1} \leq {(\frac{R_{0}}{R})}^{- \log_{2} θ} \int_{Ω ∖ B_{R_{0}}} {| u |}^{q} w_{1},

and the result is proved for

τ := - (1 / q) \log_{2} θ > 0

Lemma 3

Suppose that $u \in D^{1, p, w_{1}} (Ω)$ is a weak solution of

- div (w_{1} {| \nabla u |}^{p - 2} \nabla u) = w_{2} {| u |}^{q - 2} u in Ω .

(42)

Then for each

s > max {q_{1}, q}

there exists

R_{0} > 0

(depending on

s

) and

C = C (p, q_{1}, q, w_{1}, w_{2}; s) > 0

such that

{‖ u ‖}_{L^{s, w_{i}} (Ω ∖ B_{2 R})} \leq \frac{C}{R^{\frac{p}{q_{1} - p} - o_{s} (1)}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})},

for both

i = 1, 2

and

R \geq R_{0}

. In the above estimate,

o_{s} (1)

is a quantity that goes to

0

s \to \infty

Proof.

Firstly notice that thanks to the $L^{s, w}$ interpolation inequality it is enough to exhibit a sequence $s_{n} \to_{n \to \infty}^{+} \infty$ such that

{‖ u ‖}_{L^{s_{n}, w_{i}} (Ω ∖ B_{2 R})} \leq \frac{C}{R^{\frac{p}{q_{1} - p} - o_{n} (1)}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})} .

Observe that in the context of (11) we can view (42) as

div A = B

where

a = 1

b = c = d_{1} = e = f = g = 0

and

d_{2} = {\bar{d}}_{2} = {| u |}^{q - p}

. The assumption

u \in D^{1, p, w_{1}} (Ω)

tells us that

φ = η^{p} G (u)

is a valid test function and we can follow the notation of the proof of Theorem 1. Moreover, since

e = f = g = 0

we can further suppose that

k > 0

is arbitrary in the definition of both

F

and

G

. Starting with (18) we now integrate over

Ω

to obtain

\int_{Ω} {| η \nabla v |}^{p} w_{1} \leq p \int_{Ω} | v \nabla η | {| η \nabla v |}^{p - 1} w_{1} + (α - 1) α^{p - 1} \int_{Ω} d_{2} {| v η |}^{p} w_{2},

where

v = F (\bar{u})

. From the above, we obtain

\int_{Ω} {| \nabla (η v) |}^{p} w_{1} \leq C_{α} (\int_{Ω} {| v \nabla η |}^{p} w_{1} + \int_{Ω} {| u |}^{q - p} {| v η |}^{p} w_{2}),

and with the help of (15) we can write

\begin{aligned} \int_{Ω} {| u |}^{q - p} {| v η |}^{p} w_{2} & \leq {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} {(\int_{Ω} {| v η |}^{q} w_{2})}^{\frac{p}{q}} \\ \leq C_{p, w_{1}, w_{2}} {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} (\int_{Ω} {| \nabla (v η) |}^{p} w_{1}), \end{aligned}

therefore, we have

\int_{Ω} {| \nabla (η v) |}^{p} w_{1} \leq C_{α} \int_{Ω} {| v \nabla η |}^{p} w_{1} + C_{p, α, w_{1}, w_{2}} {(\int_{supp η} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} (\int_{Ω} {| \nabla (v η) |}^{p} w_{1}) .

Because $u \in D^{1, p, w_{1}} (Ω)$ and that (15) holds then we know that $u \in L^{q, w_{2}} (Ω)$ , therefore for any given $ν > 0$ we can find $R_{0} = R_{0} (ν) > 0$ such that

\int_{Ω ∖ B_{R}} {| u |}^{q} w_{2} \leq ν, \forall R \geq R_{0},

therefore, we can find

R_{0} = R_{0} (α) > 0

such that

C_{p, α, w_{1}, w_{2}} {(\int_{Ω ∖ B_{R}} {| u |}^{q} w_{2})}^{1 - \frac{p}{q}} \leq \frac{1}{2},

and if we suppose that

R \geq R_{0}

then for any

η

such that

supp η \subset B_{R}^{c}

we obtain

\int_{Ω} {| \nabla (η v) |}^{p} w_{1} \leq C_{α} \int_{Ω} {| v \nabla η |}^{p} w_{1} .

By using (14), (15), and passing to the limits

l \to + \infty

k \to 0^{+}

, we deduce that

\begin{aligned} {(\int_{Ω} {| η u^{α} |}^{q_{1}} w_{1})}^{\frac{1}{q_{1}}} & \leq C_{α} {(\int_{Ω} {| u^{α} \nabla η |}^{p} w_{1})}^{\frac{1}{p}}, \end{aligned}

(43)

\begin{aligned} {(\int_{Ω} {| η u^{α} |}^{q} w_{2})}^{\frac{1}{q}} & \leq C_{α} {(\int_{Ω} {| u^{α} \nabla η |}^{p} w_{1})}^{\frac{1}{p}} . \end{aligned}

(44)

We now select $η$ : for $n \geq 0$ we consider $R_{n} = R (2 - 2^{- n})$ and a smooth function $η$ such that $0 \leq η \leq 1$ , $η (x) = 0$ for $| x | \leq R_{n}$ , $η (x) = 1$ for $| x | \geq R_{n + 1}$ and satisfies $| \nabla η | \leq C 2^{n} / R$ ,

\begin{aligned} supp η & \subseteq Ω ∖ B_{R_{n}}, \\ supp \nabla η & \subseteq Ω \cap B_{R_{n}} ∖ B_{R_{n + 1}} . \end{aligned}

Hence, for

n \geq 1

we let

α_{n} = {(q_{1} / p)}^{n}

in (43) and we obtain

{(\int_{Ω ∖ B_{R_{n + 1}}} {| u |}^{\frac{q_{1}^{n + 1}}{p^{n}}} w_{1})}^{\frac{p^{n}}{q_{1}^{n + 1}}} \leq {(\frac{C_{n}}{R})}^{\frac{p^{n}}{q_{1}^{n}}} {(\int_{Ω ∖ B_{R_{n}}} {| u |}^{\frac{q_{1}^{n}}{p^{n - 1}}} w_{1})}^{\frac{p^{n - 1}}{q_{1}^{n}}},

or equivalently

\begin{aligned} U_{n + 1} \leq \frac{{\tilde{C}}_{n}}{R^{\frac{p^{n}}{q_{1}^{n}}}} U_{n}, \end{aligned}

for

s_{n} = q_{1}^{n} / p^{n - 1}

and

U_{n} = {‖ u ‖}_{L^{s_{n}, w_{1}} (Ω ∖ B_{R_{n}})}

and

{\tilde{C}}_{n} = C_{n}^{{(p / q_{1})}^{n}}

. After iterating the above inequality, we obtain

U_{n} \leq (\frac{\prod_{i = 1}^{n - 1} {\tilde{C}}_{i}}{R^{\sum_{i = 1}^{n - 1} {(\frac{p}{q_{1}})}^{i}}}) U_{1},

and because

\begin{aligned} \sum_{i = 1}^{n - 1} {(\frac{p}{q_{1}})}^{i} = \frac{p}{q_{1} - p} - \frac{q_{1}}{q_{1} - p} {(\frac{p}{q_{1}})}^{n} = \frac{p}{q_{1} - p} - o_{n} (1), \end{aligned}

for

q_{1} > p

, we obtain that for any

s > q_{1}

\begin{aligned} {‖ u ‖}_{L^{s, w_{1}} (Ω ∖ B_{2 R})} \leq \frac{C_{s}}{R^{\frac{p}{q_{1} - p} - o_{s} (1)}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})}, \end{aligned}

because

U_{1} \leq {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})}

U_{n} \geq {‖ u ‖}_{L^{s_{n}, w_{1}} (B_{2 R})}

With the same choice of $η$ and $α$ in (44) we also have

\begin{aligned} {(\int_{Ω ∖ B_{R_{n + 1}}} {| u |}^{\frac{q_{1}^{n} q}{p^{n}}} w_{2})}^{\frac{p^{n}}{q_{1}^{n} q}} & \leq {(\frac{C_{n}}{R})}^{\frac{p^{n}}{q_{1}^{n}}} {(\int_{Ω ∖ B_{R_{n}}} {| u |}^{\frac{q_{1}^{n}}{p^{n - 1}}} w_{1})}^{\frac{p^{n - 1}}{q_{1}^{n}}} \\ = {(\frac{C_{n}}{R})}^{\frac{p^{n}}{q_{1}^{n}}} U_{n} \\ \leq (\frac{\prod_{i = 1}^{n} {\tilde{C}}_{i}}{R^{\sum_{i = 1}^{n} {(\frac{p}{q_{1}})}^{i}}}) U_{1}, \end{aligned}

and just as before, we deduce that

{‖ u ‖}_{L^{s, w_{2}} (Ω ∖ B_{2 R})} \leq \frac{C_{s}}{R^{\frac{p}{q_{1} - p} - o_{s} (1)}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})}

this time for

s > q

Now we are in a position to prove Theorem 4:

Proof of Theorem 4. Proof of Theorem 4.

Consider the value of $R_{0} > 0$ given in Lemma 2, and suppose that $x \in Ω ∖ B_{2 R_{0}}$ . Fix $0 < r < R_{0} / 4$ so that $B_{2 r} (x) \subseteq Ω$ and use Lemma 1 to obtain

| u (x) | \leq {‖ u ‖}_{L^{\infty} (B_{r} (x))} \leq C_{r} [u]_{p, B_{2 r}} \leq C_{r} [{(\int_{B_{2 r}} {| u |}^{s} w_{1})}^{\frac{1}{s}} + {(\int_{B_{2 r}} {| u |}^{s} w_{2})}^{\frac{1}{s}}],

for any

s > p

. If we consider

R = | x | / 4

, then by geometric considerations we deduce that

B_{2 r} (x) \subseteq Ω ∖ B_{2 R}

hence

{(\int_{B_{2 r}} {| u |}^{s} w_{i})}^{\frac{1}{s}} \leq {(\int_{Ω ∖ B_{2 R}} {| u |}^{s} w_{i})}^{\frac{1}{s}} .

Now we fix

s

large enough so that

o_{s} (1) \leq τ / 2

in Lemma 3, where

τ > 0

is taken from Lemma 2, by doing that we obtain

\begin{aligned} {‖ u ‖}_{L^{s, w_{2}} (Ω ∖ B_{2 R})} + {‖ u ‖}_{L^{s, w_{1}} (Ω ∖ B_{2 R})} & \leq \frac{C}{R^{\frac{p}{q_{1} - p} - o_{s} (1)}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})} \\ \leq \frac{C}{R^{\frac{p}{q_{1} - p} - \frac{τ}{2}}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R})} \\ \leq \frac{C}{R^{\frac{p}{q_{1} - p} - \frac{τ}{2}}} {(\frac{R_{0}}{R})}^{τ} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R_{0}})}, \end{aligned}

therefore, by putting all the above estimates together, we obtain

| u (x) | \leq \frac{C R_{0}^{τ}}{R^{\frac{p}{q_{1} - p} + \frac{τ}{2}}} {‖ u ‖}_{L^{q_{1}, w_{1}} (Ω ∖ B_{R_{0}})} = \frac{C}{{| x |}^{\frac{p}{q_{1} - p} + λ}},

for some constant

C > 0

independent of

| x | \geq 2 R_{0}

, and the result is proved for

\tilde{R} = 2 R_{0}

4. Monomial Type Weights

In this section, we show that we can apply the main results of this work to a pair of monomial weights: for $A = (a_{1}, \dots, a_{N}), B = (b_{1}, \dots, b_{N}) \in R^{N}$ we define

w_{1} (x) = x^{A} := \prod_{i = 1}^{N} x_{i}^{a_{i}}, w_{2} (x) = x^{B} := \prod_{i = 1}^{N} x_{i}^{b_{i}},

for

x = (x_{1}, \dots, x_{N}) \in [0, \infty)^{N}

Observe that, for $a_{i} > - 1$ , the one-dimensional weight $x_{i}^{a_{i}}$ belongs to $A_{\infty} = ⋃_{p \geq 1} A_{p}$ , in particular, $x_{i}^{a_{i}}$ is doubling (with doubling constant $γ = 2^{1 + a_{i}}$ ) and $p$ -admissible if $a_{i} < p - 1$ (see Castro & Cornejo, 2023 for the case $a_{i} \geq p - 1$ ). From this observation, it is not difficult to see that $w_{1} (x) = x^{A}$ is doubling (with doubling constant $2^{N + a}$ , for $a = a_{1} + \dots + a_{N}$ ) and $p$ -admissible, and that $x^{B}$ is also doubling (with doubling constant $2^{N + b}$ for $b = b_{1} + \dots + b_{N}$ ).

From Cabré and Ros-Oton (2013) and Castro (2017), we know that the following global Sobolev inequality is satisfied:

{(\int_{(R_{+})^{N}} {| u |}^{q} x^{B} d x)}^{\frac{1}{q}} \leq C {(\int_{(R_{+})^{N}} {| \nabla u |}^{p} x^{A} d x)}^{\frac{1}{p}}

for

q = [(N + b) p / (N + a - p)] > p \geq 1

a_{i} > 0

for all

i

, and

B \in R^{N}

satisfying suitable compatibility conditions (see Castro, 2017 for the details). Therefore in order to use Theorems 1 to 4, we only need to verify that (8) is satisfied by these weights. We will check this by using Remark 4, that is, we will show that the pair of weights

w_{1} (x) = x^{A}, w_{2} (x) = x^{B}

satisfies the local Sobolev inequality (10) for suitable

A, B \in R^{N}

We begin with some lemmas in dimension 1. In what follows $I_{R}$ will denote an interval contained in $[0, \infty)$ of length $R > 0$ .

Lemma 4
Let $a > 0$ , $v (x) = x^{a - 1}$ , and $w (x) = x^{a}$ . Then for every $u \in C^{1} (I_{R})$ one has
$\frac{1}{v (I_{R})} \int_{I_{R}} | u - {\bar{u}}_{v, I_{R}} | v \leq C \frac{R}{w (I_{R})} \int_{I_{R}} | u^{'} | w .$

Proof.
Write $I_{R} = I = (α, β)$ for $β - α = R$ and let $x, y \in I$ . Starting from
$u (x) - u (y) = \int_{x}^{y} u^{'} (s) d s$
multiply by $y^{a - 1}$ and integrate over $I$ in the $y$ variable and then multiply by $x^{a - 1}$ and integrate over $I$ in the $x$ variable to obtain
$\begin{aligned} v (I) \int_{I} | u (x) - {\bar{u}}_{v, I} | x^{a - 1} d x \leq \int_{I} \int_{I} | \int_{x}^{y} u^{'} (s) y^{a - 1} x^{a - 1} d s | d y d x . \end{aligned}$
Use Tonelli’s theorem a couple of times on the right-hand side to obtain
$\begin{aligned} \int_{I} \int_{I} | \int_{x}^{y} u^{'} (s) y^{a - 1} x^{a - 1} d s | d y d x \leq \frac{2 v (I)}{a} \int_{I} | u^{'} (s) | (s^{a} - α^{a}) d s, \end{aligned}$
so that
$\int_{I} | u (x) - {\bar{u}}_{v, I} | x^{a - 1} d x \leq \frac{2}{a} \int_{I} | u^{'} (x) | (x^{a} - α^{a}) d x .$
By studying the function $f (x) = 1 - {(α / x)}^{a}$ for $a > 0$ we deduce that if $0 \leq α < x < β < \infty$ then
$x^{a} - α^{a} \leq R (a + 1) x^{a} \frac{β^{a} - α^{a}}{β^{a + 1} - α^{a + 1}} = a R \frac{v (I)}{w (I)} w (x),$
and the result follows.
Lemma 5
Let $a > 0$ and suppose that $v (x) = w (x) = x^{a}$ . Then
${(\frac{1}{v (I_{R})} \int_{I_{R}} {| u - {\bar{u}}_{v, I_{R}} |}^{q} v)}^{\frac{1}{q}} \leq C \frac{R}{w (I_{R})} \int_{I_{R}} | u^{'} | w$
for $q = (1 + a) / a$ and all $u \in C^{1} (I_{R})$ .
Proof.
This follows since the weight $w (x) = x^{a}$ is doubling and it satisfies
$\frac{w (I_{2 R})}{w (I_{R})} \leq 2^{1 + a} \Leftrightarrow {(\frac{w (I_{2 R})}{w (I_{R})})}^{1 - \frac{1}{χ}} \leq 2$
for $χ = (1 + a) / a$ , therefore, we can use Lemma 8 to deduce that
${(\frac{w (I_{2 R})}{w (I_{R})})}^{1 - \frac{1}{χ}} \leq 2 \Leftrightarrow \frac{r}{R} {(\frac{w_{2} (I_{r})}{w_{2} (I_{R})})}^{\frac{1}{χ}} \leq C (\frac{w_{1} (I_{r})}{w_{1} (I_{R})}),$
from which the local Poincaré–Sobolev inequality follows for $p = 1$ and $q = 1 / χ$ .
Corollary 1
Let $a > 0$ , $v_{0} (x) = x^{a - 1}$ , and $v_{1} (x) = w (x) = x^{a}$ . Then
$\frac{1}{v_{0} (I_{R})} \int_{I_{R}} | u | v_{0} \leq C \frac{R}{w (I_{R})} \int_{I_{R}} | u^{'} | w,$
and for $q_{1} = (1 + a) / a$
${(\frac{1}{v_{1} (I_{R})} \int_{I_{R}} {| u |}^{q_{1}} v_{1})}^{\frac{1}{q_{1}}} \leq C \frac{R}{w (I_{R})} \int_{I_{R}} | u^{'} | w,$
for every $u \in C_{c}^{1} (I)$ .
Proof.
This follows directly from the fact that the local Poincaré–Sobolev inequality implies (8), which in turn implies the local Sobolev inequality (see Remark 4).

The following lemma will allow us to interpolate between the cases $v_{0} (x) = x^{a - 1}$ and $v_{1} (x) = x^{a}$ .
Lemma 6
Suppose that $a > 0$ and that $λ \in (0, 1)$ then for $v_{0} (x) = x^{a - 1}$ , $v_{1} (x) = x^{a}$ , and $v (x) = x^{a - λ}$ there exists a constant depending solely on $a$ and $λ$ such that
$v_{0} (I_{R})^{λ} v_{1} (I_{R})^{1 - λ} \leq C v (I_{R}),$
for any interval $I_{R} \subseteq [0, \infty)$ .
Proof.
Write $v (x) = v_{0} {(x)}^{λ} v_{1} {(x)}^{1 - λ}$ and if $I = (α, β)$ then the inequality is equivalent to proving that
$(β^{a} - α^{a})^{λ} (β^{a + 1} - α^{a + 1})^{1 - λ} \leq C (β^{a - λ + 1} - α^{a - λ + 1})$
for every $0 \leq α < β < \infty$ . By homogeneity, it suffices to prove that
$f (x) := \frac{(1 - x^{a})^{λ} (1 - x^{a + 1})^{1 - λ}}{1 - x^{a - λ + 1}} = {(\frac{1 - x^{a}}{1 - x^{a + 1}})}^{λ} \cdot \frac{1 - x^{a + 1}}{1 - x^{a - λ + 1}}$
is bounded in $[0, 1)$ , which follows by continuity of $f$ and because $f (0) = 1$ and
$lim_{x \to 1^{-}} f (x) = {(\frac{a}{a + 1})}^{λ} \cdot \frac{a + 1}{a - λ + 1},$
since $λ \in (0, 1)$ and $a > 0$ .

We now consider monomial weights in $R^{N}$ . In what follows, $Q_{R}$ will be a cube with sides parallel to the coordinate axes and edge-length $R > 0$ , that is,
$Q_{R} = \prod_{i = 1}^{N} I_{R}^{i},$
with $I_{R}^{i}$ an interval in $[0, \infty)$ of length $R > 0$ . Also, from now on we will write $A = (a_{1}, \dots, a_{N})$ and suppose that $a_{i} > 0$ for all $i$ .
Lemma 7
For $Λ = (λ_{1}, \dots, λ_{N}) \in R^{N}$ with $λ_{i} \in [0, 1]$ and $\sum_{i = 1}^{N} λ_{i} = 1$ consider $v_{i} (x) = x^{A - e_{i}}$ , where $e_{i}$ is the $i$ th element of the canonical basis of $R^{N}$ , and $v_{0} (x) = x^{A - Λ}$ . There exists a constant $C = C (A, Λ)$ such that for any cube $Q_{R}$ we have
$\prod_{i = 1}^{N} v_{i} (Q_{R})^{λ_{i}} \leq C v_{0} (Q_{R}) .$
Additionally, if for $w (x) = x^{A}$ and $δ \in (0, 1)$ we define
$v (x) = v_{0} {(x)}^{δ} w {(x)}^{1 - δ},$
then there exists a constant $C = C (A, Λ, δ)$ such that
$v_{0} (Q_{R})^{δ} w (Q_{R})^{1 - δ} \leq C v (Q_{R}) .$

Proof.
The proof is similar to the proof of Lemma 6, we omit the details.
Proposition 1
For each $i \in {1, \dots, N}$ consider $v (x) = x^{A - e_{i}}$ and $w (x) = x^{A}$ then we have
$⨏_{Q_{R}} | u | v \leq C R ⨏_{Q_{R}} | \nabla u | w \forall u \in C_{c}^{1} (Q_{R}) .$
(45)
Proof.
It is enough to consider the case $i = 1$ , and in this case observe that if $Q_{R} = \prod_{i = 1}^{N} I_{R}^{i}$ , where each $I_{R}^{i}$ is an interval of length $R > 0$ contained in $[0, \infty)$ , then by Corollary 1 we have
$\int_{I_{R}^{1}} | u | x_{1}^{a_{1} - 1} {d x}_{1} \leq C R \frac{v_{1} (I_{R}^{1})}{w_{1} (I_{R}^{1})} \int_{I_{R}^{1}} | \partial_{1} u | x_{1}^{a_{1}} {d x}_{1} \leq C R \frac{v_{1} (I_{R}^{1})}{w_{1} (I_{R}^{1})} \int_{I_{R}^{1}} | \nabla u | x_{1}^{a_{1}} {d x}_{1},$
where $v_{1} (x_{1}) = x_{1}^{a_{1} - 1}$ and $w_{1} (x_{1}) = x_{1}^{a_{1}}$ . If we multiply both sides by $\prod_{i = 2}^{N} x^{a_{i}}$ and integrate over the remaining coordinates we get to
$\int_{Q_{R}} | u | v \leq C R \frac{v_{1} (I_{R}^{1})}{w_{1} (I_{R}^{1})} \int_{Q_{R}} | \nabla u | w .$
(46)
Finally, observe that
$\frac{v_{1} (I_{R}^{1})}{w_{1} (I_{R}^{1})} = \frac{v (Q_{R})}{w (Q_{R})} .$

Proposition 2 Local $(1, 1)$ Hardy inequality

For $Λ \in R^{N}$ with $λ_{i} \in [0, 1]$ satisfying $\sum_{i = 1}^{N} λ_{i} = 1$ consider $v (x) = x^{A - Λ}$ and $w (x) = x^{A}$ then we have

⨏_{Q_{R}} | u | v \leq C R ⨏_{Q_{R}} | \nabla u | w \forall u \in C_{c}^{1} (Q_{R}) .

(47)

Proof.

If $v (x) = x^{A - λ}$ then we can write $v (x) = \prod_{i = 1}^{N} v_{i} {(x)}^{λ_{i}}$ for $v_{i} (x) = x^{A - e_{i}}$ , hence by Hölder’s inequality, Proposition 1 and Lemma 7, we obtain

\begin{aligned} \int_{Q_{R}} | u | v & \leq \prod_{i = 1}^{n} {(\int_{Q_{R}} | u | v_{i})}^{λ_{i}} \\ \leq \prod_{i = 1}^{n} {(C R v_{i} (Q_{R}) ⨏_{Q_{R}} | \nabla u | w)}^{λ_{i}} \\ \leq C R v (Q_{R}) ⨏_{Q_{R}} | \nabla u | w . \end{aligned}

Proposition 3

If $v (x) = w (x) = x^{A}$ then

{(⨏_{Q_{R}} {| u |}^{\frac{N + a}{N + a - 1}} v)}^{\frac{N + a - 1}{N + a}} \leq C R ⨏_{Q_{R}} | \nabla u | w \forall u \in C_{c}^{1} (Q_{R}),

(48)

where

a = \sum_{i = 1}^{N} a_{i}

Proof.

This follows since the weight $w (x) = x^{A}$ is doubling and it satisfies

\frac{w (B_{2 R})}{w (B_{R})} \leq 2^{N + a} \Leftrightarrow {(\frac{w (B_{2 R})}{w (B_{R})})}^{1 - \frac{1}{χ}} \leq 2

for

χ = (N + a) / (N + a - 1)

, therefore we can use Lemma 8 to deduce that

{(\frac{w (B_{2 R})}{w (B_{R})})}^{1 - \frac{1}{χ}} \leq 2 \Leftrightarrow \frac{r}{R} {(\frac{w_{2} (B_{r})}{w_{2} (B_{R})})}^{\frac{1}{χ}} \leq C (\frac{w_{1} (B_{r})}{w_{1} (B_{R})}),

from which the local Sobolev inequality follows.

Proposition 4 Local

(((N + a) / (N + a - 1)), 1)

Sobolev inequality

For $B \in R^{N}$ with $a_{i} - 1 \leq b_{i} \leq a_{i}$ consider $v (x) = x^{B}$ and $w (x) = x^{A}$ . If $a = \sum_{i = 1}^{N} a_{i}$ , $b = \sum_{i = 1}^{N} b_{i}$ and $0 \leq a - b \leq 1$ then we have

{(⨏_{Q_{R}} {| u |}^{\frac{N + b}{N + a - 1}} v)}^{\frac{N + a - 1}{N + b}} \leq C R ⨏_{Q_{R}} | \nabla u | w \forall u \in C_{c}^{1} (Q_{R}) .

(49)

Proof.

Since $a_{i} - b_{i} \in (0, 1)$ we can write $a_{i} - b_{i} = (a - b) λ_{i}$ for suitable $λ_{i} \in [0, 1]$ with $\sum_{i = 1}^{N} λ_{i} = 1$ . Therefore, if we write $v_{0} (x) = x^{A - Λ}$ then

v (x) = v_{0} {(x)}^{δ} w {(x)}^{1 - δ},

where

δ = a - b

. Notice that from this choices we can write

q = (N + b) / (N + a - 1) = δ + (1 - δ) [(N + a) / (N + a - 1)]

so that

\begin{aligned} \int_{Q_{R}} {| u |}^{q} v & = \int_{Q_{R}} (| u | v_{0})^{δ} ({| u |}^{\frac{N + a}{N + a - 1}} w)^{1 - δ} \\ \leq {(\int_{Q_{R}} | u | v_{0})}^{δ} {(\int_{Q_{R}} {| u |}^{\frac{N + a}{N + a - 1}} w)}^{1 - δ} \\ \leq {(C R v_{0} (Q_{R}) ⨏_{Q_{R}} | \nabla u | w)}^{δ} [w (Q_{R})^{1 - δ} {(C R ⨏_{Q_{R}} | \nabla u | w)}^{(1 - δ) \frac{N + a - 1}{N + a}}] \\ \leq C R^{q} (v_{0} (Q_{R})^{δ} w (Q_{R})^{1 - δ}) {(\int_{Q_{R}} | \nabla u | w)}^{q} \\ \leq C R^{q} v (Q_{R}) {(\int_{Q_{R}} | \nabla u | w)}^{q} \end{aligned}

by Lemma 7 and Propositions 2 and 3

To generalize Proposition 4 to the case $p > 1$ , we do the following: from Remark 4 and Proposition 4, we deduce that the weights $w_{1} = x^{A}$ and $w_{2} = x^{B}$ satisfy (8) for $p = 1$ and $q = (N + b) / (N + a - 1)$ , that is,

\frac{r}{R} {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{N + a - 1}{N + b}} \leq C (\frac{w_{1} (Q_{r})}{w_{1} (Q_{R})}) .

The weight

w_{2}

is doubling with doubling constant

2^{N + b}

, therefore

\frac{w_{2} (Q_{R})}{w_{2} (Q_{r})} \leq C {(\frac{R}{r})}^{N + b} \Leftrightarrow {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{1 - p}{N + b}} \leq C {(\frac{r}{R})}^{1 - p},

for

p > 1

. Hence, for

N + a > p

, we can write

\begin{aligned} {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{N + a - p}{N + b}} & = {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{N + a - 1}{N + b}} {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{1 - p}{N + b}} \\ \leq C [\frac{R}{r} (\frac{w_{1} (Q_{r})}{w_{1} (Q_{R})})] {(\frac{r}{R})}^{1 - p} \\ = C {(\frac{R}{r})}^{p} (\frac{w_{1} (Q_{r})}{w_{1} (Q_{R})}), \end{aligned}

that is,

\begin{aligned} \frac{r}{R} {(\frac{w_{2} (Q_{r})}{w_{2} (Q_{R})})}^{\frac{N + a - p}{(N + b) p}} \leq C {(\frac{w_{1} (Q_{r})}{w_{1} (Q_{R})})}^{\frac{1}{p}}, \end{aligned}

(50)

thus obtaining the cube version of (8) for

q = (N + b) p / N + a - p

. Finally because

B_{R} \subseteq Q_{2 R} \subseteq B_{\sqrt{N} R}

and by using the doubling property of the weights we can replace the cubes

Q_{R}

by balls

B_{R}

in (50) to get (8) for balls.

Footnotes

Acknowledgments

The author would like to thank the anonymous referees for their suggestions to improve this manuscript.

ORCID iD

Hernán Castro

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Notes

A Simpler Version of ( 8 )

References

Björn

(2001). Poincaré inequalities for powers and products of admissible weights. Annales Academiae Scientiarum Fennicae Mathematica, 26(1), 175–188. http://eudml.org/doc/122274

Cabré

Ros-Oton

(2013). Sobolev and isoperimetric inequalities with monomial weights. Journal of Differential Equations, 255(11), 4312–4336. http://dx.doi.org/10.1016/j.jde.2013.08.010

Castro

(2017). Hardy–Sobolev-type inequalities with monomial weights. Annali di Matematica Pura ed Applicata, 196(2), 579–598. http://doi.org/10.1007/s10231-016-0587-2

Castro

(2021). Extremals for Hardy–Sobolev type inequalities with monomial weights. Journal of Mathematical Analysis and Applications, 494(2), 124645. https://doi.org/10.1016/j.jmaa.2020.124645

Castro

Cornejo

(2023). Poincaré’s inequality and Sobolev spaces with monomial weights. Mathematische Nachrichten, 296(10), 4500–4522. https://doi.org/10.1002/mana.202200100

Chanillo

Wheeden

R. L.

(1985). Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. American Journal of Mathematics, 107(5), 1191–1226. https://doi.org/10.2307/2374351

Fabes

E. B.

Kenig

C. E.

Serapioni

R. P.

(1982). The local regularity of solutions of degenerate elliptic equations. Communications in Partial Differential Equations, 7(1), 77–116. https://doi.org/10.1080/03605308208820218

Franchi

Gutiérrez

C. E.

Wheeden

R. L.

(1994). Weighted Sobolev–Poincaré inequalities for Grushin type operators. Communications in Partial Differential Equations, 19(3–4), 523–604. https://doi.org/10.1080/03605309408821025

(2025). The double phase problems on lattice graphs. Journal of Differential Equations, 438, 113380. https://doi.org/10.1016/j.jde.2025.113380

10.

Heinonen

Kilpeläinen

Martio

(2006). Nonlinear potential theory of degenerate elliptic equations. Dover Publications, Inc. Unabridged republication of the 1993 original.

11.

Papageorgiou

N. S.

Pudełko

Rădulescu

V. D.

(2023). Non-autonomous

(p, q)

-equations with unbalanced growth. Mathematische Annalen, 385(3–4), 1707–1745. https://doi.org/10.1007/s00208-022-02381-0

12.

Serrin

(1964). Local behavior of solutions of quasi-linear equations. Acta Mathematica, 111, 247–302. https://doi.org/10.1007/BF02391014