Existence Results for a Class of Nonlocal p -Laplacian Equations

Abstract

In this paper, we focus our attention on the nonlocal equation ${\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u), in Ω, \\ u = 0, on \partial Ω, \end{cases}$ getting nontrivial weak solutions for some classes of functions $f$ , where $Ω \subset R^{N}$ is a bounded smooth domain, $2 < p < N$ and $a$ , $ψ$ , $φ$ , $f$ are functions whose properties will be timely introduced. In fact, to obtain our results we will use variational and nonvariational methods, generalizing results already existing in the published literature for the case $p = 2$ .

Keywords

non-local problems variational methods integral operators

1. Introduction and Main Results

This paper establishes existence results for the class of nonlocal $p$ -Laplacian problems

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u), in Ω, \\ u = 0, on \partial Ω, \end{cases} (P)

where

Ω \subset R^{N}

is a bounded smooth domain,

2 < p < N

and

a

ψ

φ

f

are functions whose properties will be timely introduced. In fact,

(P)

is a

p

-Laplacian counterpart of the equation

{\begin{cases} - div (a (x) \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u), in Ω, \\ u = 0, on \partial Ω, \end{cases} (P_{1})

this class of problems has been studied in previous works, such as Allegretto and Barabanova (1996) and Barabanova (1996). These papers address several important questions regarding the linear version of problem

(P)

, where

f (x, u) = f (x)

, including existence and uniqueness of solutions, the validity (or failure) of maximum principles, and linear eigenvalue problems.

A natural extension is to investigate whether these results remain valid when the linear operator $div (a (x) \nabla u)$ is replaced by its nonlinear counterpart, the $p$ -Laplacian type operator $div (a (x) | \nabla u |^{p - 2} \nabla u)$ . To the best of our knowledge, the techniques developed in Allegretto and Barabanova (1996) and Barabanova (1996) do not directly apply to this more general setting.

As a result, we adopt a different approach to study qualitative properties of $(P)$ , particularly those related to the positiveness of solutions. Heuristically, when $η > 0$ is sufficiently small, the nonlocal operator

- div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u,

can be seen as a perturbation of the local

p

-Laplacian operator

- div (a (x) | \nabla u |^{p - 2} \nabla u)

. Since the latter is much better understood, we may recover certain properties, such as the maximum principle, which is not generally available in the nonlocal case, via a limiting argument as

η \to 0^{+}

It is worth emphasizing that problem

{\begin{cases} - div (a (x) \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u), in Ω, \\ u = 0, on \partial Ω \end{cases} (P_{1})

has been extensively studied in recent years under various hypotheses, see for example, Alves et al. (2022), Bahia et al. (2024), Corrêa et al. (2023), Corrêa, Lima, et al. (n.d.), de Lima et al. (2024), dos Santos et al. (2023a, 2023b), Guefaifia and Boulaaras (2024), and the references therein. In Alves et al. (2022), the authors consider the sublinear and superlinear cases. First, the authors consider the variational case, namely

ψ \equiv φ

, where questions of existence are treated through minimization and Mountain Pass techniques. In case

η > 0

small, the authors get a positive solution and when

η

is large it is obtained a sign changing solution. In nonvariational case, namely

ψ ≢ φ

, it is used the Minty–Browder’s Theorem for a perturbed problem. In this case we can not guarantee the positiviness of the solution. In Corrêa et al. (2023), the authors studied problem

(P_{1})

for the cases when nonlinearity is sublinear or concave–convex, by using different methods variational and nonvariational. More recently, in dos Santos et al. (2023a), the authors focused their attention on the following elliptic problem with singular terms

{\begin{cases} - div (a (x) \nabla u) + η ψ (x) \int_{Ω} φ u = \frac{λ}{(u^{+})^{α}} + (u^{+})^{p} in Ω, \\ u = 0 on \partial Ω, \end{cases} (P_{0})

where

α \in (0, 1)

and

p \in (0, 2^{*})

. There, the authors proved results of existence and multiplicity of solutions via Galerkin’s Method and an approximation argument (to control the singular term) and also use an appropriate local minimization (via Ekeland Variational Principle) and the Mountain Pass Theorem with an approximation argument. The same authors in dos Santos et al. (2023b), complemented the previously mentioned work by considering the problem

{\begin{cases} - div (a (x) \nabla u) + η φ (x) \int_{Ω} φ u = h (x) u^{- α} + k (x) u^{p} in Ω, \\ u > 0 in Ω and u = 0 on \partial Ω, \end{cases} (P_{0}^{'})

where

p \in (0, 1)

and

α > 1

. In this paper, the authors proved results of existence of solutions combining the variational method and the Nehari Manifold. These works, along with numerous others, demonstrate a rapidly growing interest in problems with these characteristics. Indeed, nonlocal elliptic problems in general have attracted significant attention in recent years, as evidenced by the many publications addressing difficulties similar to those considered here. Notable contributions include works by Alreshidi and Hai (2024), Araújo et al. (2021), Bueno et al. (2022), Buccheri et al. (2021), Cintra et al. (2022), Chipot and Mikayelyan (2022), Corrêa, dos Santos, et al. (n.d.), Dai (2016), dos Reis Costa and da Silva (2024), Duarte and Souto (2019), Duarte et al. (2025), Ferraz and Silva (2025), Ling and Zhao (2022), Molica Bisci et al. (2024), among many others cited in the literature.

For readers new to this field, we recommend the comprehensive treatment in the book by Kavallaris and Takashi (2018) as an excellent starting point for the study of nonlocal problems.

Moreover, such equations arise in various mathematical models stemming from physical and biological phenomena, including superconductivity, plasma reactions, thermal processes, population growth, among others. As noted by Furter and Grinfeld (1989):

In the ecological context, there is no real justification for assuming that the interactions are local. There are many (hypothetical) examples where such an assumption is clearly untenable, such as: (1) a population in which individuals compete for a shared rapidly equilibrated (e.g. by convection) resource; (2) a population in which individuals communicate either visually or by chemical means.

In particular, problem $(P_{1})$ is motivated by systems of nonlinear equations that model the behavior of microsensor thermistors in the presence of a gas. See, for example, Allegretto and Barabanova (1996), Allegretto et al. (1994), Barabanova (1996), and the references therein.

The first class of problems we address is

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η φ (x) \int_{Ω} φ u = (u^{+})^{α}, in Ω, \\ u = 0, on \partial Ω, \end{cases} (Q)

which is the variational counterpart

φ = ψ

of problem

(P)

, with

f (x, u) = (u^{+})^{α}

and

α \in (0, p - 1) \cup (p - 1, p^{*} - 1)

. Suppose that

$(A_{1})$
There exist $a_{0}, a_{\infty} \in (0, \infty)$ such that $0 < a_{0} \leq a (x) \leq a_{\infty} < \infty$ , for all $x \in Ω$ ;
$(H_{1})$
$φ \in C^{0, β} (\bar{Ω}) ∖ {0}$ , $β \in (0, 1)$ , and there exists $v \in W_{0}^{1, p} (Ω) ∖ {0}$ such that $v^{+} ≢ 0$ and $\int_{Ω} φ v d x = 0$ .

Consequently, our results associated with this problem are:
Theorem 1.1
Suppose $(A_{1})$ , $(H_{1})$ , and $α \in (0, p - 1)$ . Then, the problem $(Q)$ has a nontrivial weak solution for $η > 0$ . Moreover, $(Q)$ has a positive solution for $η > 0$ small enough.
Theorem 1.2
Suppose $(A_{1})$ , $(H_{1})$ , and $α \in (p - 1, p^{} - 1)$ . Then, the problem $(Q)$ has a nontrivial weak solution for $η > 0$ . Moreover, $(Q)$ has a positive solution for $η > 0$ small enough.

To obtain sign-changing solutions, we will assume that an additional hypothesis:
$(H_{2})$
There is $b_{0} > 0$ , such that $b_{0} \leq φ (x)$ for all $x \in Ω$ .

Then, we will prove that
Theorem 1.3
Suppose $(A_{1})$ , $(H_{1})$ , $(H_{2})$ , and $α \in (p - 1, p^{} - 1)$ . Then, problem $(Q)$ has a sign-changing weak solution for $η > 0$ large enough.

For nonvariational situations, we will consider the class of sublinear problems
${\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) {(\int_{Ω} φ u^{+})}^{p - 1} = f (x, u) + h (x) in Ω, \\ u = 0 on \partial Ω, \end{cases} (N V)$
where $a, ψ, φ, h : Ω \to R$ are measurable, $φ \geq 0$ , a.e. in $Ω$ , and $f : Ω \times R \to R$ is a continuous function verifying the following assumptions:
$(f_{1})$
$f (\cdot, 0) \neq 0$ ;
$(f_{2})$
$| f (x, t) | \leq C (| t |^{α} + 1)$ , with $α \in (0, 1)$ ;
$(A_{1})$
There are $a_{0}, a_{\infty} \in (0, \infty)$ such that $0 < a_{0} \leq a (x) \leq a_{\infty} < \infty$ , for all $x \in Ω$ ;
$(B_{1})$
$φ, ψ, h \in L^{p / p - 1} (Ω)$ .

The sign assumption on $φ$ guarantees that the term ${(\int_{Ω} φ u^{+})}^{p - 1}$ is well-defined.

The main result for this situation is the following.
Theorem 1.4
Suppose that the hypotheses $(f_{1}) - (f_{2})$ , $(A_{1})$ , $(B_{1})$ , and $φ \geq 0$ a.e. in $Ω$ hold true. Then, there exists $η_{0} > 0$ such that $(N V)$ has a nontrivial weak solution for $η \in [0, η_{0})$ . Furthermore, if we assume the condition
$| f (x, t) - f (x, s) | \leq γ | t - s |, for some γ > 0, (f_{3})$
there exists $γ_{0} > 0$ such that the solution is unique if $γ \in (0, γ_{0}]$ .

Finally, we will use the Galerkin method to ensure that there is a solution to the superlinear problem
${\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) {(\int_{Ω} φ u^{+})}^{p - 1} = h (x) in Ω, \\ u = 0 on \partial Ω, \end{cases}$
where $h, φ, ψ \in L^{p / p - 1} (Ω)$ , $φ \geq 0$ a.e. in $Ω$ .

For this, our approach relies on the following variant of the Brouwer Fixed Point Theorem, which can be found in Lions (1969):
Proposition 1.5
Suppose that $F : R^{m} \to R^{m}$ is a continuous function such that $⟨ F (ξ), ξ ⟩ \geq 0$ on $| ξ | = r$ , where $⟨ \cdot, \cdot ⟩$ is the usual inner product in $R^{m}$ and $| \cdot |$ is its corresponding norm. Then, there exists $z_{0} \in \bar{B_{r} (0)}$ such that $F (z_{0}) = 0$ .
1.1. Organization of the Paper

This paper is structured as follows. Section 2 addresses the variational framework, considering both the sublinear case $α \in (0, p - 1)$ and the superlinear case $α \in (p - 1, p^{*} - 1)$ . For the sublinear regime, we establish Theorem 1.1 via direct minimization, while Theorem 1.2 in the superlinear case is proved using the Mountain Pass Theorem. In Section 3, we turn to nonvariational settings, treating sublinear and superlinear nonlinearities separately. The sublinear case is handled through the Minty–Browder Theorem, whereas the superlinear case relies on Galerkin’s method.

1.2. Some Notations

In this paper, we use the following notations:

For $q \in [1, + \infty)$ , $q^{'}$ denotes the conjugate exponent of $q$ , that is, $q^{'} = q / (q - 1) .$

The usual norm of the Lebesgue spaces $L^{t} (Q)$ for $t \in [1, \infty]$ , will be denoted by $| . |_{t}$ .

$C$ and $C_{i}$ denote (possibly different) positive constants, whose values are not relevant.

If $A \subset R^{N}$ is a measurable set, we denote by $| A |$ its Lebesgue measure.

2. The Variational Case

In this section, we study the case where $f (x, u) = (u^{+})^{α}$ , with $α \in (0, p^{*} - 1)$ . For clarity, the analysis is divided into two subsections, corresponding to distinct ranges of $α$ .

2.1. The Case $α \in (0, p - 1)$

Through this section we will be concerned with the following problem

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η φ (x) \int_{Ω} φ u = (u^{+})^{α}, in Ω, \\ u = 0, on \partial Ω, \end{cases} (Q)

which corresponds to the variational case of problem

(P)

with

φ = ψ

, where

f (x, u) = (u^{+})^{α}

and

α \in (0, p - 1)

. Here, suppose that

(A_{1})

and

(H_{1})

occur.

We have to point out that there are several classes of functions that satisfy assumption $(H_{1})$ . We list below some of them:

Let us suppose that $Ω = (0, π), N = 1$ , $φ (x) = \cos x$ and $v (x) = \sin x$ . Clearly, such functions belong to the respective spaces and $\int_{0}^{π} \cos x \sin x d x = 0$ .

Let $φ \geq 0, φ ≢ 0$ a function whose support sppt $(φ)$ is contained in a closed ball $B \subset Ω$ . Clearly, we may take a function $v \in W_{0}^{1, p} (Ω) ∖ {0}$ such that $v \equiv 0$ in $B$ , $v^{+} ≢ 0$ and $\int_{Ω} φ v d x = 0$ .

To begin with, we consider, for each fixed $η > 0$ , the energy functional $I_{η} : W_{0}^{1, p} (Ω) ⟶ R$ defined by

I_{η} (u) := \frac{1}{p} \int_{Ω} a (x) | \nabla u |^{p} d x + \frac{η}{2} {(\int_{Ω} φ u)}^{2} - \frac{1}{α + 1} \int_{Ω} (u^{+})^{α + 1} d x .

This energy functional is of class $C^{1}$ with

I_{η}^{'} (u) (v) := \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla v d x + η (\int_{Ω} φ u) (\int_{Ω} φ v) - \int_{Ω} (u^{+})^{α} v d x, \forall v \in H_{0}^{1} (Ω) .

Remark 2.1
Thanks to condition $(A_{1})$ , $d μ = a (x) d x$ is a Lebesgue positive measure in $R^{N}$ , thus
$‖ u ‖_{1, a} = {(\int_{Ω} a (x) | \nabla u |^{p} d x)}^{1 / p},$
is an equivalent norm to the usual one of $W_{0}^{1, p} (Ω)$ given by
$| u |_{1, p} = {(\int_{Ω} | \nabla u |^{p} d x)}^{1 / p} .$

To establish Theorem 1.1, we now prove several auxiliary results. The following lemma guarantees the existence of a solution to problem $(Q)$ , while Lemma 2.8 ensures the existence of a strictly positive solution.
Lemma 2.2
$I_{η}$ is coercive.
Proof.
Indeed, observe that
$I_{η} (u) = \frac{1}{p} \int_{Ω} a (x) | \nabla u |^{p} d x + \frac{η}{2} {(\int_{Ω} φ u d x)}^{2} - \frac{1}{α + 1} \int_{Ω} (u^{+})^{α + 1} d x \geq \frac{a_{0}}{p} | u |_{1, p}^{p} - C | u |_{1, p}^{α + 1},$
and so, as $α + 1 < p$ , we have that $I_{η} (u) \to \infty$ , when $| u |_{1, p} \to \infty$ .
Lemma 2.3
$I_{η}$ is bounded from below.
Proof.
First, note that there is $R > 0$ such that $I_{η} (u) \geq 1$ for $| u |_{1, p} \geq R$ . If $| u |_{1, p} \leq R$ , we have
$\begin{aligned} | I_{η} (u) | & \leq \frac{a_{\infty}}{p} | u |_{1, p}^{p} + \frac{η}{2} M_{1} | u |_{1, p}^{2} + \frac{C}{α + 1} | u |_{1, p}^{α + 1}, \\ \leq C_{0} R^{p} + C_{1} η R^{2} + C_{2} R^{α + 1} =: K, \end{aligned}$
thus $I_{η} (u) \geq - K$ , for all $u \in W_{0}^{1, p} (Ω)$ . Hence, there is
$I_{\infty} = inf_{u \in W_{0}^{1, p} (Ω)} I_{η} (u) \in R .$
(2.1)
Lemma 2.4
There exists $u_{η} \in W_{0}^{1, p} (Ω) ∖ {0}$ such that $I_{η} (u_{η}) = I_{\infty}$ .
Proof.
In view of (2.1) there is a minimizing sequence $(u_{n}) \subset W_{0}^{1, p} (Ω)$ with $I_{η} (u_{n}) \to I_{\infty}$ . Since $I_{η}$ is coercive, the sequence $(u_{n})$ is bounded in $W_{0}^{1, p} (Ω)$ . We now observe that, perhaps for a subsequence, $u_{n} ⇀ u_{η}$ in $W_{0}^{1, p} (Ω)$ . Thus
$\underset{n \to \infty}{lim inf} ‖ u_{n} ‖_{1, a}^{p} \geq ‖ u_{η} ‖_{1, a}^{p}, lim_{n \to + \infty} \int_{Ω} φ u_{n} d x = \int_{Ω} φ u_{η} d x, and lim_{n \to + \infty} \int_{Ω} (u_{n}^{+})^{α + 1} d x = \int_{Ω} (u_{η}^{+})^{α + 1} d x,$
consequently,
$I_{\infty} = \underset{n \to + \infty}{lim inf} I_{η} (u_{n}) \geq I_{η} (u_{η}) \geq I_{\infty},$
and, since $I_{\infty}$ is the global minimum of $I_{η}$ in $W_{0}^{1, p} (Ω)$ , we get $I_{η} (u_{η}) = I_{\infty}$ and $I_{η}^{'} (u_{η}) = 0.$ Now, if $t \geq 0$ and considering $v$ as $(H_{1})$ , then $I_{η} (t v) < 0$ for $t \approx 0^{+}$ which implies that $I_{\infty} < 0$ and $u_{η} \neq 0$ .
Lemma 2.5
$u_{η}^{+} \neq 0$ , for all $η > 0$ .
Proof.
Suppose that, there exists $η > 0$ such that $u_{η}^{+} = 0$ , that is $u_{η} \leq 0$ . As $I_{η}^{'} (u_{η}) (u_{η}) = 0$ , it follows
$\int_{Ω} a (x) | \nabla u_{η} |^{p} d x + η {(\int_{Ω} φ u_{η} d x)}^{2} - \int_{Ω} (u_{η}^{+})^{α + 1} d x = 0,$
that is,
$\int_{Ω} a (x) | \nabla u_{η} |^{p} d x + η {(\int_{Ω} φ u_{η} d x)}^{2} = 0.$
Remember that $a > 0$ in $Ω$ and $η > 0$ , its clear that $\int_{Ω} a (x) | \nabla u_{η} |^{p} d x = 0$ , that is we have $u_{η} = 0$ which is a contradiction, because $I_{η} (u_{η}) < 0$ . Therefore, $u_{η}^{+} \neq 0$ for all $η > 0$ .
Lemma 2.6
The family ${u_{η}}_{η > 0}$ , for $η > 0$ small enough, is bounded in $W_{0}^{1, p} (Ω)$ .
Proof.
Note that $I_{η}^{'} (u_{η}) (u_{η}) = 0$ , and we obtain
$a_{0} \int_{Ω} | \nabla u_{η} |^{p} d x \leq \int_{Ω} a (x) | \nabla u_{η} |^{p} d x \leq - η {(\int_{Ω} φ u_{η} d x)}^{2} + \int_{Ω} (u_{η}^{+})^{α + 1} d x,$
(2.2)
implies that
$a_{0} \int_{Ω} | \nabla u_{η} |^{p} d x \leq η {(\int_{Ω} | φ u_{η} |)}^{2} + \int_{Ω} (u_{η}^{+})^{α + 1} d x,$
and, consequently
$| u_{η} |_{1, p}^{p} \leq η C | u_{η} |_{1, p}^{2} + C_{1} | u_{η} |_{1, p}^{α + 1} .$
(2.3)
From the above inequality, and using the fact that $2 < p$ and $α + 1 < p$ , if $η > 0$ is small enough, we have that the family ${u_{η}}_{η > 0}$ is bounded in $W_{0}^{1, p} (Ω)$ .
Lemma 2.7
$u_{η} > 0$ , for $η > 0$ small enough.
Proof.
From Lemma 2.6, as $W_{0}^{1, p} (Ω)$ is reflexive, there is $u_{0} \in W_{0}^{1, p} (Ω)$ such that $u_{η_{i}} ⇀ u_{0}$ in $W_{0}^{1, p} (Ω)$ , where $η_{i} \to 0^{+}$ . Thus,
$\int_{Ω} φ u_{η_{i}} \to \int_{Ω} φ u_{0} and \int_{Ω} a (x) | \nabla u_{η_{i}} |^{p - 2} \nabla u_{η_{i}} \nabla v d x \to \int_{Ω} a (x) | \nabla u_{0} |^{p - 2} \nabla u_{0} \nabla v d x, \forall v \in W_{0}^{1, p} (Ω),$
and, by Lebesgue’s Theorem, we have
$\int_{Ω} (u_{η_{i}}^{+})^{α} v d x \to \int_{Ω} (u_{0}^{+})^{α} v d x, \forall v \in W_{0}^{1, p} (Ω) .$
Hence, as
$\int_{Ω} a (x) | \nabla u_{η_{i}} |^{p - 2} \nabla u_{η_{i}} \nabla v d x + η_{i} \int_{Ω} φ v \int_{Ω} φ u_{η_{i}} = \int_{Ω} (u_{η_{i}}^{+})^{α} v d x, \forall v \in W_{0}^{1, p} (Ω),$
(2.4)
if $η_{i} \to 0$ , when $i \to \infty$ , we get
$\int_{Ω} a (x) | \nabla u_{0} |^{p - 2} \nabla u_{0} \nabla v d x = \int_{Ω} (u_{0}^{+})^{α} v d x, \forall v \in W_{0}^{1, p} (Ω),$
that is, $u_{0} \in W_{0}^{1, p} (Ω)$ is a weak solution for
${\begin{cases} - div (a (x) | \nabla u_{0} |^{p - 2} \nabla u_{0}) = (u_{0}^{+})^{α}, in Ω, \\ u_{0} = 0, on \partial Ω . \end{cases} (Q A)$
Now, observe that, taking $v \in W_{0}^{1, p} (Ω) ∖ {0}$ as in $(H_{1})$ , we have
$I_{η} (s v) = \frac{s^{p}}{p} ‖ v ‖_{1, p}^{p} - \frac{s^{α + 1}}{α + 1} \int_{Ω} (v^{+})^{α + 1} d x, with s > 0,$
since, $\int_{Ω} φ v d x = 0$ . Hence, as $\int_{Ω} (v^{+})^{α + 1} d x > 0$ , we get $s^{} > 0$ (small enough) such that
$I_{η} (s^{} v) < 0.$
On the other hand, as $I_{η} (s^{*} v) = K < 0$ does not depend on $η$ , we obtain $I_{η} (u_{η}) \leq K < 0$ and $I_{η_{i}} (u_{η_{i}}) \to I_{0} (u_{0})$ , that is $I_{0} (u_{0}) \leq K < 0$ which yields $u_{0} \neq 0$ , where $I_{0}$ is the energy functional of problem $(Q A)$ . From the Maximum Principle and invoking problem $(Q A)$ , $u_{0} > 0$ in $Ω$ .

Moreover, from (2.4), if $v = u_{η_{i}}$ we have
$\int_{Ω} a (x) | \nabla u_{η_{i}} |^{p} d x = o_{η_{i}} (1) + \int_{Ω} (u_{η_{i}}^{+})^{α + 1} d x,$
and, for $η_{i} \to 0$ , we get
$\int_{Ω} a (x) | \nabla u_{η_{i}} |^{p} d x \to \int_{Ω} (u_{0}^{+})^{α + 1} d x = \int_{Ω} a (x) | \nabla u_{0} |^{p} d x,$
consequently, $u_{η_{i}} \to u_{0}$ in $W_{0}^{1, p} (Ω)$ .

On the other hand, as $u_{0}$ is a solution of $(Q A)$ , observing that Peral (1997, Theorem E.0.20, Lemma A.08) and Guedda and Veron (1989, Corollary 1.1), we have $u_{0} \in C^{1, γ} (\bar{Ω})$ , for some $γ \in (0, 1)$ , and $\partial u_{0} / \partial ν < 0$ on $\partial Ω$ . Related to this subject, the interested reader may also consult (Diaz & Saa, 1987). Moreover, similarly, as $u_{η}$ is a solution of problem
${\begin{cases} - div (a (x) | \nabla u_{η} |^{p - 2} \nabla u_{η}) = f (x, u_{η}), in Ω, \\ u_{η} = 0, on \partial Ω, \end{cases}$
where
$f (x, u_{η}) = (u_{η}^{+})^{α} - η φ (x) \int_{Ω} φ u_{η},$
and $| f (x, u_{η}) | \leq C + | u_{η} |^{α}$ , we get $(u_{η}) \subset C^{1, γ} (\bar{Ω})$ , for some $γ \in (0, 1)$ , and $\partial u_{η} / \partial ν < 0$ on $\partial Ω$ . Consequently, since $(u_{η})$ is bounded in $C_{0}^{1, γ} (\bar{Ω})$ and $C_{0}^{1, γ} (\bar{Ω}) ↪ C_{0}^{1} (\bar{Ω})$ compactly, we get
$u_{η_{i}} \to u_{0} in C_{0}^{1} (\bar{Ω}) .$
Hence, since $u_{0} >$ in $Ω$ and $\partial u_{0} / \partial ν < 0$ on $\partial Ω$ , we have that $u_{0}$ belongs to the interior of the positive cone of $C_{0}^{1} (\bar{Ω})$ and so $u_{η}$ belongs to this interior, for $η \approx 0^{+}$ . Consequently, for such values of small $η > 0$ the following holds true: $u_{η} > 0$ in $Ω$ and $\partial u_{η} / \partial ν < 0$ on $\partial Ω$ .
Proof Proof of Theorem 1.1.

Consequently, gathering the above results, we have the proof of the theorem.

2.2. The Case $α \in (p - 1, p^{*} - 1)$

We are going to study the problem

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η φ (x) \int_{Ω} φ u = (u^{+})^{α}, in Ω, \\ u = 0, on \partial Ω, \end{cases} (Q)

in the case

α \in (p - 1, p^{*} - 1)

. In order to prove Theorem 1.2, we need the results below:

Lemma 2.8
The functional $I_{η}$ enjoys the Mountain Pass Geometry, that is, $(i)$
There exist $r, ρ > 0$ such that
$I_{η} (u) \geq ρ, \forall u \in W_{0}^{1, p} (Ω) with ‖ u ‖_{1, a} = r .$

$(i i)$
There exists $e \in W_{0}^{1, p} (Ω)$ with $‖ u ‖_{1, a} > r$ , such that $I_{η} (e) < 0$ .

Proof.
Note that, for $u \in W_{0}^{1, p} (Ω)$ ,
$\begin{aligned} I_{η} (u) & \geq \frac{1}{p} ‖ u ‖_{1, a}^{p} - c_{0} \int_{Ω} | u |^{α + 1} d x \\ \geq \frac{1}{p} ‖ u ‖_{1, a}^{p} - c_{1} ‖ u ‖_{1, a}^{α + 1} \\ \geq \frac{1}{p} ‖ u ‖_{1, a}^{p} - c_{1} ‖ u ‖_{1, a}^{α + 1}, \end{aligned}$
consequently $I_{η} (u) \geq I (u)$ , where
$I (u) = \frac{1}{p} ‖ u ‖_{1, a}^{p} - \frac{1}{α + 1} \int_{Ω} (u^{+})^{α + 1} d x .$
Noticing that for some $ρ, r > 0$ , we have that $I_{η} (u) \geq ρ, if ‖ u ‖_{1, a} = r > 0$ , that is, $(i)$ is proved.

Let $v \in W_{0}^{1, p} (Ω)$ as in the assumption $(H_{1})$ , that is, $\int_{Ω} φ v d x = 0$ and $v^{+} ≢ 0$ . For $t > 0$ , we have,
$I_{η} (t v) = \frac{1}{p} t^{p} ‖ v ‖_{1, a}^{p} + \frac{η t^{2}}{2} {(\int_{Ω} φ v)}^{2} - t^{α + 1} \int_{Ω} (v^{+})^{α + 1} d x .$
Hence, because $p < α + 1$ , there is $t_{} > 0$ such that $v_{} = t_{} v \in W_{0}^{1, p} (Ω)$ satisfies $‖ v_{} ‖_{1, a} > r$ and $I_{η} (v_{*}) < 0$ . So $(i i)$ is proved. Therefore, $I_{η}$ verifies the Mountain Pass Geometry, consequently its well-defined the mountain pass level
$c_{η} := inf_{γ \in Γ_{η}} max_{t \in [0, 1]} I_{η} (γ (t)) > 0,$
where $Γ_{η} := {γ \in C ([0, 1], W_{0}^{1, p} (Ω)) : γ (0) = 0 and I_{η} (γ (1)) < 0}$ .

We now remember the following definitions:
Definition 2.1 (Palais-Smale $(P S)$ Sequence)

Let $X$ be a Banach space and $Ψ \in C^{1} (X, R)$ . We say that a sequence $(x_{n}) \subset X$ is a $(P S)$ sequence at the level $c$ , and we write $(P S)_{c}$ sequence, when

Ψ (x_{n}) \to c and Ψ^{'} (x_{n}) \to 0.

(2.5)

Definition 2.2 (

(P S)

Condition)

We say that $Ψ$ verifies the $(P S)$ condition when every $(P S)_{c}$ sequence, for $c \in R$ , possesses a strongly convergent subsequence in $X$ , that is,

Ψ (x_{n}) \to c and Ψ^{'} (x_{n}) \to 0,

(2.6)

implies that there exists

(x_{n_{j}}) \subset (x_{n})

and

x_{0} \in X

such that

x_{n_{j}} \to x_{0} in X .

(2.7)

Lemma 2.9

Let $(w_{n})$ be a $(P S)_{c}$ sequence for $I_{η}$ . Then, $(w_{n})$ is bounded in $W_{0}^{1, p} (Ω)$ .

Proof.

Let $(w_{n}) \subset W_{0}^{1, p} (Ω)$ with

I_{η} (w_{n}) \to c and I_{η}^{'} (w_{n}) \to 0.

Thus

I_{η} (w_{n}) - \frac{1}{α + 1} I_{η}^{'} (w_{n}) (w_{n}) \leq c + 1 + ‖ w_{n} ‖_{1, a} .

On the other hand,

I_{η} (w_{n}) - \frac{1}{α + 1} I_{η}^{'} (w_{n}) (w_{n}) = (\frac{1}{p} - \frac{1}{α + 1}) ‖ w_{n} ‖_{1, a}^{p} + (\frac{1}{2} - \frac{1}{α + 1}) {(\int_{Ω} φ w_{n})}^{2} .

Consequently, and because

α > 1

, we have

c + 1 + ‖ w_{n} ‖_{1, a} \geq (\frac{1}{p} - \frac{1}{α + 1}) ‖ w_{n} ‖_{1, a}^{p} .

Therefore,

(w_{n})

is bounded.

Lemma 2.10

The functional $I_{η}$ verifies the $(P S)$ condition.

Proof.

Let $(w_{n})$ a $(P S)_{c}$ sequence, then

I_{η} (w_{n}) \to c and I_{η}^{'} (w_{n}) \to 0.

(w_{n})

is bounded, for some subsequence of

(w_{n})

, still denoted by itself, there is

w \in W_{0}^{1, p} (Ω)

such that

w_{n} ⇀ w

W_{0}^{1, p} (Ω)

. Thus, as

I_{η}^{'} (w_{n}) \to 0

, we have

I_{η}^{'} (w_{n}) (w_{n} - w) \to 0.

Now, from

w_{n} ⇀ w

W_{0}^{1, p} (Ω)

, we have

w_{n} \to w

L^{t} (Ω)

t \in [1, p^{*})

. In this case, it is easy to see that

\int_{Ω} (w_{n}^{+})^{α} (w_{n} - w) d x, \int_{Ω} φ (w_{n} - w) d x \to 0

(2.8)

and

\int_{Ω} φ w_{n}^{+} d x

is bounded, so

\int_{Ω} a (x) | \nabla w_{n} |^{p - 2} \nabla w_{n} \nabla (w_{n} - w) d x \to 0.

(2.9)

Also, as

w_{n} ⇀ w

W_{0}^{1, p} (Ω)

, by Hölder inequality

\int_{Ω} a (x) | \nabla w |^{p - 2} \nabla w \nabla (w_{n} - w) d x \to 0.

Then

\int_{Ω} a (x) (| \nabla w_{n} |^{p - 2} \nabla w_{n} - | \nabla w |^{p - 2} \nabla w) \nabla (w_{n} - w) d x \to 0.

Consequently, by Tartar’s inequality, see Sakaguchi (1985, Lemma 3.1), we have that

w_{n} \to w

W_{0}^{1, p} (Ω)

Therefore, $I_{η}$ verifies the $(P S)$ condition.

The existence of solution for $(Q)$ follows now from the classical Mountain Pass Theorem. So, for all $η > 0$ , there exists $u_{η} \in W_{0}^{1, p} (Ω)$ such that $I_{η} (u_{η}) = c_{η} > 0$ , where $c_{η}$ is the Mountain Pass Level, and $I_{η}^{'} (u_{η}) = 0$ . The following remark completes the proof of Theorem 1.2

Remark 2.11

The problem $(Q)$ admits a positive weak solution for $η \approx 0^{+}$ . To prove the existence of a positive solution in the superlinear case, we can adapt the arguments from the previous section with the necessary modifications.

2.3. Sign-Changing Solutions

Now, we will prove the Theorem 1.3.

It follows from definition of $c_{η}$ and the calculations made in the first geometry of the Mountain Pass Theorem that $ρ$ does not depend on $η$ , and $c_{η} \geq ρ > 0$ , for all $η > 0.$

Lemma 2.12
The sequence $(c_{η})$ of mountain pass levels is bounded.
Proof.
Observe that, by $(H_{1})$ , there is $v \in W_{0}^{1, p} (Ω) ∖ {0}$ such that $v^{+} ≢ 0$ and $\int_{Ω} φ v d x = 0$ , thus
$c_{η} \leq max_{s \geq 0} I_{η} (s v) \leq max_{s \geq 0} {\frac{s^{p}}{p} | v |_{1, p}^{p} - \frac{s^{α + 1}}{α + 1} \int_{Ω} (v^{+})^{α + 1} d x} = A < \infty .$
Hence, $(c_{η})$ is bounded.
Lemma 2.13
The family $(u_{η})$ is bounded in $W_{0}^{1, p} (Ω)$ .
Proof.
In fact,
$c_{η} = I_{η} (u_{η}) = I_{η} (u_{η}) - \frac{1}{α + 1} I_{η}^{'} (u_{η}) (u_{η}) = (\frac{1}{p} - \frac{1}{α + 1}) ‖ u_{η} ‖_{1, a}^{p} + η (\frac{1}{2} - \frac{1}{α + 1}) {(\int_{Ω} φ u_{η} d x)}^{2},$
and so, as $(c_{η})$ is bounded, we have that $(u_{η})$ is bounded in $W_{0}^{1, p} (Ω)$ .
Proof Proof of Theorem 1.3.

We will split the proof in two steps.

Step 1: $(u_{η})$ is either nonnegative or sign-changing if $η > 0$ is large enough.

Let us suppose, by contradiction, that there exists $η_{j} \to + \infty$ such that $u_{η_{j}} \leq 0$ in $Ω$ . Then,

I_{η_{j}}^{'} (u_{η_{j}}) (u_{η_{j}}) = 0,

if, and only if,

‖ u_{η_{j}} ‖_{1, a}^{p} + η {(\int_{Ω} φ u_{η_{j}})}^{2} = \int_{Ω} (u_{η_{j}}^{+})^{α} u_{η_{j}} d x = 0,

hence

u_{η_{j}} = 0

, which is a contradiction.

Step 2: $(u_{η})$ is sign-changing for $η > 0$ large enough.

Let us suppose, on the contrary, that there exists $η_{j} \to + \infty$ such that $u_{η_{j}} \geq 0$ in $Ω$ , for all $j \in N$ . We are going to prove that there exists $δ > 0$ such that

\int_{Ω} u_{η_{j}} d x \geq δ > 0,

(2.10)

j

is sufficiently large. On the contrary,

\int_{Ω} u_{η_{j}} d x \to 0,

that is

u_{η_{j}} \to 0

L^{1} (Ω)

. As

(u_{η})

is bounded in

W_{0}^{1, p} (Ω)

, we have

u_{η_{j}} ⇀ 0

W_{0}^{1, p} (Ω)

, thus

u_{η_{j}} \to 0

L^{α + 1} (Ω)

. However,

0 \leq ρ \leq c_{η_{j}} = I_{η_{j}} (u_{η_{j}}) - \frac{1}{p} I_{η_{j}}^{'} (u_{η_{j}}) (u_{η_{j}}) = η_{j} (\frac{1}{2} - \frac{1}{p}) {(\int_{Ω} φ u_{η_{j}})}^{2} + (\frac{1}{p} - \frac{1}{α + 1}) \int_{Ω} (u_{η_{j}}^{+})^{α + 1} d x .

By Lemma 2.12, we can suppose, without loss fo generality, that

I_{η_{j}} (u_{η_{j}}) = c_{η_{j}} \to c \geq ρ > 0

. Then, as

u_{n_{j}} \to 0

L^{α + 1} (Ω)

, we have

η_{j} {(\int_{Ω} φ u_{η_{j}})}^{2} \to \frac{2 p c}{p - 2} .

Making

j \to \infty

I_{η_{j}} (u_{η_{j}}) := \frac{1}{p} \int_{Ω} a (x) | \nabla u_{n_{j}} |^{p} d x + \frac{η_{j}}{2} {(\int_{Ω} φ u_{η_{j}} d x)}^{2} - \frac{1}{α + 1} \int_{Ω} (u_{η_{j}}^{+})^{α + 1} d x,

we obtain

lim_{j \to \infty} \frac{1}{p} \int_{Ω} a (x) | \nabla u_{n_{j}} |^{p} d x = c - \frac{p c}{p - 2} < 0,

and, clearly we have a contradiction. Hence, (2.10) is true. On the other hand, as

I_{η_{j}}^{'} (u_{η_{j}}) (u_{η_{j}}) = 0

, we get

\int_{Ω} a (x) | \nabla u_{η_{j}} |^{p} d x + η_{j} {(\int_{Ω} φ u_{η_{j}})}^{2} = \int_{Ω} (u_{η_{j}}^{+})^{α + 1} d x \leq M < \infty,

(2.11)

and

η_{j} {(\int_{Ω} φ u_{η_{j}})}^{2} \leq \int_{Ω} (u_{η_{j}}^{+})^{α + 1} d x \leq M < \infty .

(H_{2})

and (2.10),

η_{j} b_{0}^{2} δ^{2} \leq \int_{Ω} (u_{η_{j}}^{+})^{α + 1} d x \leq M < \infty .

That is a contradiction. Therefore we complete the proof of theorem.

3. The Nonvariational Case

In this final section, we introduce two classes of nonvariational problems, sublinear and superlinear, associated with the operator previously discussed.

3.1. The Sublinear Case

In this subsection, we are interested in studying the general nonvariational problem

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) {(\int_{Ω} φ u^{+} d y)}^{p - 1} = f (x, u) + h (x) in Ω, \\ u = 0 on \partial Ω, \end{cases} (N V)

where

a, ψ, φ, h : Ω \to R

are mensurable, with

φ \geq 0

a.e. in

Ω

, and

f : Ω \times R \to R

is a continuous function verifying the assumptions

(f_{1}) - (f_{2})

(A_{1})

and

(B_{1})

We are going to prove Theorem 1.4.

Proof.
Initially, we consider the operator $B : W_{0}^{1, p} (Ω) \to (W_{0}^{1, p} (Ω))^{'}$ , given by
$⟨ B (u), v ⟩ = \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla v d x + η {(\int_{Ω} φ u^{+} d y)}^{p - 1} (\int_{Ω} ψ v d x) - \int_{Ω} f (x, u) v d x - \int_{Ω} h (x) v d x,$
for all $u, v \in W_{0}^{1, p} (Ω)$ .

The proof consists in using the Minty–Browder’s Theorem to guarantee the existence of a function $u \in W_{0}^{1, p} (Ω)$ such that $B (u) = 0$ . For that, we are going to prove some properties of operator $B$ .

Step 1: $B$ is coercive, that is,
$\frac{⟨ B (u), u ⟩}{| u |_{1, p}} \to \infty$
when $| u |_{1, p} \to \infty$ . Indeed, we have
$⟨ B (u), u ⟩ = \int_{Ω} a (x) | \nabla u |^{p} d x + η {(\int_{Ω} φ u^{+} d y)}^{p - 1} (\int_{Ω} ψ u d x) - \int_{Ω} f (x, u) u d x - \int_{Ω} h (x) u d x,$
and so, as
$\begin{aligned} \int_{Ω} a (x) | \nabla u |^{p} d x \geq a_{0} | u |_{1, p}^{p}, {(\int_{Ω} φ u^{+} d y)}^{p - 1} \leq | φ |_{p / (p - 1)}^{p - 1} | u |_{p}^{p - 1}, \int_{Ω} ψ (\pm u) d x \leq | ψ |_{p / (p - 1)} | u |_{p}, \end{aligned}$

$\begin{aligned} \int_{Ω} f (x, u) u d x \leq C (| u |_{α + 1}^{α + 1} + | u |_{1}) and \int_{Ω} h (x) u d x \leq | h |_{p / (p - 1)} | u |_{p}, \end{aligned}$
we get
$⟨ B (u), u ⟩ \geq a_{0} | u |_{1, p}^{p} - η C | φ |_{p / (p - 1)}^{p - 1} | ψ |_{p / (p - 1)} | u |_{1, p}^{p} - C | u |_{1, p}^{α + 1} - C | u |_{1, p} .$
Fixing $η^{} > 0$ such that
$0 < η C | φ |_{p / (p - 1)}^{p - 1} | ψ |_{p / (p - 1)} < \frac{a_{0}}{2}, \forall η \in (0, η^{}),$
we have that
$⟨ B (u), u ⟩ \geq \frac{a_{0}}{2} | u |_{1, p}^{p} - C | u |_{1, p}^{α + 1} - C | u |_{1, p}, \forall u \in W_{0}^{1, p} (Ω), \forall η \in (0, η^{}),$
(3.1)
consequently,
$\frac{⟨ B (u), u ⟩}{| u |_{1, p}} \geq \frac{a_{0}}{2} | u |_{1, p}^{p - 1} - C | u |_{1, p}^{α} - C \to \infty, when | u |_{1, p} \to \infty,$
that is, $B$ is coercive.

Step 2: $B$ is continuous.

Let $(u_{n}) \subset W_{0}^{1, p} (Ω)$ and $u \in W_{0}^{1, p} (Ω)$ , with $u_{n} \to u$ in $W_{0}^{1, p} (Ω)$ . We have that, for all $v \in W_{0}^{1, p} (Ω)$
$\begin{aligned} ⟨ B (u_{n}) - B (u), v ⟩ = & \int_{Ω} a (x) [| \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u] \nabla v d x \\ + η (\int_{Ω} ψ v d x) [{(\int_{Ω} φ u_{n}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u^{+} d y)}^{p - 1}] \\ - \int_{Ω} [f (x, u_{n}) - f (x, u)] v d x, \end{aligned}$
and, observing that $| | \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u | \leq C | \nabla u_{n} - \nabla u | (| \nabla u_{n} | + | \nabla u |)^{p - 2}$ , we get
$\int_{Ω} a (x) [| \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u] \nabla v d x \leq C | u_{n} - u |_{1, p} | v |_{1, p},$
(3.2)
moreover, it is true that
$\begin{aligned} \int_{Ω} | f (x, u_{n}) - f (x, u) | v d x \leq C | f (\cdot, u_{n}) - f (\cdot, u) |_{\frac{α + 1}{α}} | v |_{1, p}, \end{aligned}$
(3.3)

$\begin{aligned} | \int_{Ω} ψ v d x | \leq C | ψ |_{\frac{p}{p - 1}} | v |_{1, p}, \end{aligned}$
(3.4)
and
$\int_{Ω} φ u_{n}^{+} d y \to \int_{Ω} φ u^{+} d y,$
(3.5)
consequently, from (3.2) to (3.5)
$\begin{aligned} | ⟨ B (u_{n}) - B (u), v ⟩ | & \leq C | u_{n} - u |_{1, p} | v |_{1, p} \\ + C η | ψ |_{\frac{p}{p - 1}} [{(\int_{Ω} φ u_{n}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u^{+} d y)}^{p - 1}] | v |_{1, p} \\ + C | f (\cdot, u_{n}) - f (\cdot, u) |_{\frac{α + 1}{α}} | v |_{1, p}, \end{aligned}$
and
$‖ B (u_{n}) - B (u) ‖_{} \leq C | u_{n} - u |_{1, p} + C η | ψ |_{\frac{p}{p - 1}} [{(\int_{Ω} φ u_{n}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u^{+} d y)}^{p - 1}] + C | f (\cdot, u_{n}) - f (\cdot, u) |_{\frac{α + 1}{α}},$
where $‖ \cdot ‖_{*}$ is the usual norm in $(W_{0}^{1, p} (Ω))^{'}$ . Hence, $B (u_{n}) \to B (u)$ in $(W_{0}^{1, p} (Ω))^{'}$ .

Step 3: $B$ is bounded, that is, $B$ takes bounded sets of $W_{0}^{1, p} (Ω)$ onto bounded sets of $(W_{0}^{1, p} (Ω))^{'}$ .

The proof is analogous to the previous step.

Step 4: $B$ is pseudo-monotone, that is, if $u_{n} ⇀ u$ in $W_{0}^{1, p} (Ω)$ and
$\underset{n \to \infty}{lim sup} ⟨ B (u_{n}), u_{n} - u ⟩ \leq 0,$
then
$\underset{n \to \infty}{lim inf} ⟨ B (u_{n}), u_{n} - u ⟩ \geq ⟨ B (u), u - v ⟩,$
for all $v \in W_{0}^{1, p} (Ω)$ .

In fact, since $u_{n} ⇀ u$ in $W_{0}^{1, p} (Ω)$
$\begin{aligned} ⟨ B (u_{n}), u_{n} - u ⟩ = & \int_{Ω} a (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \nabla (u_{n} - u) d x \\ + η {(\int_{Ω} φ u_{n}^{+} d y)}^{p - 1} (\int_{Ω} ψ (x) (u_{n} - u) d x) \\ - \int_{Ω} f (x, u_{n}) (u_{n} - u) d x \\ - \int_{Ω} h (x) (u_{n} - u) d x, \end{aligned}$
as, its true that
$\begin{aligned} \int_{Ω} φ u_{n}^{+} d y \leq | φ |_{\frac{p}{p - 1}} | u_{n} |_{p} \leq C < \infty, \end{aligned}$
(3.6)

$\begin{aligned} \int_{Ω} ψ (x) (u_{n} - u) d y \leq C | ψ |_{\frac{p}{p - 1}} | u_{n} - u |_{p} \to 0, \end{aligned}$
(3.7)

$\begin{aligned} \int_{Ω} f (x, u_{n}) (u_{n} - u) d x \leq | f (\cdot, u_{n}) |_{\frac{p}{p - 1}} | u_{n} - u |_{p} \to 0, \end{aligned}$
(3.8)
and
$\int_{Ω} h (x) (u_{n} - u) d x \leq | h |_{\frac{p}{p - 1}} | u_{n} - u |_{p} \to 0,$
(3.9)
we have that, from (3.6) to (3.9)
$⟨ B (u_{n}), u_{n} - u ⟩ = \int_{Ω} a (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \nabla (u_{n} - u) d x + o_{n} (1) .$
(3.10)
Since that $\int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla (u_{n} - u) d x = o_{n} (1)$ , we get
$\begin{aligned} ⟨ B (u_{n}), u_{n} - u ⟩ & = \int_{Ω} a (x) | \nabla u_{n} |^{p - 2} \nabla u_{n} \nabla (u_{n} - u) d x - \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla (u_{n} - u) d x + o_{n} (1) \\ = \int_{Ω} a (x) (| \nabla u_{n} |^{p - 2} \nabla u_{n} - | \nabla u |^{p - 2} \nabla u, \nabla (u_{n} - u)) d x + o_{n} (1) \\ \geq C \int_{Ω} | \nabla (u_{n} - u) |^{p} d x + o_{n} (1) \\ = C_{0} | u_{n} - u |_{1, p}^{p} + o_{n} (1), \end{aligned}$
and so,
$0 \leq C_{0} \underset{n \to \infty}{lim inf} | u_{n} - u |_{1, p}^{p} \leq C_{0} \underset{n \to \infty}{lim sup} | u_{n} - u |_{1, p}^{p} \leq \underset{n \to \infty}{lim sup} ⟨ B (u_{n}), u_{n} - u ⟩ \leq 0.$
(3.11)
Hence, $u_{n} \to u$ in $W_{0}^{1, p} (Ω)$ and the pseudo-monotonicity follows now from the continuity of the operator $B$ . The existence follows now from the previous steps and the Minty–Browder’s Theorem.

Now, to prove the uniqueness, suppose that there exist weak solutions $u_{1}$ and $u_{2}$ , with $u_{1} \neq u_{2}$ , for $(N V)$ . Choosing $u_{1} - u_{2}$ as a test function and subtracting both identities, we have
$\begin{aligned} 0 = & \int_{Ω} a (x) [| \nabla u_{1} |^{p - 2} \nabla u_{1} - | \nabla u_{2} |^{p - 2} \nabla u_{2}] \nabla (u_{1} - u_{2}) d x \\ + η \int_{Ω} ψ (x) (u_{1} - u_{2}) d x [{(\int_{Ω} φ u_{1}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u_{2}^{+} d y)}^{p - 1}] \\ - \int_{Ω} [f (x, u_{1}) - f (x, u_{2})] (u_{1} - u_{2}) d x, \end{aligned}$
thus, as
$\int_{Ω} a (x) [| \nabla u_{1} |^{p - 2} \nabla u_{1} - | \nabla u_{2} |^{p - 2} \nabla u_{2}] \nabla (u_{1} - u_{2}) d x \geq C_{0} \int_{Ω} | \nabla (u_{1} - u_{2}) |^{p} d x$
we get
$\begin{aligned} C_{0} \int_{Ω} | \nabla (u_{1} - u_{2}) |^{p} d x & \leq η \int_{Ω} ψ (x) (u_{1} - u_{2}) d x [{(\int_{Ω} φ u_{1}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u_{2}^{+} d y)}^{p - 1}] \\ + \int_{Ω} [f (x, u_{1}) - f (x, u_{2})] (u_{1} - u_{2}) d x \end{aligned}$
and, by $(f_{3})$ , we obtain
$C_{0} | u_{1} - u_{2} |_{1, p}^{p} \leq η | ψ |_{\frac{p}{p - 1}} | u_{1} - u_{2} |_{p} | {(\int_{Ω} φ u_{1}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u_{2}^{+} d y)}^{p - 1} | + C γ | u_{1} - u_{2} |_{p}^{2} .$
Observe that, if $η \approx 0^{+}$
$C_{1} = C_{0} | u_{1} - u_{2} |_{1, p}^{p} - η | ψ |_{\frac{p}{p - 1}} | u_{1} - u_{2} |_{p} | {(\int_{Ω} φ u_{1}^{+} d y)}^{p - 1} - {(\int_{Ω} φ u_{2}^{+} d y)}^{p - 1} | > 0,$
and so, $C_{1} - C γ | u_{1} - u_{2} |_{p}^{2} \leq 0$ . Moreover, since $| u_{1} - u_{2} |_{p} > 0$ and $C, C_{1} > 0$ , there is $γ_{0} \approx 0^{+}$ such that
$0 < C_{1} - C γ | u_{1} - u_{2} |_{p}^{2} \leq 0, \forall γ \in (0, γ_{0}] .$
That is contradiction. Therefore, the uniqueness result follows.
3.2. The Superlinear Case

We will use the Galerkin method to ensure there is a weak solution to the problem

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) {(\int_{Ω} φ u^{+} d y)}^{p - 1} = h (x) in Ω, \\ u = 0 on \partial Ω, \end{cases}

where

h, φ, ψ \in L^{p / p - 1} (Ω)

For this, our approach relies on the Brouwer Fixed Point Theorem (Proposition 1.5).

Let $B = {e_{1}, \dots, e_{m}, \dots}$ be a Schauder basis of $W_{0}^{1, p} (Ω)$ . For each $m \in N$ , let

V_{m} = span {e_{1}, \dots, e_{m}} \subset W_{0}^{1, p} (Ω) .

And so, each

u \in V_{m}

is represented solely by

u = \sum_{j = 1}^{m} ξ_{j} e_{j} .

Now, we define, in

V_{m}

, a norm

‖ u ‖_{m} = \sum_{j = 1}^{m} | ξ_{j} | .

Observe that

(V_{m}, ‖ \cdot ‖_{m})

and

(R^{m}, ‖ \cdot ‖_{s})

are isometrically isomorphic finite-dimensional vector spaces, by application

\begin{array}{cccc} T : & (V_{m}, ‖ \cdot ‖_{m}) & \to & (R^{m}, ‖ \cdot ‖_{s}), \\ u = \sum_{j = 1}^{m} ξ_{j} e_{j} & \mapsto & T (u) = ξ = (ξ_{1}, \dots, ξ_{m}), \end{array}

where

‖ ξ ‖_{s} = \sum_{j = 1}^{m} | ξ_{j} |

, we can observe that

‖ u ‖_{m} = ‖ ξ ‖_{s} = ‖ T (u) ‖_{s}

. We will identify

u \in V_{m}

with

ξ \in R^{m}

via isometrics

T

Since, for each $m$ , $V_{m}$ is a finite dimension space, and so the norms $‖ \cdot ‖_{m}$ and $| \cdot |_{1, p}$ in $V_{m}$ are equivalents. Consequently, there exist $C (m), K (m) > 0$ such that

C (m) ‖ u ‖_{m} \leq | u |_{1, p} \leq K (m) ‖ u ‖_{m}, \forall u \in V_{m} .

Now, we will consider, for each

m \in N

, the application

\begin{array}{cccc} F : & R^{m} & \to & R^{m} \\ ξ & \mapsto & F (ξ) = (F_{1} (ξ), \dots, F_{m} (ξ)), \end{array}

where

F_{j} (ξ) = \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla e_{j} d x + η \int_{Ω} ψ e_{j} d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} - \int_{Ω} h (x) e_{j} d x,

and we have identified

u

and

ξ

by application of

T

, where

T (u) = ξ

. See that,

F_{j} (ξ) ξ_{j} = \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla (ξ_{j} e_{j}) d x + η \int_{Ω} ψ (ξ_{j} e_{j}) d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} - \int_{Ω} h (x) (ξ_{j} e_{j}) d x,

and

\begin{aligned} ⟨ F (ξ), ξ ⟩ = & \sum_{j = 1}^{m} F_{j} (ξ) ξ_{j}, \\ = & \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla (\sum_{j = 1}^{m} ξ_{j} e_{j}) d x + η \int_{Ω} ψ (\sum_{j = 1}^{m} ξ_{j} e_{j}) d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} \\ - \int_{Ω} h (x) (\sum_{j = 1}^{m} ξ_{j} e_{j}) d x, \\ = & \int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla u d x + η \int_{Ω} ψ u d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} - \int_{Ω} h (x) u d x, \end{aligned}

remembering that

u = \sum_{j = 1}^{m} ξ_{j} e_{j}

. Hence,

⟨ F (ξ), ξ ⟩ = ‖ u ‖_{1, a}^{p} + η \int_{Ω} ψ u d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} - \int_{Ω} h (x) u d x .

On the other hand, see that

- \int_{Ω} ψ u d x {(\int_{Ω} φ u^{+} d y)}^{p - 1} \leq | ψ |_{\frac{p}{p - 1}} | φ |_{\frac{p}{p - 1}} | u |_{p}^{p} \leq C | ψ |_{\frac{p}{p - 1}} | φ |_{\frac{p}{p - 1}} ‖ u ‖_{1, a}^{p},

thus,

⟨ F (ξ), ξ ⟩ \geq ‖ u ‖_{1, a}^{p} - η C | ψ |_{\frac{p}{p - 1}} | φ |_{\frac{p}{p - 1}} ‖ u ‖_{1, a}^{p} - C | h |_{\frac{p}{p - 1}} ‖ u ‖_{1, a},

and

⟨ F (ξ), ξ ⟩ \geq (1 - η C | ψ |_{\frac{p}{p - 1}} | φ |_{\frac{p}{p - 1}}) ‖ u ‖_{1, a}^{p} - C | h |_{\frac{p}{p - 1}} ‖ u ‖_{1, a},

thus, if

η \approx 0^{+}

⟨ F (ξ), ξ ⟩ > > 0, if ‖ ξ ‖_{s} \approx + \infty,

or, by equivalent norms

⟨ F (ξ), ξ ⟩ > > 0, if ‖ ξ ‖_{1, a} \approx + \infty .

Therefore, from Brower’s fixed point Theorem, there is

u_{m} \in V_{m}

such that

F (u_{m}) = 0

, that is,

F_{j} (u_{m}) = 0

, for

j = 1, \dots, m

, where

T (u_{m}) = ξ^{m} = (ξ_{1}, \dots, ξ_{m})

, and so

\int_{Ω} a (x) | \nabla u_{m} |^{p - 2} \nabla u_{m} \nabla e_{j} d x + η \int_{Ω} ψ e_{j} d x {(\int_{Ω} φ u_{m}^{+} d y)}^{p - 1} - \int_{Ω} h (x) e_{j} d x = 0, j = 1, \dots, m .

e_{j} \in V_{m}

j = 1, \dots, m

u_{m} \in V_{m}

, we have that

\int_{Ω} a (x) | \nabla u_{m} |^{p} d x + η \int_{Ω} ψ u_{m} d x {(\int_{Ω} φ u_{m}^{+} d y)}^{p - 1} = \int_{Ω} h (x) u_{m} d x,

and

\int_{Ω} a (x) | \nabla u_{m} |^{p} d x = - η \int_{Ω} ψ u_{m} d x {(\int_{Ω} φ u_{m}^{+} d y)}^{p - 1} + \int_{Ω} h (x) u_{m} d x,

that is, we have that

‖ u_{m} ‖_{1, a}^{p} \leq C η ‖ u_{m} ‖_{1, a}^{p} + C ‖ u_{m} ‖_{1, a} \Rightarrow ‖ u_{m} ‖_{1, a} \leq C,

because

p > 2

and

η \approx 0^{+}

. As a consequence,

u_{m} ⇀ u in W_{0}^{1, p} (Ω),

and

u_{m} \to u in L^{q} (Ω), for 1 \leq q < p^{*} .

l \leq m

, we have

V_{l} \subseteq V_{m}

. Consider

σ \in V_{l}

\int_{Ω} a (x) | \nabla u_{m} |^{p - 2} \nabla u_{m} \nabla σ d x + η \int_{Ω} ψ σ d x {(\int_{Ω} φ u_{m}^{+} d y)}^{p - 1} = \int_{Ω} h (x) σ d x,

is hold for all

σ \in V_{l}

. As

l \leq m

, we can do

m \to \infty

and, consequently

\begin{aligned} u_{m} ⇀ u in W_{0}^{1, p} (Ω), \end{aligned}

\begin{aligned} u_{m} \to u in L^{q} (Ω), for 1 \leq q < p^{*}, \end{aligned}

\begin{aligned} u_{m} (x) \to u (x) a.e. in Ω, \end{aligned}

hence

\int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla σ d x + η \int_{Ω} ψ σ d x {(\int_{Ω} φ σ^{+} d y)}^{p - 1} = \int_{Ω} h (x) σ d x, \forall σ \in V_{l} .

l

is arbitrary, we have

\int_{Ω} a (x) | \nabla u |^{p - 2} \nabla u \nabla σ d x + η \int_{Ω} ψ σ d x {(\int_{Ω} φ σ^{+} d y)}^{p - 1} = \int_{Ω} h (x) σ d x, \forall σ \in W_{0}^{1, p} (Ω),

that is,

u

is weak solution for the problem.

4. Open Problems

In what follows we will list some problems that may be studied.

In Allegretto and Barabanova (1996) and Barabanova (1996), the authors study the linear eigenvalue problem

{\begin{cases} - div (a (x) \nabla u) + η ψ (x) \int_{Ω} φ u = λ u, in Ω, \\ u = 0, on \partial Ω . \end{cases} (L E V)

We believe that it is an interesting question to analyze the following nonlinear eigenvalue problem

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u = λ | u |^{p - 2} u, in Ω, \\ u = 0, on \partial Ω, \end{cases} (N E V)

We have studied problem $(Q)$ , variational formulation, in the subcrtical case: $α \in (0, p - 1) \cup (p - 1, p * - 1)$ . It would be interesting to consider the critical case $α = p * - 1$ . Perhaps, we may use Moser iteration technique in order to obtain a priori bounds.

Another interesting question is related with nonlinear Neumann boundary condition. Perhaps it is possible to consider problems like

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u) & in & Ω, \\ u > 0 & in & Ω, \\ | \nabla u |^{p - 2} \frac{\partial u}{\partial η} = \int_{\partial Ω} g (u) d σ & on & \partial Ω, \end{cases}

(4.1)

and

{\begin{cases} - div (a (x) | \nabla u |^{p - 2} \nabla u) + η ψ (x) \int_{Ω} φ u = f (x, u) & in & Ω, \\ u > 0 & in & Ω, \\ | \nabla u |^{p - 2} \frac{\partial u}{\partial η} = \int_{Ω} h (u) d x & on & \partial Ω, \end{cases}

(4.2)

We believe even in the case $p = 2$ , these problems have not been studied in great generality.

Footnotes

Acknowledgments

The authors would like to express our gratitude to an anonymous reviewer for his suggestions that improved this paper.

ORCID iDs

Francisco JSA Corrêa

Natan de Assis Lima

Ronaldo Duarte

Romildo Lima

Author Contributions

The authors wrote the article together.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: R. Lima was partially supported by CNPq/Brazil 306.411/2022-9.

Declaration of Conflicting Interests

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

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Existence Results for a Class of Nonlocal p -Laplacian Equations

Abstract

Keywords

1. Introduction and Main Results

1.2. Some Notations

2. The Variational Case

2.1. The Case α ∈ ( 0 , p − 1 )

2.2. The Case α ∈ ( p − 1 , p * − 1 )

3.1. The Sublinear Case

4. Open Problems

Footnotes

Acknowledgments

ORCID iDs

Author Contributions

Funding

Declaration of Conflicting Interests

References

2.1. The Case $α \in (0, p - 1)$

2.2. The Case $α \in (p - 1, p^{*} - 1)$