Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity

Abstract

In this paper, we consider the following Kirchhoff-type diffusion problem involving the fractional Laplacian and logarithmic nonlinearity at high initial energy level: $\begin{array}{l} \{\begin{matrix} u_{t} + {[u]}_{s}^{2 (θ - 1)} {(- Δ)}^{s} u = | u |^{q - 2} u ln | u | & (x, t) \in Ω \times R^{+}, \\ u (x, t) = 0 & (x, t) \in (R^{N} ∖ Ω) \times R^{+}, \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}$ where ${(- Δ)}^{s}$ is the fractional Laplacian with $s \in (0, 1), N > 2 s$ , ${[u]}_{s}$ is the Gagliardo seminorm of u, $Ω \subset R^{N}$ is a bounded domain with Lipschitz boundary, $1 ⩽ θ < N / (N - 2 s)$ , $2 θ < q < 2_{s}^{*}$ . Based on the potential well theory, a sufficient condition is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time under some appropriate assumptions. In particular, the existence of ground state solutions for the above stationary problem is obtained by restricting the related discussion on Nehari manifold.

Keywords

Fractional Kirchhoff problems logarithmic nonlinearity global existence blow up

1. Introduction

This paper deals with the global existence and finite time blow-up of solutions for the following initial boundary value problem involving the fractional Laplacian and logarithmic nonlinearity: $\begin{array}{l} (1.1) & \{\begin{matrix} u_{t} + {[u]}_{s}^{2 (θ - 1)} {(- Δ)}^{s} u = | u |^{q - 2} u ln | u | & (x, t) \in Ω \times R^{+}, \\ u (x, t) = 0 & (x, t) \in (R^{N} ∖ Ω) \times R^{+}, \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}$ where $s \in (0, 1), N > 2 s$ , Ω is a bounded domain in $R^{N} (N ⩾ 2)$ with Lipschitz boundary, $2 θ < q < 2_{s}^{*}$ , ${[u]}_{s}$ is the Gagliardo seminorm of u defined by $\begin{array}{l} {[u]}_{s} = {(\iint_{R^{2 N}} \frac{| u (x, t) - u (y, t) |^{2}}{| x - y |^{N + 2 s}} d x d y)}^{1 / 2}, \end{array}$ ${(- Δ)}^{s}$ is the fractional Laplacian which, up to a normalization constant, is defined for any $x \in R^{N}$ as $\begin{matrix} {(- Δ)}^{s} φ (x) = 2 lim_{δ \to 0} \int_{R^{N} ∖ B_{δ} (x)} \frac{φ (x) - φ (y)}{| x - y |^{N + 2 s}} d y \end{matrix}$ for any $φ \in C_{0}^{\infty} (R^{N})$ . Here $B_{δ} (x)$ denotes the ball in $R^{N}$ centered at x with radius δ. For basic properties of the fractional Laplacian, we refer the readers to [6] for a simple introduction. Actually, fractional Laplace equations and related nonlocal operator equations have an increasingly wide utilization in many important fields, as explained by Caffarelli in [3], Laskin in [11] and Vázquez in [25]. The literature on elliptic type problems involving fractional Laplacian and its variant is very rich and vast, see for example [6,16,17,24] and the references therein.

It is worthy pointing out that a stationary Kirchhoff-type fractional Laplacian model was proposed by Fiscella and Valdinoci in [8] to study the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. Indeed, the stationary problem of (1.1) is a fractional version of a model, the so-called stationary Kirchhoff equation, which was introduced by Kirchhoff in [10] as a model to study elastic string vibrations. It is called that problem (1.1) is non-degenerate if the coefficient of ${(- Δ)}^{s}$ is positive on every point. If there exists a point such that the coefficient of ${(- Δ)}^{s}$ is zero on this point, then it is named that problem (1.1) is degenerate. In recent years, there are a lot of results concerning the Kirchhoff-type fractional Laplacian problems in the non-degenerate and degenerate case, see, for instance, [13–15,18–20,28,29].

Problem (1.1) is a class of nonlocal fractional diffusion problem, which is related to the anomalous diffusion theory. More precisely, as stated in [7], the function $u (x, t)$ may be used to denote the density of population at the point x and time t, the term ${(- Δ)}^{s} u$ represents the diffusion of density, $| u |^{q - 2} ln | u |$ is the external source. The coefficient $M ({[u]}_{s}^{2})$ denotes the possible changes of total population in the whole space. This implies that the behavior of individuals is subject to total population, such as the diffusion process of bacteria, see for example [12,21].

Recently, the fractional diffusion problems with power type nonlinearities have been studied by many researchers. For example, Pucci et al. [21] studied the following initial boundary value problem: $\begin{array}{l} (1.2) & u_{t} + M ({[u]}_{s, p}^{p}) {(- Δ)}_{p}^{s} u = f (x, t) in Ω \times R^{+}, \end{array}$ where the Kirchhoff term $M : R_{0}^{+} \to R^{+}$ is a continuous and nondecreasing function, and ${[u]}_{s, p}$ is the Gagliardo seminorm of u defined by $\begin{array}{l} {[u]}_{s, p} = {(\iint_{R^{2 N}} \frac{| u (x, t) - u (y, t) |^{p}}{| x - y |^{N + p s}} d x d y)}^{1 / p} . \end{array}$ Under some suitable conditions, the well-posedness of solutions for (1.2) was obtained by applying the subdifferential approach. Moreover, the authors obtained the large-time behavior and extinction of solutions. Pan et al. [20] considered the following initial boundary value problem: $\begin{array}{l} u_{t} + {[u]}_{s, p}^{(λ - 1) p} {(- Δ)}_{p}^{s} u = | u |^{q - 2} u in Ω \times R^{+}, \end{array}$ where $1 ⩽ λ < N / (N - s p)$ and $p < q < N p / (N - s p)$ with $1 < p < N / s$ . They obtained the existence of a global solution by the Galerkin method combined with the potential well theory. Mingqi et al. [12] discussed the following initial boundary value problem: $\begin{array}{l} u_{t} + M ({[u]}_{s}^{2}) {(- Δ)}^{s} u = | u |^{p - 2} u in Ω \times R^{+}, \end{array}$ where $M : R_{0}^{+} \to R^{+}$ is a continuous function satisfying

there exists $m_{0} > 0$ and $θ > 1$ such that $M (σ) ⩾ m_{0} σ^{θ - 1}$ for all $σ ⩾ 0$ ;

there exists $θ \in [1, N / N - 2 s)$ such that $\begin{array}{l} θ M (t) : = θ \int_{0}^{t} M (τ) d τ ⩾ M (t) t, \forall t ⩾ 0 . \end{array}$

Under some appropriate hypotheses, the authors obtained the local existence by using the Galerkin method and showed that the local solutions blow up in finite time. In [26], Xiang and Yang investigated the following Kirchhoff-type problem:

\begin{array}{l} \{\begin{matrix} u_{t} + M ({[u]}_{s, p}^{p}) {(- Δ)}_{p}^{s} u = λ | u |^{q - 2} u - μ | u |^{r - 2} u & (x, t) \in Ω \times (0, \infty), \\ u (x, t) = 0 & (x, t) \in (R^{N} ∖ Ω) \times (0, \infty), \\ u (x, 0) = u_{0} (x) & x \in Ω, \end{matrix} \end{array}

where

1 < p < 2

q, r > 0

, λ and μ are positive parameters. The authors obtained the extinction and non-extinction properties of solutions by using differential inequalities techniques.

To the best of our knowledge, there are a few papers that dealt with the global existence and blow-up results for fractional Laplacian problems involving logarithmic nonlinearities. For example, Zhang and Hu in [33] studied the following problem: $\begin{array}{l} \{\begin{matrix} i u_{t} - {(- Δ)}^{s} u + u ln | u |^{2} = 0 & x \in Ω, t > 0, \\ u (x, t) = 0 & x \in \partial Ω, t ⩾ 0, \\ u (x, 0) = u_{0} (x) & x \in Ω . \end{matrix} \end{array}$ They obtained the global existence of solutions by combining the potential well method with the fractional logarithmic Sobolev inequality. In [1], Ardila considered the Cauchy problem of the following Schrödinger equation $\begin{array}{l} i u_{t} - {(- Δ)}^{s} u + u ln | u |^{2} = 0 x \in Ω, t > 0 . \end{array}$ The existence of global solutions was obtained by using a compactness method. Moreover, the author proved the existence of ground states as minimizers of the action on the Nehari manifold. For more details about the existence and multiplicity of solutions to fractional Schrödinger equations involving logarithmic nonlinearity, we refer to [5].

If $s ↑ 1$ , $θ = 1$ and $q = 2$ , then problem (1.1) reduces to the following equation: $\begin{array}{l} u_{t} - Δ u = u ln | u | in Ω . \end{array}$ In [4], Chen et al. studied the global existence and finite time blow-up of solutions for the above equation by using the logarithmic Sobolev inequality and the potential well method. For more results related to logarithmic nonlinearity, we refer to [4,9,30,31] and the references therein.

However, to the best of our knowledge, there are no papers to deal with the global existence and blow-up results for problems like (1.1). Inspired by the above works, we study the existence of global solutions that vanish at infinity or solutions that blow up in finite time for problem (1.1) at high initial energy level. Since our problem is nonlocal and the diffusion coefficient ${[u]}_{s}^{2 (θ - 1)}$ is a function, our discussions are more elaborate than those appearing in the literature.

The rest of the paper is organized as follows. In Section 2, we give some related definitions and properties of the fractional Sobolev spaces and some preliminary results. In Section 3, we prove the existence of ground state solutions for problem (1.1) in the stationary case. In Section 4, we gave some useful lemmas, which will be employed in the next section. In Section 5, we prove the existence of global solutions that vanish at infinity or solutions that blow up in finite time for problem (1.1).

2. Preliminaries

In this section, we first recall some necessary definitions and properties of the fractional Sobolev spaces, see [8,15,22,23,27] for further details.

Let Ω be a bounded domain in $R^{N}$ ( $N > 2 s$ ) with Lipschitz boundary and let $0 < s < 1$ be real number. For convenience, we shortly denote by $‖ \cdot ‖_{ν}$ the norm of Lebesgue space $L^{ν} (Ω)$ with $ν ⩾ 1$ . The fractional critical exponent $2_{s}^{*}$ is defined as $2_{s}^{*} = \frac{2 N}{N - 2 s}$ . The fractional Sobolev space $W^{s, 2} (Ω)$ is a linear space of Lebesgue measurable functions from $R^{N}$ to $R$ such that the restriction to Ω of any function u in $W^{s, 2} (Ω)$ belongs to $L^{2} (Ω)$ and $\begin{array}{l} \iint_{R^{2 N}} \frac{∣ u (x) - u (y) ∣^{2}}{| x - y |^{N + 2 s}} d x d y < \infty . \end{array}$ The space $W^{s, 2} (Ω)$ is equipped with the norm $\begin{array}{l} ‖ u ‖_{W^{s, 2} (Ω)} = ‖ u ‖_{2} + {(\iint_{R^{2 N}} \frac{∣ u (x) - u (y) ∣^{2}}{| x - y |^{N + 2 s}} d x d y)}^{1 / 2} . \end{array}$ We shall work in the closed linear subspace $\begin{array}{l} W_{0}^{s, 2} (Ω) = {u \in W^{s, 2} (Ω) : u (x) = 0 a.e. in R^{N} ∖ Ω} . \end{array}$ By the result in [27], one can deduce that $\begin{matrix} {[u]}_{s} = {(\iint_{R^{2 N}} \frac{∣ u (x) - u (y) ∣^{2}}{| x - y |^{N + 2 s}} d x d y)}^{1 / 2} \end{matrix}$ is an equivalent norm of $W_{0}^{s, 2} (Ω)$ .

Since our problem contains the logarithmic nonlinearity term, the following lemmas will be used to obtain the key estimates which are related to problem (1.1). More precisely, the proof of Lemma 2.1 is straightforward. Lemma 2.2 is a fractional version of the famous Gagliardo–Nirenberg’s inequality, which was proved in [26].

Lemma 2.1.
Let σ be a positive number. Then $\begin{array}{l} ln t ⩽ \frac{e^{- 1}}{σ} t^{σ}, \forall t \in [1, + \infty) . \end{array}$
Proof.
The proof is simple. For convenience, we give its proof. Let $h (t) : = ln t - \frac{e^{- 1}}{σ} t^{σ}$ for all $t ⩾ 1$ . Clearly, one can obtain that $t_{} = e^{\frac{1}{σ}}$ is the unique maximum point of function h. Thus, $h (t) ⩽ h (t_{}) = 0$ for all $t ⩾ 1$ . This completes the proof. □
Lemma 2.2 (Fractional Gagliardo–Nirenberg inequality, see [26, Lemma 2.2]).

Suppose that $β ⩾ 0, N > 2 s$ and $1 ⩽ r ⩽ q ⩽ (β + 1) 2_{s}^{*}$ , then for u such that $| u |^{β} u \in W_{0}^{s, 2} (Ω)$ , we have $\begin{array}{l} ‖ u ‖_{q} ⩽ C_{2_{s}^{*}}^{\frac{1 - ϑ}{β + 1}} ‖ u ‖_{r}^{ϑ} {[| u |^{β} u]}_{s}^{\frac{1 - ϑ}{β + 1}} \end{array}$ with $ϑ = \frac{[(β + 1) 2_{s}^{*} - q] r}{[(β + 1) 2_{s}^{*} - r] q}$ , where $C_{2_{s}^{*}} > 0$ is the best constant of embedding from $W_{0}^{s, 2} (Ω)$ to $L^{2_{s}^{*}} (Ω)$ .

To order to investigate the global existence and blow-up property of solutions for problem (1.1) with high initial energy, we next introduce some notations.

For $u \in W_{0}^{s, 2} (Ω)$ , set $\begin{array}{l} (2.1) & E (u) = \frac{1}{2 θ} {[u]}_{s}^{2 θ} - \frac{1}{q} \int_{Ω} | u |^{q} ln | u | d x + \frac{1}{q^{2}} ‖ u ‖_{q}^{q} \end{array}$ and $\begin{array}{l} (2.2) & I (u) = {[u]}_{s}^{2 θ} - \int_{Ω} | u |^{q} ln | u | d x . \end{array}$ Then $\begin{array}{l} (2.3) & E (u) = \frac{1}{2 θ} I (u) + (\frac{1}{2 θ} - \frac{1}{q}) {[u]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u ‖_{q}^{q} . \end{array}$ From Lemma 2.1, for any $σ > 0$ with $2 θ < q + σ ⩽ 2_{s}^{*}$ we can get that $\begin{array}{l} I (u) & = {[u]}_{s}^{2 θ} - \int_{Ω} | u |^{q} ln | u | d x \\ ⩾ {[u]}_{s}^{2 θ} - \frac{e^{- 1}}{σ} \int_{Ω} | u |^{q + σ} d x \\ ⩾ {[u]}_{s}^{2 θ} - \frac{e^{- 1}}{σ} | Ω |^{\frac{2_{s}^{*} - q - σ}{2_{s}^{*}}} S^{- \frac{q + σ}{2}} {[u]}_{s}^{q + σ} \\ = (1 - \frac{e^{- 1}}{σ} | Ω |^{\frac{2_{s}^{*} - q - σ}{2_{s}^{*}}} S^{- \frac{q + σ}{2}} {[u]}_{s}^{q + σ - 2 θ}) {[u]}_{s}^{2 θ}, \end{array}$ where $\begin{array}{l} S : = inf_{u \in W_{0}^{s, 2} (Ω) ∖ {0}} \frac{{[u]}_{s}^{2}}{‖ u ‖_{2_{s}^{*}}^{2}} . \end{array}$ If $I (u) ⩽ 0$ , then $\begin{array}{l} (2.4) & {[u]}_{s} ⩾ γ_{σ} : = {(σ e | Ω |^{\frac{q + σ - 2_{s}^{*}}{2_{s}^{*}}} S^{\frac{q + σ}{2}})}^{\frac{1}{q + σ - 2 θ}} . \end{array}$ Define $\begin{array}{l} N = {u \in W_{0}^{s, 2} (Ω) ∖ {0} | I (u) = 0}, \\ N_{+} = {u \in W_{0}^{s, 2} (Ω) | I (u) > 0}, \\ N_{-} = {u \in W_{0}^{s, 2} (Ω) | I (u) < 0} . \end{array}$

Lemma 2.3.
Let $1 < q < 2_{s}^{}$ holds. Then the functionals* $E (u)$ and $I (u)$ are well-defined and continuous on $W_{0}^{s, 2} (Ω)$ . Moreover, $E \in C^{1} (W_{0}^{s, 2} (Ω), R)$ and $I (u) = ⟨ E^{'} (u), u ⟩$ for al $u \in W_{0}^{s, 2} (Ω)$ .
Proof.
By Lemma 2.1, we have $\begin{array}{l} \int_{Ω} | u |^{q} | ln | u | | d x & = - \int_{{| u | ⩽ 1}} | u |^{q} ln | u | d x + \int_{{| u | > 1}} | u |^{q} ln | u | d x \\ ⩽ \frac{e^{- 1}}{q} | Ω | + \frac{e^{- 1}}{σ} \int_{Ω} | u |^{q + σ} d x \\ ⩽ \frac{e^{- 1}}{q} | Ω | + \frac{e^{- 1}}{σ} {[u]}_{s}^{q + σ}, \end{array}$ for all $u \in W_{0}^{s, 2} (Ω)$ , where ${| u | ⩽ 1} : = {x \in Ω : | u (x) | ⩽ 1}$ and ${| u | > 1} : = {x \in Ω : | u (x) | > 1}$ . In view of the definitions of E and I, we know that $E (u)$ and $I (u)$ are well-defined. The continuity of E and I follows from the facts that $1 < q < 2_{s}^{}$ and the embedding $W_{0}^{s, 2} (Ω) ↪ L^{q} (Ω)$ is compact ([6, Corollary 7.2]). By using a similar discussion as in [27], one verify that $E \in C^{1} (W_{0}^{s, 2} (Ω))$ and $I (u) = ⟨ E^{'} (u), u ⟩$ . □
Lemma 2.4.
Let* $u \in W_{0}^{s, 2} (Ω) ∖ {0}$ and define $h (k) = E (k u)$ for all $k > 0$ . Then
${lim}_{k \to 0^{+}} h (k) = 0$ and ${lim}_{k \to \infty} h (k) = - \infty$ ;

there exists a unique $k_{} > 0$ such that* $h^{'} (k_{}) = 0$ , $h (k)$ is increasing on* $(0, k_{})$ , decreasing on* $(k_{}, \infty)$ and attains its maximum at* $k_{}$ ;

$I (k u) > 0$ for* $0 < k < k_{}$ , $I (k u) < 0$ for all* $k > k_{}$ and* $I (k_{} u) = 0$ .

Proof.
It follows from the definition of E that $\begin{array}{l} h (k) = \frac{k^{2 θ}}{2 θ} {[u]}_{s}^{2 θ} - \frac{k^{q}}{q} \int_{Ω} | u |^{q} ln | u | d x - \frac{k^{q}}{q} ln k ‖ u ‖_{q}^{q} + \frac{k^{q}}{q^{2}} ‖ u ‖_{q}^{q}, \end{array}$ which together with $2 θ < q$ and $‖ u ‖_{q} > 0$ yields that (i) holds true.

A simple calculation yields that $\begin{array}{l} h^{'} (k) = k^{2 θ - 1} ({[u]}_{s}^{2 θ} - k^{q - 2 θ} ln k \int_{Ω} | u |^{q} d x - k^{q - 2 θ} \int_{Ω} | u |^{q} ln | u | d x) . \end{array}$ Set $j (k) = k^{1 - 2 θ} h^{'} (k)$ . A direct calculation gives $\begin{array}{l} j^{'} (k) = - k^{q - 2 θ - 1} [(q - 2 θ) ln k \int_{Ω} | u |^{q} d x + (q - 2 θ) \int_{Ω} | u |^{q} ln | u | d x + ‖ u ‖_{q}^{q}], \end{array}$ which implies that there exists a $k_{1} > 0$ such that $j^{'} (k) > 0$ on $(0, k_{1})$ , $j^{'} (k) < 0$ on $(k_{1}, \infty)$ and $j^{'} (k_{1}) = 0$ . Hence j is increasing on $(0, k_{1})$ , decreasing on $(k_{1}, \infty)$ . Note that $j (0) = {[u]}_{s}^{2 θ} > 0$ and ${lim}_{k \to \infty} j (k) = - \infty$ . Thus, there exists a unique $k_{} > 0$ such that $j (k_{}) = 0$ , that is, $h^{'} (k_{}) = 0$ . It follows from $h^{'} (k) = k^{2 θ - 1} j (k)$ that (ii) holds. By virtue of the fact that $I (k u) = k h^{'} (k)$ , we can deduce that (iii) holds. □

3. Ground state solutions for the stationary problem

In this section, we shall investigate the existence of ground state solutions for problem (1.1) in the steady case. For this, we recall the potential well theory related to our problem.

Similar to [30 ,31], we define the potential well $W$ and the set outside the potential well $V$ as $\begin{array}{l} W = {u \in W_{0}^{s, 2} (Ω) | I (u) = 0, E (u) < d} \cup {0}, \\ V = {u \in W_{0}^{s, 2} (Ω) | I (u) > 0, E (u) < d}, \end{array}$ where $\begin{array}{l} d : = inf_{u \in N} E (u) = inf_{u \in N} [(\frac{1}{2 θ} - \frac{1}{q}) {[u]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u ‖_{q}^{q}] \end{array}$ is the depth of the potential well $W$ .

Lemma 3.1.
Assume $2 θ < q < 2_{s}^{}$ holds. Then d is attained at some* $u_{} \in N$ and* $d ⩾ (\frac{1}{2 θ} - \frac{1}{q}) γ_{σ}^{2 θ}$ for any $σ > 0$ with $q + σ ⩽ 2_{s}^{}$ .
Proof.
By (2.3) and (2.4), one can obtain that d is positive. More precisely, $d ⩾ (\frac{1}{2 θ} - \frac{1}{q}) γ_{σ}^{2 θ}$ for any $σ > 0$ with $q + σ ⩽ 2_{s}^{}$ .

Next we show that there is a $u_{} \in N$ such that $d = E (u_{})$ . Let ${u_{n}}$ be a minimizing sequence of d, i.e. ${u_{n}} \subset N$ and $E (u_{n}) \to d$ as $n \to \infty$ . By (2.3), we have $\begin{array}{l} E (u_{n}) & = \frac{1}{2 θ} I (u_{n}) + (\frac{1}{2 θ} - \frac{1}{q}) {[u_{n}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u_{n} ‖_{q}^{q} \\ = (\frac{1}{2 θ} - \frac{1}{q}) {[u_{n}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u_{n} ‖_{q}^{q}, \end{array}$ which together with $2 θ < q < 2_{s}^{}$ implies that ${u_{n}}$ is bounded in $W_{0}^{s, 2} (Ω)$ . Moreover, $\begin{array}{l} (3.1) & {[u_{n}]}_{s} ⩾ γ_{σ} > 0 . \end{array}$ Going if necessary to a subsequence, we assume that $\begin{array}{l} \{\begin{matrix} u_{n} ⇀ u_{} & weakly in W_{0}^{s, 2} (Ω), \\ u_{n} ⇀ u_{} & strongly in L^{q} (Ω), \\ u_{n} \to u_{} & a.e. in Ω . \end{matrix} \end{array}$ Now we claim that $\begin{array}{l} (3.2) & lim_{n \to \infty} \int_{Ω} | u_{n} |^{q} ln | u_{n} | d x = \int_{Ω} | u_{} |^{q} ln | u_{} | d x . \end{array}$ For any measurable subset $U \subset Ω$ , according to the fractional Sobolev embedding theorem, we have $\begin{array}{l} \int_{U} | u_{n} |^{q} ln | u_{n} | d x & ⩽ \frac{1}{q e} | U | + \frac{1}{σ e} | U |^{1 - \frac{q + σ}{2_{s}^{}}} {(\int_{Ω} | u_{n} |^{2_{s}^{}} d x)}^{\frac{q + σ}{2_{s}^{}}} \\ ⩽ \frac{1}{q e} | U | + C \frac{1}{σ e} | U |^{1 - \frac{q + σ}{2_{s}^{}}}, \end{array}$ which implies that ${| u_{n} |^{q} ln | u_{n} |}$ is uniformly bounded and equi-integrable in $L^{1} (Ω)$ . Note that $| u_{n} |^{q} ln | u_{n} | \to | u_{} |^{q} ln | u_{} |$ a.e. in Ω. Thus, the Vitali convergence theorem yields the claim (3.2). By (3.1) and $I (u_{n}) = 0$ , one can obtain $\begin{array}{l} γ_{σ}^{2 θ} ⩽ {[u_{n}]}_{s}^{2 θ} = \int_{Ω} | u_{n} |^{q} ln | u_{n} | d x, \end{array}$ which together with (3.2) yields that $\begin{array}{l} γ_{σ}^{2 θ} ⩽ \int_{Ω} | u_{} |^{q} ln | u_{} | d x . \end{array}$ This implies that $u_{} \neq 0$ . From (3.2) and the weak lower semicontinuity of the norm, we have $\begin{array}{l} (3.3) & E (u_{}) ⩽ \underset{n \to \infty}{lim inf} E (u_{n}) = d \end{array}$ and $\begin{array}{l} I (u_{}) ⩽ \underset{n \to \infty}{lim inf} I (u_{n}) = 0 . \end{array}$ Since $u_{} \in W_{0}^{s, 2} (Ω)$ , it follows from Lemma 2.4 that there exists $k_{} > 0$ such that $I (k_{} u_{}) = 0$ . If $I (u_{}) < 0$ , then by the definition of h given in the proof of Lemma 2.4 we have $\begin{matrix} h^{'} (1) = {[u_{}]}_{s}^{2 θ} - \int_{Ω} | u_{} |^{q} ln | u_{} | d x = I (u_{}) < 0 . \end{matrix}$ It follows from Lemma 2.4(2) that $k_{} \in (0, 1)$ . Thus, we deduce $\begin{array}{l} d & ⩽ E (k_{} u_{}) = (\frac{1}{2 θ} - \frac{1}{q}) {[k_{} u_{}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ k_{} u_{} ‖_{q}^{q} \\ ⩽ k_{}^{2 θ} [(\frac{1}{2 θ} - \frac{1}{q}) {[u_{}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u_{} ‖_{q}^{q}] \\ ⩽ k_{}^{2 θ} \underset{n \to \infty}{lim inf} [(\frac{1}{2 θ} - \frac{1}{q}) {[u_{n}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u_{n} ‖_{q}^{q}] \\ = k_{}^{2 θ} \underset{n \to \infty}{lim inf} E (u_{n}) \\ = k_{}^{2 θ} d < d, \end{array}$ which is absurd. Consequently, we have $I (u_{}) = 0$ . Thus, $u_{} \in N$ . Therefore, we conclude from (3.3) and the minimality of d that $E (u_{}) = d$ . □
Theorem 3.1.
Assume $1 ⩽ θ < 2_{s}^{} / 2$ and* $2 θ < q < 2_{s}^{}$ hold. Then* $u_{}$ obtained by Lemma* 3.1 is a ground state (least energy) solution of the following elliptic type problem: $\begin{array}{l} \{\begin{matrix} {[u]}_{s}^{2 (θ - 1)} {(- Δ)}^{s} u = | u |^{q - 2} u ln | u | & x \in Ω, \\ u = 0 & x \in R^{N} ∖ Ω . \end{matrix} \end{array}$
Proof.
By Lemma 3.1, it is enough to show that $E^{'} (u_{}) = 0$ . The proof is similar to Theorem 2.3 in [2]. For completeness, we give the detailed treatment.

Since $d = E (u_{})$ and $I (u_{}) = 0$ , by the theory of Lagrange multipliers, there exists $μ \in R$ such that $E^{'} (u_{}) = μ I^{'} (u_{})$ . Thus $\begin{array}{l} (3.4) & I (u_{}) = ⟨ E^{'} (u_{}), u_{} ⟩ = μ ⟨ I^{'} (u_{}), u_{} ⟩ . \end{array}$ Gathering $I (u_{}) = 0$ , $q > 2 θ$ with $u_{} \neq 0$ , we arrive at $\begin{array}{l} ⟨ I^{'} (u_{}), u_{} ⟩ = 2 θ {[u]}_{s}^{2 θ} - q \int_{Ω} | u |^{q} ln | u | d x - \int_{Ω} | u |^{q} d x = - [(q - 2 θ) {[u]}_{s}^{2 θ} + ‖ u ‖_{q}^{q}] < 0, \end{array}$ which together with (3.4) implies that $μ = 0$ . Therefore, $E^{'} (u_{*}) = 0$ . □

4. Some technical lemmas

To introduce some technical lemmas with respect to the global existence and blow-up of solutions, we first give the definition of (weak) solutions.

Definition 4.1.
A function $u \in L^{2} (0, T; W_{0}^{s, 2} (Ω)) \cap C (0, T; L^{2} (Ω))$ is said to be a (weak) solution of problem (1.1), if $u_{t} \in L^{2} (0, T; L^{2} (Ω))$ and for a.e. $t \in (0, T)$ , $\begin{array}{l} \int_{Ω} u_{t} φ d x + {[u]}_{s}^{2 (θ - 1)} {⟨ u, φ ⟩}_{s} = \int_{Ω} | u |^{q - 2} u ln | u | φ d x, \end{array}$ for all $φ \in W_{0}^{s, 2} (Ω)$ , where $\begin{matrix} {⟨ u, φ ⟩}_{s} = \iint_{R^{2 N}} \frac{[u (x, t) - u (y, t)] [φ (x) - φ (y)]}{∣ x - y ∣^{N + 2 s}} d x d y . \end{matrix}$
Remark 4.1.
Note that the existence of solutions for problem (1.1) can be obtained by using the Galerkin method, see for example [12,20].

For any $k > d$ , define the (open) sublevels of E by $\begin{array}{l} E^{k} = {u \in W_{0}^{s, 2} (Ω) | E (u) < k} . \end{array}$ By (2.3), (2.4) and the definitions of $N, E^{k}$ and d, we see that $\begin{array}{l} (4.1) & N_{k} ≜ N \cap E^{k} \neq \emptyset, \forall k > d . \end{array}$ We define two variational numbers $\begin{array}{l} (4.2) & λ_{k} = inf {‖ u ‖_{2} : u \in N_{k}}, Λ_{k} = sup {‖ u ‖_{2} : u \in N_{k}} . \end{array}$ Obviously, $λ_{k}$ is nonincreasing with respect to k and $Λ_{k}$ is nondecreasing. Moreover, we have the following result. Lemma 4.1.
For any $k > d, λ_{k}$ and $Λ_{k}$ defined in ( 4.2 ) satisfy $\begin{array}{l} 0 < λ_{k} ⩽ Λ_{k} < + \infty . \end{array}$
Proof.
For any $u \in W_{0}^{s, 2} (Ω)$ , by Lemma 2.2 we have $\begin{array}{l} (4.3) & ‖ u ‖_{q + σ}^{q + σ} ⩽ C_{2_{s}^{}}^{(1 - ϑ) (q + σ)} ‖ u ‖_{2}^{ϑ (q + σ)} {[u]}_{s}^{(1 - ϑ) (q + σ)}, \end{array}$ where $ϑ = \frac{2 (2_{s}^{} - q - σ)}{(2_{s}^{} - 2) (q + σ)}$ and $2 < q + σ ⩽ 2_{s}^{}$ . Obviously, $ϑ \in (0, 1)$ by $q + σ > 2$ . Now choose $σ > 0$ such that $q + σ < \frac{2 (2_{s}^{} - 1)}{2_{s}^{}}$ . Then $(1 - ϑ) (q + σ) < 2$ . For any $u \in N$ , from (4.3) and Lemma 2.1, we deduce that $\begin{array}{l} {[u]}_{s}^{2 θ} = \int_{Ω} | u |^{q} ln | u | d x ⩽ \frac{e^{- 1}}{σ} ‖ u ‖_{q + σ}^{q + σ} ⩽ \frac{e^{- 1}}{σ} C_{2_{s}^{}}^{(1 - ϑ) (q + σ)} ‖ u ‖_{2}^{ϑ (q + σ)} {[u]}_{s}^{(1 - ϑ) (q + σ)} . \end{array}$ It follows that $\begin{array}{l} (4.4) & {[u]}_{s}^{2 θ - (1 - ϑ) (q + σ)} ⩽ \frac{e^{- 1}}{σ} C_{2_{s}^{}}^{(1 - ϑ) (q + σ)} ‖ u ‖_{2}^{ϑ (q + σ)} . \end{array}$ Note that $N_{k} \subset N$ . Thus, (4.4) and (2.4) yield that $\begin{array}{l} λ_{k} & = inf_{u \in N_{k}} ‖ u ‖_{2} ⩾ inf_{u \in N} ‖ u ‖_{2} \\ ⩾ {[\frac{e σ}{C_{2_{s}^{}}^{(1 - ϑ) (q + σ)}}]}^{\frac{1}{ϑ (q + σ)}} {(inf_{u \in N} {[u]}_{s})}^{\frac{2 θ - (1 - ϑ) (q + σ)}{ϑ (q + σ)}} \\ ⩾ {[\frac{e σ}{C_{2_{s}^{}}^{(1 - ϑ) (q + σ)}}]}^{\frac{1}{ϑ (q + σ)}} γ_{σ}^{\frac{2 θ - (1 - ϑ) (q + σ)}{ϑ (q + σ)}} > 0 . \end{array}$ On the other hand, it follows from (2.3) and $E (u) < k$ for all $u \in N_{k}$ that $\begin{array}{l} {[u]}_{s} ⩽ {(\frac{k}{\frac{1}{2 θ} - \frac{1}{q}})}^{\frac{1}{2 θ}} . \end{array}$ Thus, we have $\begin{array}{l} Λ_{k} & = sup_{u \in N_{k}} ‖ u ‖_{2} ⩽ C_{2_{s}^{}} | Ω |^{\frac{1}{2} - \frac{1}{2_{s}^{}}} sup_{u \in N_{k}} {[u]}_{s} \\ ⩽ C_{2_{s}^{}} | Ω |^{\frac{1}{2} - \frac{1}{2_{s}^{}}} {(\frac{k}{\frac{1}{2 θ} - \frac{1}{q}})}^{\frac{1}{2 θ}} < \infty . \end{array}$ Hence, the lemma is proved. □
Lemma 4.2.

let $T_{max} ⩽ + \infty$ be the maximum existence time of the solution $u (x, t)$ to problem ( 1.1 ) with initial data $u_{0}$ . Then $\begin{array}{l} \int_{0}^{t} {‖ u_{τ} (x, τ) ‖}_{2}^{2} d τ + E (u (x, t)) = E (u_{0}) t \in [0, T_{max}) . \end{array}$

Both $N$ and $N_{-}$ are away from 0, i.e., dist $(0, N) > 0$ , dist $(0, N_{-}) > 0$ .

For any $k > d$ , the set $E^{k} \cap N_{+}$ is bounded in $W_{0}^{s, 2} (Ω)$ .
Proof.
(1) Note that the solution of problem ( 1.1 ) can be obtained as the limit of the following sequence of Galerkin’s approximation (see [ 20 , 26 ]) $\begin{array}{l} v_{n} (x, t) = \sum_{j = 1}^{n} {(ξ_{n} (t))}_{j} e_{j} (x), n = 1, 2, \dots, \end{array}$ where $ξ_{n} (t) \in C^{1} [0, T_{max})$ and ${e_{j}} \subset C_{0}^{\infty} (Ω)$ is an orthonormal basis in $L^{2} (Ω)$ . Let u be a sufficiently smooth solution to problem ( 1.1 ) (or the approximate solution $v_{n}$ ). Choosing $ϕ = u_{t}$ in Definition 4.1 , we have $\begin{array}{l} \int_{Ω} u_{t}^{2} d x + {[u]}_{s}^{2 (θ - 1)} {⟨ u, u_{t} ⟩}_{s} = \int_{Ω} | u |^{q - 2} u ln | u | u_{t} d x, \end{array}$ which implies that $\begin{array}{l} ‖ u_{t} ‖_{2}^{2} + \frac{1}{2 θ} \frac{d}{d t} {[u]}_{s}^{2 θ} - \frac{1}{q} \frac{d}{d t} \int_{Ω} | u |^{q} ln | u | d x + \frac{1}{q^{2}} \frac{d}{d t} ‖ u ‖_{q}^{q} = 0 . \end{array}$ Furthermore, we obtain $\begin{array}{l} ‖ u_{t} ‖_{2}^{2} + \frac{d}{d t} E (u (x, t)) = 0, \end{array}$ Next we integrate with respect to time variable on $[0, t]$ . Then we get the equality $\begin{array}{l} \int_{0}^{t} {‖ u_{τ} (x, τ) ‖}^{2} d τ + E (u (x, t)) = E (u_{0}), \end{array}$

(2) For any $u \in N$ , we have $\begin{array}{l} I (u) = {[u]}_{s}^{2 θ} - \int_{Ω} | u |^{q} ln | u | d x = 0, \end{array}$ By virtue of ( 2.4 ), one can deduce $\begin{array}{l} {[u]}_{s}^{2 θ} ⩾ \frac{γ_{σ}}{(\frac{1}{2 θ} - \frac{1}{q})} = {(σ e | Ω |^{\frac{q + σ - 2_{s}^{}}{2_{s}^{}}} S^{\frac{q + σ}{2}})}^{\frac{2}{q + σ - 2 θ}}, \end{array}$ which implies $\begin{array}{l} dist (0, N) = inf_{u \in N} {[u]}_{s} ⩾ {(\frac{γ_{σ}}{\frac{1}{2 θ} - \frac{1}{q}})}^{\frac{1}{2 θ}} > 0 . \end{array}$ Similarly, we can deduce from ( 2.4 ) that $dist (0, N_{-}) > 0$ .

(3) For any $k > d$ and $u \in E^{k} \cap N_{+}$ , i.e., $E (u) < k$ and $I (u) > 0$ . ( 2.3 ) yields $\begin{array}{l} \frac{1}{2 θ} I (u) + (\frac{1}{2 θ} - \frac{1}{q}) {[u]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u ‖_{q}^{q} ⩽ E (u) < k . \end{array}$ Since $q > 2 θ$ and $I (u) > 0$ , it follows that $\begin{array}{l} {[u]}_{s}^{2 θ} < \frac{k}{(\frac{1}{2 θ} - \frac{1}{q})}, \end{array}$ which ends the proof. □

5. Global existence and blow-up of solutions

In this section, we consider the global existence and blow-up of solutions. Inspired by some ideas from [4,30–32], we give a sufficient condition for the existence of global solutions that vanish at infinity or solutions that blow up in finite time as $E (u_{0}) > d$ . For fixed time $t \in [0, T_{max})$ , we think of the function $u (x, t)$ with respect to space variable x as an element of $W_{0}^{s, 2} (Ω)$ , and briefly denote by $u (t)$ . Hence $u (t) \in W_{0}^{s, 2} (Ω)$ . If now we vary the time variable t over the interval $[0, T_{max})$ , then we obtain a function $t \to u (t)$ . If the solution is global, i.e. $T_{max} = \infty$ , we denote by $\begin{array}{l} ω (u_{0}) = ⋂_{t ⩾ 0} {\overline{u (τ) : τ ⩾ t}}^{W_{0}^{s, 2} (Ω)}, \end{array}$ the ω-limit set of $u_{0}$ . We define two sets $\begin{array}{l} B = {u_{0} \in W_{0}^{s, 2} (Ω) : the solutions of problem (1.1) blow up in finite time}, \\ V = {u_{0} \in W_{0}^{s, 2} (Ω) : the solutions of problem (1.1) vanish at \infty} . \end{array}$

Theorem 5.1.

If $u_{0} \in W_{0}^{s, 2} (Ω)$ and $E (u_{0}) > d$ , then there hold

If $u_{0} \in N_{+}$ and $‖ u_{0} ‖_{2} ⩽ λ_{E (u_{0})}$ , then $u_{0} \in V$ ;

If $u_{0} \in N_{-}$ and $‖ u_{0} ‖_{2} ⩾ Λ_{E (u_{0})}$ , then $u_{0} \in B$ .

Proof.

(a) Suppose that $u_{0} \in N_{+}$ and $‖ u_{0} ‖_{2} ⩽ λ_{E (u_{0})}$ , where $λ_{E (u_{0})}$ is defined in (4.2) with $k = E (u_{0})$ . We first claim that $u (t) \in N_{+}$ for all $t \in [0, T_{max})$ . Arguing by contradiction, we assume that there exists a $t_{0} \in (0, T)$ such that $u (t) \in N_{+}$ for $t \in [0, t_{0})$ and $u (t_{0}) \in N$ . Then it follows from $I (u (t)) = - \int_{Ω} u_{t} (t) u (t) d x$ that $u_{t} (x, t) \neq 0$ for $(x, t) \in Ω \times (0, t_{0})$ .

In view of Lemma 4.2(1), we have $E (u (t_{0})) < E (u_{0})$ , which implies that $u (t_{0}) \in E^{E (u_{0})}$ . Hence, $u (t_{0}) \in N^{E (u_{0})}$ . According to the definition of $λ_{E (u_{0})}$ , we have $\begin{array}{l} (5.1) & {‖ u (t_{0}) ‖}_{2} ⩾ λ_{E (u_{0})} . \end{array}$ Taking $ϕ = u$ as a test function in Definition 4.1, we obtain $\begin{array}{l} (5.2) & \frac{1}{2} \frac{d}{d t} {‖ u (t) ‖}_{2}^{2} = \int_{Ω} u_{t} (t) u (t) d x = - {[u (t)]}_{s}^{2 θ} + \int_{Ω} {| u (t) |}^{q} ln | u (t) | d x = - I (u (t)) . \end{array}$ By Lemma 4.2(1), for $t \in [0, T_{max})$ , $E (u (t))$ is continuous. Furthermore, one can we get $I (u (t))$ is continuous for any $t \in [0, T_{max})$ . Since $I (u (t)) > 0$ for $t \in [0, t_{0})$ , (5.2) shows that $‖ u (t) ‖_{2}^{2}$ is strictly decreasing. Thus, we conclude that $\begin{matrix} {‖ u (t_{0}) ‖}_{2} < ‖ u_{0} ‖_{2} ⩽ λ_{J (u_{0})} \end{matrix}$ which contradicts (5.1) and the claim follows immediately.

Lemma 4.2(3) shows that the orbit ${u (t)}$ is bounded in $W_{0}^{s, 2} (Ω)$ for all $t \in [0, T_{max})$ so that $T_{max} = \infty$ . For any $ω \in ω (u_{0})$ , the definition of $ω (u_{0})$ implies that for all $t > 0$ , there exists $τ_{n} ⩾ t$ such that $u (τ_{n}) \to ω$ as $n \to \infty$ . Then by (5.2), we get $‖ u (τ_{n}) ‖_{2} \to ‖ ω ‖_{2}$ as $n \to \infty$ . Thus, $\begin{matrix} {‖ u (τ_{n}) ‖}_{2} ⩽ {‖ u (t) ‖}_{2} < {‖ u (0) ‖}_{2}, \end{matrix}$ which implies that $\begin{array}{l} (5.3) & ‖ ω ‖_{2} < λ_{E (u_{0})}, E (ω) < E (u_{0}), \forall ω \in ω (u_{0}) . \end{array}$ Then it follow from the definition of $λ_{E (u_{0})}$ that $ω (u_{0}) \cap N = \emptyset$ . Indeed, if there exists a $ω_{0} \in ω (u_{0})$ such that $I (ω_{0}) = 0$ , then (5.3) yields that $ω_{0} \in N_{E (u_{0})}$ . This together with the definition of $λ_{E (u_{0})}$ deduces that $‖ ω_{0} ‖_{2} ⩾ λ_{E (u_{0})}$ , which is absurd. Hence $ω (u_{0}) \cap N = \emptyset$ .

Next, we prove that $\begin{array}{l} (5.4) & ω (u_{0}) = {0} . \end{array}$ In fact, it follows from $u (t) \in N_{+}$ and the definitions of E and $N_{+}$ that $\begin{array}{l} E (u (t)) & = \frac{1}{2 θ} {[u (t)]}_{s}^{2 θ} - \frac{1}{q} \int_{Ω} {| u (t) |}^{q} ln | u (t) | d x + \frac{1}{q^{2}} {‖ u (t) ‖}_{q}^{q} \\ = \frac{1}{2 θ} I (u (t)) + (\frac{1}{2 θ} - \frac{1}{q}) {[u (t)]}_{s}^{2 θ} + \frac{1}{q^{2}} {‖ u (t) ‖}_{q}^{q} > 0, \end{array}$ which means that $E (u (t))$ is bounded from below. This together with the fact that $E (u (t))$ is nonincreasing with respect to t implies that there exists a constant C such that $\begin{array}{l} lim_{t \to + \infty} J (u (t)) = C . \end{array}$ Thus, for any $ω \in ω (u_{0})$ , we have $E (u_{ω} (t)) = c$ for all $t ⩾ 0$ , where $u_{ω} (t)$ is the solution of problem (1.1) with initial value ω. So we get from Lemma 4.2(1) that $u_{ω} (t) \equiv C$ , and then it follows from (5.2) that $\begin{array}{l} (5.5) & I (ω) = 0, \forall ω \in ω (u_{0}) . \end{array}$ Gathering (5.5), $ω \notin N$ and the definition of N, we get (5.4). In other words, the solution $u (t) \to 0$ as $t \to + \infty$ . Thus, the claim is proved.

(b) Assume $u_{0} \in N_{-}$ and $‖ u_{0} ‖_{2} ⩾ Λ_{E (u_{0})}$ , where $Λ_{E (u_{0})}$ is defined in (4.2) with $k = E (u_{0})$ . By applying a similar argument as above, we can obtain that $u (t) \in N_{-}$ for all $t \in [0, T_{max})$ . Arguing by contradiction, we assume that $T_{max} = \infty$ . Then for every $ω \in ω (u_{0})$ , it follows from (5.2) and Lemma 4.2(1) that $\begin{matrix} ‖ ω ‖_{2} > Λ_{E (u_{0})}, E (ω) < E (u_{0}) . \end{matrix}$ Going back to the definition of $Λ_{E (u_{0})}$ again, we then infer that $ω (u_{0}) \cap N = \emptyset$ . Therefore, it must hold that $ω (u_{0}) = 0$ , which contradicts the fact that both $dist (0, N_{-})$ and $dist (0, N)$ are strictly positive. Hence, $T_{max} < \infty$ as desired. Therefore, the proof is complete. □

Theorem 5.2.

If $u_{0} \in W_{0}^{s, 2} (Ω)$ , $E (u_{0}) > d$ and $q^{2} | Ω |^{\frac{q - 2}{q}} E (u_{0}) < ‖ u_{0} ‖_{2}^{q}$ , then $u_{0} \in N_{-} \cap B$ .

Proof.

Assume $E (u_{0}) > d$ and $u_{0} \in W_{0}^{s, 2} (Ω)$ . By assumption and the Hölder inequality, we have $\begin{array}{l} (5.6) & q^{2} | Ω |^{\frac{q - 2}{q}} E (u_{0}) < ‖ u_{0} ‖_{2}^{q} ⩽ | Ω |^{\frac{q - 2}{2}} \int_{Ω} | u_{0} |^{q} d x . \end{array}$ It follows from the definitions of $E (u_{0})$ and $I (u_{0})$ that $\begin{array}{l} E (u_{0}) & = \frac{1}{2 θ} ‖ u ‖^{2 θ} - \frac{1}{q} \int_{Ω} | u_{0} |^{q} ln | u_{0} | d x + \frac{1}{q^{2}} ‖ u_{0} ‖_{q}^{q} \\ = \frac{1}{2 θ} I (u_{0}) + (\frac{1}{2 θ} - \frac{1}{q}) {[u_{0}]}_{s}^{2 θ} + \frac{1}{q^{2}} ‖ u_{0} ‖_{q}^{q} \\ > \frac{1}{2 θ} I (u_{0}) + E (u_{0}), \end{array}$ which implies that $I (u_{0}) < 0$ , i.e., $u_{0} \in N_{-}$ .

To prove that $u_{0} \in B$ , we need only to show $‖ u_{0} ‖_{2} ⩾ Λ_{E (u_{0})}$ by Theorem 5.1(b). To this aim, for all $u \in N_{E (u_{0})}$ , we deduce from the Hölder inequality that $\begin{array}{l} | Ω |^{\frac{2 - q}{2}} ‖ u ‖_{2}^{q} & ⩽ (\frac{q}{2 θ} - 1) {[u]}_{s}^{2 θ} + ‖ u ‖_{q}^{q} \\ ⩽ \frac{q}{2} {[u]}_{s}^{2 θ} - \int_{Ω} | u |^{q} ln | u | d x + ‖ u ‖_{q}^{q} \\ ⩽ q^{2} E (u) ⩽ q^{2} E (u_{0}) . \end{array}$ Then we get $\begin{array}{l} ‖ u ‖_{2}^{q} ⩽ q^{2} | Ω |^{\frac{q - 2}{2}} E (u_{0}) . \end{array}$ Taking supremum with respect to u over $N_{E (u_{0})}$ on the left hand side of the above inequality, we obtain $\begin{array}{l} Λ_{E (u_{0})}^{q} ⩽ q^{2} | Ω |^{\frac{q - 2}{2}} E (u_{0}), \end{array}$ which together with (5.6) implies that $u_{0} \in N_{-} \cap B$ . This ends the proof. □

Remark 5.1.

The method applied in this paper can be used to deal with the following fractional p-Laplacian problem of Kirchhoff type: $\begin{array}{l} \{\begin{matrix} u_{t} + {[u]}_{s, p}^{p (θ - 1)} {(- Δ)}_{p}^{s} = | u |^{q - 2} u ln | u |, & x \in Ω, t > 0, \\ u (x, t) = 0, & x \in R^{N} ∖ Ω, t > 0, \\ u (x, 0) = u_{0} (x), & x \in Ω, \end{matrix} \end{array}$ where $N > p s$ , $1 ⩽ θ < N / (N - p s)$ and $θ p < q < N p / (N - p s)$ . Hence the corresponding main results obtained in this paper could be extended, we leave the proofs to the interested readers.

Footnotes

Acknowledgements

M. Xiang was supported by the Natural Science Foundation of China (No. 11601515) and the Fundamental Research Funds for the Central Universities (No. 3122017080). B. Zhang was supported by the Natural Science Foundation of China (No. 11871199) and Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q18109) and Research Foundation of Heilongjiang Educational Committee (No. 12531546).

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