Fractional p -Kirchhoff problems involving critical exponents and sign-changing weight functions

Abstract

In this paper, we study the following Kirchhoff type problems involving fractional p-Laplacian and critical exponents: $\begin{matrix} \{\begin{matrix} M ({[u]}_{s, p}^{p}) {(- Δ)}_{p}^{s} u = λ | u |^{p_{s}^{*} - 2} u + b (x) | u |^{p - 2} u + f (x, u), & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{matrix}$ where Ω is a bounded domain in $R^{N}$ ( $N ⩾ 3$ ) with Lipshcitz boundary, $1 < p < N / s$ with $s \in (0, 1)$ , $p_{s}^{*} = N p / (N - p s)$ is the fractional critical Sobolev exponent, λ is a positive parameter, $M : [0, + \infty) \to R^{+}$ and $f : \overline{Ω} \times R \to R$ are continuous functions, $b : Ω \to R$ is a sign-changing function. By using the fractional version of concentration compactness principle together with mountain pass theorem, we obtained the multiplicity of solutions for the above problem.

Keywords

Kirchhoff type problem mountain pass theorem critical exponent concentration–compactness principle

1. Introduction and main result

In this paper we deal with the multiplicity of solutions to the following fractional p-Kirchhoff equations involving the critical exponents: $\begin{matrix} (1.1) & \{\begin{matrix} M ({[u]}_{s, p}^{p}) {(- Δ)}_{p}^{s} u = λ | u |^{p_{s}^{*} - 2} u + b (x) | u |^{p - 2} u + f (x, u), & in Ω, \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{matrix}$ where $N > s p$ with $s \in (0, 1)$ , $p > 1$ , and $\begin{array}{rcl} {[u]}_{s, p}^{p} = \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y, \end{array}$ $p_{s}^{*} = N p / (N - p s)$ is the fractional critical Sobolev exponent, $Ω \subset R^{N}$ ( $N ⩾ 3$ ) is a bounded domain with Lipshcitz boundary, ${(- Δ)}_{p}^{s}$ is the associated fractional p-Laplacian operator which, up to a normalization constant, is defined as $\begin{matrix} {(- Δ)}_{p}^{s} φ (x) = 2 lim_{ε \to 0^{+}} \int_{R^{N} ∖ B_{ε} (x)} \frac{| φ (x) - φ (y) |^{p - 2} (φ (x) - φ (y))}{| x - y |^{N + p s}} d y, x \in R^{N}, \end{matrix}$ along functions $φ \in C_{0}^{\infty} (R^{N})$ . Henceforward $B_{ε} (x)$ denotes the ball of $R^{N}$ centered at $x \in R^{N}$ and radius $ε > 0$ . Let $b \in L^{\infty} (Ω)$ with $Ω_{b}^{+} : = {x \in Ω : b (x) > 0}$ . Throughout the paper, without explicit mention, we assume that $M : R_{0}^{+} \to R^{+}$ is a continuous and increasing function that satisfies:

There exists $m_{0} > 0$ such that $M (t) ⩾ m_{0}$ for all $t \in [0, + \infty)$ ;

There exists $σ \in (p / p_{s}^{*}, 1]$ such that $\hat{M} (t) ⩾ σ M (t) t$ for all $t ⩾ 0$ , where $\hat{M} (t) = \int_{0}^{t} M (s) d s$ . Moreover, it holds that $(σ - 1) b (x) = 0$ for all $x \in Ω_{b}^{+}$ .

Moreover, the functions

b (x)

and f satisfy the following conditions:

There exists a constant $α > 0$ such that $\begin{matrix} m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x ⩾ α \int_{Ω} | u |^{p} d x, \forall u \in W_{0}^{s, p} (Ω), \end{matrix}$ where $m_{0}$ is the number given by the hypothesis $(M_{0})$ .

$f \in C (Ω \times R, R)$ , odd with respect to t and $\begin{array}{l} f (x, t) = o (| t |^{p_{s}^{*} - 1}), as | t | \to + \infty uniformly in x \in Ω, \\ f (x, t) = o (| t |^{p - 1}), as | t | \to 0 uniformly in x \in Ω . \end{array}$

$p F (x, t) ⩽ σ f (x, t) t$ for all $x \in Ω$ and $t \in R$ , where $F (x, t) = \int_{0}^{t} f (x, τ) d τ$ and the number σ is given by the condition $(M_{1})$ .

A typical example of f is given by

f (x, t) = | t |^{q - 2} t

for all

x \in Ω

and

t \in R

, where

p < q < p_{s}^{*}

. Note that if

x \in Ω_{b}^{+}

then we have that

σ = 1

, and hence the condition

(M_{1})

becomes that the inequality

\hat{M} (t) ⩾ M (t) t

holds for all

t ⩾ 0

. If

x \in Ω ∖ Ω_{b}^{+}

and

σ = \frac{1}{2}

, then the function

M (t) = m_{0} + m_{1} t

with

m_{1} ⩾ 0

obviously satisfies the conditions

(M_{0})

–

(M_{1})

In the last years, a great attention has been attracted to the study of fractional and non-local problems involving critical nonlinearities. For example, some of the recent contributions on the existence of solutions for critical fractional Laplacian equations are given in [2,11–15,21,27–29]. It is worthy mentioning that the interest in nonlocal problems goes beyond the mathematical curiosity. Indeed, this type of operators arises in a quite natural way in many different applications, such as, continuum mechanics, phase transition phenomena, population dynamics, minimal surfaces and game theory, see for example [1,4,6,9] and the references therein. For the basic properties of fractional Sobolev spaces and the fractional Laplacian, we refer the readers to [6,22].

Our motivation for studying problem (1.1) is two-folds. On the one hand, Li et al. [10] showed the existence of multiple solutions for the p-Kirchhoff problem with the subcritical exponent and sign-changing weight functions. By using Nehari manifold, Chen et al. [5] obtained the existence of two positive solutions for the Kirchhoff type problem involving sign-changing weight functions. On the other hand, Fiscella and Valdinoci in [7] first proposed a stationary Kirchhoff variational equation which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. More precisely, they established a model given by the following formulation: $\begin{array}{rcl} (1.2) & \{\begin{matrix} M ({[u]}_{s, 2}^{2}) {(- Δ)}^{s} u = λ f (x, u) + | u |^{2_{s}^{*} - 2} u & in Ω \\ u = 0 & in R^{N} ∖ Ω, \end{matrix} \end{array}$ where $M : R_{0}^{+} \to R^{+}$ is an increasing and continuous function satisfying the assumption $(M_{0})$ . Obviously, $M (t) = a + 2 b t$ with $b ⩾ 0$ is a prototype meeting the above hypotheses. Note that if M is this type, problem (1.2) is called non-degenerate if $a > 0$ and $b ⩾ 0$ , while it is named degenerate if $a = 0$ and $b > 0$ , see [25] for some physical explanations about non-degenerate Kirchhoff problems. Hence our problem (1.1) and problem (1.2) are non-degenerate. For some motivation in the physical background for the fractional Kirchhoff model, we refer to [7, Appendix A]. In [30], Xiang et al. investigated the existence of solutions for Kirchhoff type problem involving the fractional p-Laplacian via variational methods, where the nonlinearity is subcritical and the Kirchhoff function is non-degenerate. By using the mountain pass theorem and Ekeland’s variational principle, Xiang et al. in [31] studied the multiplicity of solutions to a nonhomogeneous Kirchhoff type problem driven by the fractional p-Laplacian, where the nonlinearity is convex-concave and the Kirchhoff function is degenerate. Using the same methods as in [31], Pucci et al. in [24] obtained the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type in the whole space. For more recent results about fractional Kirchhoff type problems, for example, we refer to [18,19]. It is worth pointing out that the weight functions in the above papers involving the fractional Laplacian just consider the constant sign. As for a critical Kirchhoff problem similar to (1.1) with sign-changing weight functions, we refer to [20] for an existence result.

Compared to the above papers, some difficulties arise in our paper when dealing with problem (1.1), because of the appearance of the p-fractional Laplacian as well as the possibly sign-changing weight function. In this paper, we study the multiplicity of solutions for the fractional p-Kirchhoff equations involving the critical exponents. Due to the presence of the critical exponents, the problem considered here is lack of compactness. To overcome this difficulty, we use the fractional version of concentration compactness principle due to Lions [16,17]. In addition, we would like to stress that the extension from the case $p = 2$ to the case $1 < p < \infty$ is not trivial. For this purpose, we will use the genus theory, introduced by Krasnoselskii and a variant of the mountain pass theorem for even functionals due to Rabinowitz [26]. We believe that there are few papers studying the existence of solutions for the fractional p-Kirchhoff type problems involving critical exponents and sign-changing weight functions.

We will seek weak solutions to problem (1.1) in $W_{0}^{s, p} (Ω)$ which is the usual Sobolev space with respect to the norm $‖ u ‖ = {[u]}_{s, p}$ . We then have that $W_{0}^{s, p} (Ω)$ is continuously and compactly embedded into the Lebesgue space $L^{r} (Ω)$ endowed the norm $| u |_{r} = {(\int_{Ω} | u |^{r} d x)}^{\frac{1}{r}}$ , $p < r < p_{s}^{*}$ . Denote by $S_{r}$ the best constant for this embedding, that is, $\begin{matrix} (1.3) & S_{r} | u |_{r} ⩽ ‖ u ‖, \forall u \in W_{0}^{s, p} (Ω) . \end{matrix}$ In particular, if S is the best constant for the embedding $W_{0}^{s, p} (Ω) ↪ L^{p_{s}^{*}} (Ω)$ , then it is defined by $\begin{matrix} (1.4) & S = inf_{u \in W_{0}^{s, p} (Ω) ∖ {0}} \frac{\iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p}}{| x - y |^{N + p s}} d x d y}{{(\int_{Ω} | u |^{p_{s}^{*}} d x)}^{\frac{p}{p_{s}^{*}}}} . \end{matrix}$ The main result of this paper can be summarized in the following theorem.

Theorem 1.1.
Assume that conditions $(M_{0})$ – $(M_{1})$ and $(F_{0})$ – $(F_{2})$ hold. Then, there exists a decreasing sequence of real numbers ${λ_{k}} \subset (0, + \infty)$ with $λ_{k + 1} < λ_{k}$ such that, for each $λ \in (λ_{k + 1}, λ_{k})$ , problem ( 1.1 ) has at least k pairs of nontrivial weak solutions.

2. The Palais–Smale condition

In this section, we denote by $c_{i}$ general positive number whose value may change from line to line. For each $λ > 0$ , let us consider the functional $J_{λ} : W_{0}^{s, p} (Ω) \to R$ associated with problem (1.1) by the formula $\begin{matrix} J_{λ} (u) = \frac{1}{p} \hat{M} ({[u]}_{s, p}^{p}) - \frac{λ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x - \frac{1}{p} \int_{Ω} b (x) | u |^{p} d x - \int_{Ω} F (x, u) d x, \end{matrix}$ where $\hat{M} (t) = \int_{0}^{t} M (s) d s$ and $F (x, t) = \int_{0}^{t} f (x, s) d s$ . Note that $J_{λ} \in C^{1} (W_{0}^{s, p} (Ω), R)$ and $\begin{array}{rcl} J_{λ}^{'} (u) (v) & = & M ({[u]}_{s, p}^{p}) \iint_{R^{2 N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N + s p}} (v (x) - v (y)) d x d y \\ - λ \int_{Ω} | u |^{p_{s}^{*} - 2} u v d x - \int_{Ω} b (x) | u |^{p - 2} u v d x - \int_{Ω} f (x, u) v d x \end{array}$ for all $u, v \in W_{0}^{s, p} (Ω)$ . This means that we can seek weak solutions of problem (1.1) as the critical points of the functional $J_{λ}$ .

Lemma 2.1.
Assume that the conditions $(M_{1})$ and $(F_{1})$ – $(F_{2})$ hold. Then, any ${(PS)}_{c}$ sequence of $J_{λ}$ is bounded in $W_{0}^{s, p} (Ω)$ .
Proof.
Let ${u_{n}} \subset W_{0}^{s, p} (Ω)$ be a ${(PS)}_{c}$ sequence for the functional $J_{λ}$ , that is $\begin{matrix} (2.1) & J_{λ} (u_{n}) \to c, J_{λ}^{'} (u_{n}) \to 0 in {(W_{0}^{s, p} (Ω))}^{}, n \to \infty . \end{matrix}$ By (2.1), conditions $(M_{1})$ , $(F_{2})$ and the definition of $J_{λ}$ , for n large enough, we deduce that $\begin{array}{l} c + 1 + ‖ u_{n} ‖ ⩾ & J_{λ} (u_{n}) - \frac{σ}{p} J_{λ}^{'} (u_{n}) (u_{n}) \\ = & \frac{1}{p} \hat{M} ({[u_{n}]}_{s, p}^{p}) - \frac{λ}{p_{s}^{}} \int_{Ω} | u_{n} |^{p_{s}^{}} d x - \frac{1}{p} \int_{Ω} b (x) | u_{n} |^{p} d x \\ - \int_{Ω} F (x, u_{n}) d x - \frac{σ}{p} M ({[u_{n}]}_{s, p}^{p}) {[u]}_{s, p}^{p} + \frac{λ σ}{p} \int_{Ω} | u_{n} |^{p_{s}^{}} d x \\ + \frac{σ}{p} \int_{Ω} b (x) | u_{n} |^{p} d x + \frac{σ}{p} \int_{Ω} f (x, u_{n}) u_{n} d x \\ ⩾ & \frac{1}{p} [\hat{M} ({[u]}_{s, p}^{p}) - σ M ({[u]}_{s, p}^{p}) {[u]}_{s, p}^{p}] + λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x \\ - \frac{1 - σ}{p} \int_{Ω} b (x) | u_{n} |^{p} d x + \int_{Ω} (\frac{σ}{p} f (x, u_{n}) u_{n} - F (x, u_{n})) d x \\ ⩾ & λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x - \frac{1 - σ}{p} | b |_{\infty} \int_{Ω} | u_{n} |^{p} d x, \end{array}$ which yields $\begin{matrix} (2.2) & λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x ⩽ \frac{1 - σ}{p} | b |_{\infty} \int_{Ω} | u_{n} |^{p} d x + c + 1 + ‖ u_{n} ‖ . \end{matrix}$ On the other hand, since $p < p_{s}^{}$ , for any $ϵ > 0$ , there exists a positive constant $C_{ϵ} > 0$ depending on ϵ such that $\begin{matrix} | t |^{p} ⩽ ϵ | t |^{p_{s}^{}} + C_{ϵ}, \forall t \in R . \end{matrix}$ From this it follows that $\begin{matrix} λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x ⩽ c + 1 + ‖ u_{n} ‖ + \frac{1 - σ}{p} ϵ | b |_{\infty} \int_{Ω} | u_{n} |^{p_{s}^{}} d x + \frac{1 - σ}{p} C_{ϵ} | b |_{\infty} | Ω |, \end{matrix}$ which implies that $\begin{matrix} [λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) - \frac{1 - σ}{p} ϵ | b |_{\infty}] \int_{Ω} | u_{n} |^{p_{s}^{}} d x ⩽ \frac{1 - σ}{p} C_{ϵ} | b |_{\infty} | Ω | + c + 1 + ‖ u_{n} ‖, \end{matrix}$ and so, if we choose $ϵ > 0$ small enough, it follows that $\begin{matrix} (2.3) & \int_{Ω} | u_{n} |^{p_{s}^{}} d x ⩽ c_{4} (1 + ‖ u_{n} ‖) . \end{matrix}$ By conditions $(F_{1})$ and $(F_{2})$ , we have $\begin{matrix} (2.4) & | F (x, t) | ⩽ \frac{ϵ}{p_{s}^{}} | t |^{p_{s}^{}} + B_{ϵ}, \forall (x, t) \in Ω \times R . \end{matrix}$ It follows from condition $(M_{0})$ , relations (2.1), (2.3) and (2.4), and the definition of the functional $J_{λ}$ that $\begin{array}{l} \frac{m_{0} α}{p (2 | b |_{\infty} + α)} {[u_{n}]}_{s, p}^{p} & ⩽ \frac{1}{p} (m_{0} {[u_{n}]}_{s, p}^{p} - \int_{Ω} b (x) | u_{n} |^{p} d x) \\ ⩽ \frac{1}{p} \hat{M} ({[u_{n}]}_{s, p}^{p}) - \frac{1}{p} \int_{Ω} b (x) | u_{n} |^{p} d x \\ = J_{λ} (u_{n}) + \frac{λ}{p_{s}^{}} \int_{Ω} | u_{n} |^{p_{s}^{}} d x + \int_{Ω} F (x, u_{n}) d x \\ ⩽ c + o_{n} (1) + \frac{λ}{p_{s}^{}} \int_{Ω} | u_{n} |^{p_{s}^{}} d x + \frac{ϵ}{p_{s}^{}} \int_{Ω} | u_{n} |^{p_{s}^{}} d x + B_{ϵ} | Ω | \\ = c + o_{n} (1) + \frac{c_{4}}{p_{s}^{*}} (λ + ϵ) (1 + ‖ u_{n} ‖) + B_{ϵ} | Ω | . \end{array}$ Hence, $\begin{matrix} ‖ u_{n} ‖^{p} ⩽ c_{5} (1 + ‖ u_{n} ‖), \end{matrix}$ where $c_{5}$ is a positive constant. This implies that the sequence ${u_{n}}$ is bounded in $W_{0}^{s, p} (Ω)$ due to $1 < p < N / s$ . □

In order to prove the Palais–Smale condition, we need the following results. Lemma 2.2 ([8, P.161]).

If ${u_{n}} \subset W_{0}^{s, p} (Ω)$ is such that $u_{n} ⇀ u$ in $W_{0}^{s, p} (Ω)$ , and $\begin{matrix} \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \to 0, \end{matrix}$ as $n \to \infty$ , then $u_{n} \to u$ in $W_{0}^{s, p} (Ω)$ as $n \to \infty$ .

Lemma 2.3 ([23, Theorem 2.5]).

Let ${u_{n}}$ be a bounded sequence in $W_{0}^{s, p} (Ω)$ , let $| D^{s} u_{n} |^{p} (x) : = \int_{R^{N}} \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d y$ , for a.e. $x \in R^{N}$ . Then, up to a subsequence, there exists $u \in W_{0}^{s, p} (Ω)$ , two Borel regular measures μ and ν, Λ denumerable, $x_{i} \in Ω$ , $ν_{i} ⩾ 0$ , $μ_{i} ⩾ 0$ with $μ_{i} + ν_{i} > 0$ , such that $\begin{array}{l} u_{n} ⇀ u weakly in W_{0}^{s, p} (Ω), and u_{n} \to u strongly in L^{p} (Ω), \\ {| D^{s} u_{n} |}^{p} \overset{w^{*}}{\to} d μ, | u_{n} |^{p_{s}^{*}} \overset{w^{*}}{\to} d ν, \\ d μ ⩾ {| D^{s} u |}^{p} + \sum_{j \in Λ} μ_{i} δ_{x_{i}}, μ_{i} : = μ ({x_{i}}), \\ d ν = | u |^{p_{s}^{*}} + \sum_{i \in Λ} ν_{i} δ_{x_{i}}, ν_{i} : = ν ({x_{i}}), \\ μ_{i} ⩾ S ν_{i}^{p / p_{s}^{*}}, \end{array}$ where S is defined by ( 1.4 ).

Lemma 2.4.
For any $i \in Λ$ we have $ν_{i} > {(\frac{m_{0} S}{λ})}^{N / (2 p)}$ or $ν_{i} = 0$ , where $m_{0}$ is given by the condition $(M_{0})$ .
Proof.
Take $ψ \in C_{0}^{\infty} (Ω)$ such that $0 ⩽ ψ ⩽ 1$ ; $ψ \equiv 1$ in $B (x_{i}, ε)$ , $ψ (x) = 0$ in $Ω ∖ B (x_{i}, 2 ε)$ and $| \nabla ψ | ⩽ 2 / ε$ in Ω. For any $ε > 0$ , define $ψ_{ε}^{i} (x) = ψ (\frac{x - x_{i}}{ε})$ , where $i \in I$ . It follows from $⟨ J_{λ}^{'} (u_{n}), u_{n} ψ_{ε}^{i} ⟩ \to 0$ , that is, $\begin{array}{l} M ({[u_{n}]}_{s, p}^{p}) \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (ψ_{ε}^{i} (x) - ψ_{ε}^{i} (y)) u_{n} (x)}{| x - y |^{N + p s}} d x d y \\ + M ({[u_{n}]}_{s, p}^{p}) \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} ψ_{ε}^{i} (y) d x d y \\ (2.5) & = λ \int_{Ω} u_{n}^{p_{s}^{}} ψ_{ε}^{i} (x) d x + \int_{Ω} b (x) | u_{n} |^{p} ψ_{ε}^{i} d x + \int_{Ω} f (x, u_{n}) u_{n} ψ_{ε}^{i} (x) d x + o (1) . \end{array}$ Using the Hölder inequality, we deduce $\begin{array}{l} lim_{ε \to 0} lim_{n \to \infty} | \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) (ψ_{ε}^{i} (x) - ψ_{ε}^{i} (y)) u_{n} (x)}{| x - y |^{N + p s}} d x d y | \\ ⩽ C lim_{ε \to 0} lim_{n \to \infty} {(\iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p}}{| x - y |^{N + p s}} d x d y)}^{(p - 1) / p} \\ \times {(\iint_{R^{2 N}} \frac{| (ψ_{ε}^{i} (x) - ψ_{ε}^{i} (y)) u_{n} (x) |^{p}}{| x - y |^{N + p s}} d x d y)}^{1 / p} \\ (2.6) & ⩽ C lim_{ε \to 0} lim_{n \to \infty} {(\iint_{R^{2 N}} \frac{| (ψ_{ε}^{i} (x) - ψ_{ε}^{i} (y)) u_{n} (x) |^{p}}{| x - y |^{N + p s}} d x d y)}^{1 / p} . \end{array}$ Similar to the proof of Lemma 2.1 in [32], we claim that $\begin{array}{rcl} (2.7) & lim_{ε \to 0} lim_{n \to \infty} \iint_{R^{2 N}} \frac{| (ψ_{ε}^{i} (x) - ψ_{ε}^{i} (y)) u_{n} (x) |^{p}}{| x - y |^{N + p s}} d x d y = 0 . \end{array}$ On the other hand, by the compactness lemma of Strauss, the boundedness of ${u_{n}}$ in $W_{0}^{s, p} (Ω)$ and Sobolev embedding, it follows that $\begin{matrix} (2.8) & \int_{Ω} f (x, u_{n}) u_{n} ψ_{ε}^{i} d x = 0, as n \to \infty, ε \to 0 \end{matrix}$ and $\begin{matrix} (2.9) & \int_{Ω} b (x) | u_{n} |^{p - 2} u_{n} ψ_{ε}^{i} d x = 0, as n \to \infty, ε \to 0 . \end{matrix}$ Letting $ε \to 0$ and using the standard theory of Radon measures, we conclude that $λ ν_{i} ⩾ m_{0} μ_{i}$ . Using the Lemma 2.3 we have $\begin{matrix} (2.10) & \frac{λ ν_{i}}{m_{0} S} ⩾ \frac{μ_{i}}{S} ⩾ ν_{i}^{\frac{N - 2 p}{N}}, \end{matrix}$ which implies that $ν_{i} = 0$ or $ν_{i} ⩾ {(\frac{m_{0} S}{λ})}^{N / (s p)}$ for all $i \in Λ$ . The proof of Lemma 2.3 is now complete. □
Lemma 2.5.
Assume that conditions* $(M_{0})$ – $(M_{1})$ and $(F_{0})$ – $(F_{2})$ hold. Then the functional $J_{λ}$ satisfies the ${(PS)}_{c}$ condition for $c \in (0, λ^{1 - N / (s p)} (\frac{σ}{p} - \frac{1}{p_{s}^{}}) {(m_{0} S)}^{N / (s p)})$ .
Proof.
Let ${u_{n}} \subset W_{0}^{s, p} (Ω)$ be a sequence such that $\begin{matrix} (2.11) & J_{λ} (u_{n}) = c + o_{n} (1), J_{λ}^{'} (u_{n}) = o_{n} (1) . \end{matrix}$ From conditions $(M_{1})$ , $(F_{2})$ and relation (2.11), we get $\begin{array}{l} o_{n} (1) + c = & J_{λ} (u_{n}) - \frac{σ}{p} J_{λ}^{'} (u_{n}) (u_{n}) \\ = & \frac{1}{p} \hat{M} ({[u_{n}]}_{s, p}^{p}) - \frac{λ}{p_{s}^{}} \int_{Ω} | u_{n} |^{p_{s}^{}} d x - \frac{1}{p} \int_{Ω} b (x) | u_{n} |^{p} d x - \int_{Ω} F (x, u_{n}) d x \\ - \frac{σ}{p} M ({[u_{n}]}_{s, p}^{p}) {[u_{n}]}_{s, p}^{p} + \frac{λ σ}{p} \int_{Ω} | u_{n} |^{p_{s}^{}} d x + \frac{σ}{p} \int_{Ω} b (x) | u_{n} |^{p} d x \\ + \frac{σ}{p} \int_{Ω} f (x, u_{n}) u_{n} d x \\ = & \frac{1}{p} [\hat{M} ({[u_{n}]}_{s, p}^{p}) - σ M ({[u_{n}]}_{s, p}^{p}) {[u_{n}]}_{s, p}^{p}] + λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x \\ + \frac{σ - 1}{p} \int_{Ω} b (x) | u_{n} |^{p} d x + \int_{Ω} (\frac{σ}{p} f (x, u_{n}) u_{n} - F (x, u_{n})) d x \\ ⩾ & λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x + \frac{σ - 1}{p} \int_{Ω} b (x) | u_{n} |^{p} d x \\ (2.12) & ⩾ & λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) \int_{Ω} | u_{n} |^{p_{s}^{}} d x, \end{array}$ since $σ = 1$ for all $x \in Ω_{b}^{+} : = {x \in Ω : b (x) > 0}$ and $σ < 1$ if $x \in Ω ∖ Ω_{b}^{+}$ . Letting $n \to \infty$ in (2.12), we get $\begin{matrix} (2.13) & c ⩾ λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{}} d x . \end{matrix}$ By Lemma 2.3, we have $\begin{matrix} lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{}} d x = \int_{Ω} | u |^{p_{s}^{}} d x + \sum_{i \in Λ} ν_{i} ⩾ ν_{i}, \forall i \in Λ . \end{matrix}$ If $ν_{s} > 0$ for some $s \in Λ$ , we deduce from (2.13) that $\begin{array}{l} c & ⩾ λ (\frac{σ}{p} - \frac{1}{p_{s}^{}}) {(\frac{m_{0} S}{λ})}^{\frac{N}{s p}} \\ = λ^{1 - \frac{N}{s p}} (\frac{σ}{p} - \frac{1}{p_{s}^{}}) {(m_{0} S)}^{\frac{N}{s p}}, \end{array}$ which is absurd. This leads to the fact that $ν_{i} = 0$ for any $i \in Λ$ , thus $\begin{matrix} (2.14) & lim_{n \to \infty} \int_{Ω} | u_{n} |^{p_{s}^{}} d x = \int_{Ω} | u |^{p_{s}^{}} d x \end{matrix}$ and by the Brézis–Lieb lemma [3], the sequence ${u_{n}}$ converges strongly to u in $L^{p_{s}^{}} (Ω)$ . For this reason, by the Hölder inequality we deduce that $\begin{array}{l} | \int_{Ω} | u_{n} |^{p_{s}^{} - 2} u_{n} (u_{n} - u) d x | & ⩽ \int_{Ω} | u_{n} |^{p_{s}^{} - 1} | u_{n} - u | d x \\ ⩽ | u_{n} |_{L^{p_{s}^{}} (Ω)}^{p_{s}^{} - 1} | u_{n} - u |_{L^{p_{s}^{}} (Ω)} \\ (2.15) & \to 0, as n \to \infty . \end{array}$ By condition $(F_{1})$ and the compactness of the embedding $W_{0}^{s, p} (Ω) ↪ L^{p} (Ω)$ , we also have that the following relations hold $\begin{array}{l} | \int_{Ω} b (x) | u_{n} |^{p - 2} u_{n} (u_{n} - u) d x | & ⩽ | b |_{\infty} \int_{Ω} | u_{n} |^{p - 1} | u_{n} - u | d x \\ ⩽ | b |_{\infty} | u_{n} |_{p}^{p - 1} | u_{n} - u |_{p} \\ (2.16) & \to 0, as n \to \infty \end{array}$ and $\begin{array}{l} | \int_{Ω} f (x, u_{n}) (u_{n} - u) d x | & ⩽ \int_{Ω} | f (x, u_{n}) | | u_{n} - u | d x \\ ⩽ c_{9} \int_{Ω} (1 + | u_{n} |^{p - 1}) | u_{n} - u | d x \\ ⩽ c_{9} (| Ω |^{\frac{p - 1}{p}} + | u_{n} |_{p}^{p - 1}) | u_{n} - u |_{p} \\ (2.17) & \to 0, as n \to \infty . \end{array}$ Since the sequence ${u_{n}}$ converges weakly to u in $W_{0}^{s, p} (Ω)$ , the sequence ${u_{n} - u}$ is bounded in $W_{0}^{s, p} (Ω)$ and $J_{λ}^{'} (u_{n}) (u_{n} - u) \to 0$ as $n \to \infty$ , that is, $\begin{array}{l} J_{λ}^{'} (u_{n}) (u_{n} - u) \\ = M ({[u_{n}]}_{s, p}^{p}) \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \\ - λ \int_{Ω} | u_{n} |^{p_{s}^{} - 2} u_{n} (u_{n} - u) d x - \int_{Ω} b (x) | u_{n} |^{p - 2} u_{n} (u_{n} - u) d x \\ (2.18) & - \int_{Ω} f (x, u_{n}) (u_{n} - u) d x \to 0, as n \to \infty . \end{array}$ From (2.15)–(2.18), we have $\begin{array}{l} lim_{n \to \infty} M ({[u_{n}]}_{s, p}^{p}) \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y \\ = 0 \end{array}$ and by $(M_{0})$ it follows that $\begin{matrix} (2.19) & lim_{n \to \infty} \iint_{R^{2 N}} \frac{| u_{n} (x) - u_{n} (y) |^{p - 2} (u_{n} (x) - u_{n} (y)) ((u_{n} - u) (x) - (u_{n} - u) (y))}{| x - y |^{N + p s}} d x d y = 0 . \end{matrix}$ By Lemma 2.2, some standard arguments help us to show that ${u_{n}}$ converges strongly to u in $W_{0}^{s, p} (Ω)$ and thus, $J_{λ}$ satisfies the ${(PS)}_{c}$ condition for $c \in (0, λ^{1 - \frac{N}{s p}} (\frac{σ}{p} - \frac{1}{p_{s}^{}}) {(m_{0} S)}^{\frac{N}{s p}})$ . □

3. Proof of the main result

In this section, we prove the main results Theorem 1.1. Let X be a real Banach space and Σ the family of set $E \subset X ∖ {0}$ such that E is closed in X and symmetric with respect to 0, that is, $\begin{matrix} Σ = {E \subset X ∖ {0} : E is closed in X and E = - E} . \end{matrix}$

For each $E \in Σ$ , we say genus of E is a number k denoted by $γ (E) = k$ if there is an odd map $h \in C (E, R^{N} ∖ {0})$ and k is the smallest integer with this property.

Now, let us recall a version of the mountain pass theorem for even functionals, the readers can consult its proof in [26]. This is also the main tool for proving Theorem 1.1.

Proposition 3.1.

Let X be an infinite dimensional Banach space with $X = V \oplus Y$ , where V is finite dimensional and let $J \in C^{1} (X, R)$ be an even functional with $J (0) = 0$ such that the following conditions hold:

There exist positive constants $β, ρ > 0$ such that $J (u) ⩾ β$ for all $u \in \partial B_{ρ} (0) \cap Y$ .

There exists $c^{*} > 0$ such that J satisfies the ${(PS)}_{c}$ condition for $0 < c < c^{*}$ .

For each finite dimensional subspace $\hat{X} \subset X$ , there exists $R = R (\hat{X})$ such that $J (u) ⩽ 0$ for all $u \in \hat{X} ∖ B_{R} (0)$ .

Suppose that V is k dimensional and $V = span {e_{1}, e_{2}, \dots, e_{k}}$ . For $n ⩾ k$ , inductively choose $e_{n + 1} \notin X_{n} : = span {e_{1}, e_{2}, \dots, e_{n}}$ . Let $R_{n} = R (X_{n})$ and $D_{n} = B_{R_{n}} (0) \cap X_{n}$ . Define $\begin{matrix} G_{n} : = {h \in C (D_{n}, X) : h is odd and h (u) = u, \forall \partial B_{R_{n}} (0) \cap X_{n}} \end{matrix}$ and $\begin{matrix} Γ_{j} : = {h (\overline{D_{n} ∖ E}) : h \in G_{n}, n ⩾ j, E \in Σ and γ (E) ⩽ n - j} . \end{matrix}$ For each $j \in N$ , let $\begin{matrix} c_{j} : = inf_{K \in Γ_{j}} max_{u \in K} J (u) . \end{matrix}$

Then, $0 < β ⩽ c_{j} ⩽ c_{j + 1}$ for $j > k$ , and if $j > k$ and $c_{j} < c^{*}$ , then we conclude that $c_{j}$ is the critical value of J. Moreover, if $c_{j} = c_{j + 1} = \dots = c_{j + l} = c < c^{*}$ for $j > k$ , then $γ (K_{c}) ⩾ l + 1$ , where $\begin{matrix} K_{c} : = {u \in E : J (u) = c and J^{'} (u) = 0} . \end{matrix}$

Lemma 3.1.

Assume that conditions $(M_{0})$ and $(F_{1})$ hold, then the functional $J_{λ}$ satisfies the hypothesis $(I_{1})$ .

Proof.

Let $δ > 0$ , for all $u \in W_{0}^{s, p} (Ω)$ , we have $\begin{array}{l} m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x = & \frac{1 + δ}{1 + δ} [m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x] \\ = & \frac{1}{1 + δ} [m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x] \\ + \frac{δ}{1 + δ} m_{0} {[u]}_{s, p}^{p} - \frac{δ}{1 + δ} \int_{Ω} b (x) | u |^{p} d x . \end{array}$ Using condition $(A_{0})$ , it follows that $\begin{array}{l} m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x & ⩾ \frac{α}{1 + δ} \int_{Ω} | u |^{p} d x + \frac{δ}{1 + δ} m_{0} {[u]}_{s, p}^{p} - \frac{δ}{1 + δ} \int_{Ω} b (x) | u |^{p} d x \\ = \frac{δ}{1 + δ} m_{0} {[u]}_{s, p}^{p} + \frac{1}{1 + δ} \int_{Ω} (α - δ b (x)) | u |^{p} d x . \end{array}$ Since $b \in L^{\infty} (Ω)$ , one has that $α - δ b (x) ⩾ α - δ | b |_{\infty}$ . Choosing $δ = \frac{α}{2 | b |_{\infty}} > 0$ , from the above information, we deduce that for all $u \in W_{0}^{s, p} (Ω)$ , $\begin{matrix} (3.1) & m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x ⩾ \frac{m_{0} α}{2 | b |_{\infty} + α} {[u]}_{s, p}^{p} + \frac{2 α | b |_{\infty}}{2 | b |_{\infty} + α} \int_{Ω} | u |^{p} d x . \end{matrix}$ By the condition $(F_{1})$ , for any $ϵ > 0$ , there exists a positive constant $C_{ϵ} > 0$ such that $\begin{matrix} (3.2) & | F (x, t) | ⩽ \frac{ϵ}{p} | t |^{p} + \frac{C_{ϵ}}{p_{s}^{*}} | t |^{p_{s}^{*}}, \forall (x, t) \in Ω \times R . \end{matrix}$ Thus, condition $(M_{0})$ and relation (3.2) imply that $\begin{array}{l} J_{λ} (u) & = \frac{1}{p} \hat{M} ({[u]}_{s, p}^{p}) - \frac{λ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x - \frac{1}{p} \int_{Ω} b (x) | u |^{p} d x - \int_{Ω} F (x, u) d x \\ ⩾ \frac{1}{p} (m_{0} {[u]}_{s, p}^{p} - \int_{Ω} b (x) | u |^{p} d x) - \frac{λ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x - \int_{Ω} (\frac{ϵ}{p} | u |^{p} + \frac{C_{ϵ}}{p_{s}^{*}} | u |^{p_{s}^{*}}) d x \\ ⩾ \frac{m_{0} α}{p (2 | b |_{\infty} + α)} {[u]}_{s, p}^{p} + (\frac{α | b |_{\infty}}{p (2 | b |_{\infty} + α)} - \frac{ϵ}{p}) \int_{Ω} | u |^{p} d x - \frac{λ + C_{ϵ}}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x . \end{array}$ Let $ϵ \in (0, \frac{α | b |_{\infty}}{2 | b |_{\infty} + α})$ be small enough, by the continuous embedding $X ↪ L^{p_{s}^{*}} (Ω)$ , we obtain that $\begin{matrix} (3.3) & S^{\frac{1}{p}} | u |_{p_{s}^{*}} ⩽ ‖ u ‖, \forall u \in W_{0}^{s, p} (Ω), \end{matrix}$ where S is given by (1.4). Consequently, for all $u \in W_{0}^{s, p} (Ω)$ with $‖ u ‖ = ρ$ , $ρ \in (0, 1)$ , it follows that $\begin{array}{l} J_{λ} (u) & ⩾ \frac{m_{0} α}{p (2 | b |_{\infty} + α)} ‖ u ‖^{p} - \frac{S^{\frac{1}{p}} (λ + C_{ϵ})}{p_{s}^{*}} ‖ u ‖^{p_{s}^{*}} \\ = (\frac{m_{0} α}{p (2 | b |_{\infty} + α)} - \frac{S^{\frac{1}{p}} (λ + C_{ϵ})}{p_{s}^{*}} ‖ u ‖^{p_{s}^{*} - p}) ‖ u ‖^{p} . \end{array}$ Since $1 < p < p_{s}^{*}$ , there exists $β > 0$ such that $J_{λ} (u) ⩾ β$ for all $u \in W_{0}^{s, p} (Ω)$ with $‖ u ‖ = ρ$ , where ρ is chosen sufficiently small. □

Lemma 3.2.

Assume that conditions $(M_{1})$ and $(F_{1})$ hold. Then the functional $J_{λ}$ satisfies the hypothesis $(I_{3})$ .

Proof.

Let E be a finite dimensional subspace of $W_{0}^{s, p} (Ω)$ . By the condition $(F_{1})$ , for $ϵ > 0$ , there exists $B_{ϵ} > 0$ such that $\begin{matrix} (3.4) & F (x, t) ⩾ - B_{ϵ} - \frac{ϵ}{p_{s}^{*}} | t |^{p_{s}^{*}}, \forall (x, t) \in Ω \times R . \end{matrix}$ From $(M_{1})$ , there exists $c_{1} > 0$ such that $\begin{matrix} (3.5) & \hat{M} (t) ⩽ \frac{M (t_{0})}{t_{0}^{\frac{1}{σ}}} t^{\frac{1}{σ}} = c_{1} t^{\frac{1}{σ}}, \forall t ⩾ t_{0} > 0 . \end{matrix}$ From (3.4) and (3.5), for any $u \in E$ with $‖ u ‖$ large enough, we have $\begin{array}{l} J_{λ} (u) & ⩽ \frac{c_{1}}{p} {[u]}_{s, p}^{\frac{p}{σ}} - \frac{λ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x + \frac{| b |_{\infty}}{p} \int_{Ω} | u |^{p} d x + \frac{ϵ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x + B_{ϵ} | Ω | \\ = \frac{c_{1}}{p} ‖ u ‖^{\frac{p}{σ}} - \frac{1}{p_{s}^{*}} (λ - ϵ) \int_{Ω} | u |^{p_{s}^{*}} d x + \frac{c_{2} | b |_{\infty}}{p} ‖ u ‖^{p} + B_{ϵ} | Ω | . \end{array}$ Since $dim (E) < + \infty$ , the norms $‖ \cdot ‖$ and $| \cdot |_{p_{s}^{*}}$ are equivalent in E. This means that there exists $c_{3} > 0$ such that $\begin{matrix} ‖ u ‖ ⩽ c_{3} | u |_{p_{s}^{*}}, \forall u \in W_{0}^{s, p} (Ω) . \end{matrix}$ Choosing $ϵ = λ / 2 > 0$ , we obtain from the above information that $\begin{matrix} J_{λ} (u) ⩽ \frac{c_{1}}{p} ‖ u ‖^{\frac{p}{σ}} - \frac{λ}{2 p_{s}^{*} c_{2}^{p_{s}^{*}}} ‖ u ‖^{p_{s}^{*}} + \frac{c_{3} | b |_{\infty}}{p} ‖ u ‖^{p} + B_{ϵ} | Ω |, \forall u \in E . \end{matrix}$ Since $1 < p / σ < p_{s}^{*}$ , we conclude that $J_{λ} (u) < 0$ for all $u \in E$ with $‖ u ‖ ⩾ R$ , where R is chosen large enough. □

Lemma 3.3.

Assume that the conditions $(M_{0})$ – $(M_{1})$ , and $(F_{1})$ – $(F_{2})$ hold. Then, there exists a sequence ${M_{n}} \subset (0, + \infty)$ independent of λ, with $M_{n} ⩽ M_{n + 1}$ , such that for any $λ > 0$ , $\begin{matrix} c_{n}^{λ} : = inf_{K \in Γ_{n}} max_{u \in K} J_{λ} (u) < M_{n} . \end{matrix}$

Proof.

Our proof is similar to that presented in [33, Lemma 5]. From the definition of $c_{n}^{λ}$ , we deduce that $\begin{array}{l} c_{n}^{λ} & = inf_{K \in Γ_{n}} max_{u \in K} J_{λ} (u) \\ = inf_{K \in Γ_{n}} max_{u \in K} {\frac{1}{p} \hat{M} ({[u]}_{s, p}^{p}) - \frac{λ}{p_{s}^{*}} \int_{Ω} | u |^{p_{s}^{*}} d x - \frac{1}{p} \int_{Ω} b (x) | u |^{p} d x - \int_{Ω} F (x, u) d x} \\ ⩽ inf_{K \in Γ_{n}} max_{u \in K} {\frac{1}{p} \hat{M} ({[u]}_{s, p}^{p}) - \frac{1}{p} \int_{Ω} b (x) | u |^{p} d x - \int_{Ω} F (x, u) d x} . \end{array}$ Set $\begin{matrix} M_{n} : = inf_{K \in Γ_{n}} max_{u \in K} {\frac{1}{p} \hat{M} ({[u]}_{s, p}^{p}) - \frac{1}{p} \int_{Ω} b (x) | u |^{p} d x - \int_{Ω} F (x, u) d x}, \end{matrix}$ then we conclude that $M_{n} < + \infty$ and $M_{n} ⩽ M_{n + 1}$ by the definition of $Γ_{n}$ . □

Proof of Theorem 1.1.

By choosing for each $k ⩾ 1$ , $λ_{k}$ is sufficiently small, we construct a sequence of positive real numbers ${λ_{k}}$ , with $λ_{k} < λ_{k - 1}$ such that $\begin{matrix} M_{k} < λ_{k}^{1 - \frac{N}{s p}} (\frac{σ}{p} - \frac{1}{p_{s}^{*}}) {(m_{0} S)}^{\frac{N}{s p}} . \end{matrix}$ Therefore, for $λ \in (λ_{k}, λ_{k - 1}]$ , by Lemma 3.3 we get $\begin{matrix} 0 < c_{1}^{λ} ⩽ c_{2}^{λ} ⩽ \dots ⩽ c_{k}^{λ} < M_{k} < λ^{1 - \frac{N}{s p}} (\frac{σ}{p} - \frac{1}{p_{s}^{*}}) {(m_{0} S)}^{\frac{N}{s p}} . \end{matrix}$ From Proposition 3.1, the levels $c_{1}^{λ} ⩽ c_{2}^{λ} ⩽ \dots ⩽ c_{k}^{λ}$ are critical values of $J_{λ}$ . So, if $c_{1}^{λ} < c_{2}^{λ} < \dots < c_{k}^{λ}$ , the functional $J_{λ}$ has at least k critical points. Now, if $c_{j}^{λ} = c_{j + 1}^{λ}$ for some $j = 1, 2, \dots, k - 1$ , again Proposition 3.1 implies that $K_{c_{j}^{λ}}$ is an infinite set [26, Chapter 7] and hence in this case, problem (1.1) has infinitely many weak solutions. Consequently, problem (1.1) has at least k pair of weak solutions. □

Footnotes

Acknowledgements

S. Liang was supported by the National Natural Science Foundation of China (Grant No.11301038), the Natural Science Foundation of Jilin Province (Grant No.20160101244JC). B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199).

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