Existence and multiplicity of solutions for Schrödinger–Kirchhoff type problems involving the fractional p ( ⋅ ) $p(\cdot )$ -Laplacian in R N $\mathbb{R}^{N}$

Abstract

Abstract We are concerned with the following elliptic equations with variable exponents: M ( [ u ] s , p ( ⋅ , ⋅ ) ) L u ( x ) + V ( x ) | u | p ( x ) − 2 u = λ ρ ( x ) | u | r ( x ) − 2 u + h ( x , u ) in R N , $$ M \bigl([u]_{s,p(\cdot,\cdot)} \bigr)\mathcal{L}u(x) +\mathcal {V}(x) \vert u \vert ^{p(x)-2}u =\lambda\rho(x) \vert u \vert ^{r(x)-2}u + h(x,u) \quad \text{in } \mathbb {R}^{N}, $$ where [ u ] s , p ( ⋅ , ⋅ ) : = ∫ R N ∫ R N | u ( x ) − u ( y ) | p ( x , y ) p ( x , y ) | x − y | N + s p ( x , y ) d x d y $[u]_{s,p(\cdot,\cdot)}:=\int_{\mathbb {R}^{N}}\int_{\mathbb {R}^{N}} \frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+sp(x,y)}} \,dx \,dy$ , the operator L $\mathcal{L}$ is the fractional p ( ⋅ ) $p(\cdot)$ -Laplacian, p , r : R N → ( 1 , ∞ ) $p, r: {\mathbb {R}^{N}} \to(1,\infty)$ are continuous functions, M ∈ C ( R + ) $M \in C(\mathbb {R}^{+})$ is a Kirchhoff-type function, the potential function V : R N → ( 0 , ∞ ) $\mathcal {V}:\mathbb {R}^{N} \to(0,\infty)$ is continuous, and h : R N × R → R $h:\mathbb {R}^{N}\times\mathbb {R} \to\mathbb {R}$ satisfies a Carathéodory condition. Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities. To do this, we use the mountain pass theorem and variant of the Ekeland variational principle as the main tools.

Keywords

• Fractional p ( x ) $p(x)$ -Laplacian
• Variable exponent fractional Sobolev spaces
• Weak solution
• Mountain pass theorem
• Ekeland’s variational principle