A Kirchhoff Type Equation in $$\pmb {\mathbb {R}}^{N}$$ Involving the fractional (p, q)-Laplacian

Abstract

AbstractIn this paper, we deal with the following class of fractional (p, q)-Laplacian Kirchhoff type problem: $$\begin{aligned} \left\{ \begin{array}{ll} \left( 1+[u]_{s,p}^{p}\right) (-\Delta )_{p}^{s}u+ \left( 1+[u]^{q}_{s, q}\right) (-\Delta )_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) &{} \text{ in } \mathbb {R}^{N}, \\ u\in W^{s, p}(\mathbb {R}^{N})\cap W^{s,q}(\mathbb {R}^{N}), \quad u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$ 1 + [ u ] s , p p ( - Δ ) p s u + 1 + [ u ] s , q q ( - Δ ) q s u + V ( ε x ) ( | u | p - 2 u + | u | q - 2 u ) = f ( u ) in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 in R N , where $$\varepsilon >0$$ ε > 0 , $$s\in (0, 1)$$ s ∈ ( 0 , 1 ) , $$1<p<q<\frac{N}{s}<2q$$ 1 < p < q < N s < 2 q , $$(-\Delta )_{t}^{s}$$ ( - Δ ) t s , with $$t\in \{p, q\}$$ t ∈ { p , q } , is the fractional t-Laplacian operator, $$V:\mathbb {R}^{N}\rightarrow \mathbb {R}$$ V : R N → R is a positive continuous potential such that $$\inf _{\partial \Lambda }V>\inf _{\Lambda } V$$ inf ∂ Λ V > inf Λ V for some bounded open set $$\Lambda \subset \mathbb {R}^{N}$$ Λ ⊂ R N , and $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R is a superlinear continuous nonlinearity with subcritical growth at infinity. By combining the method of Nehari manifold, a penalization technique, and the Lusternik–Schnirelman category theory, we study the multiplicity and concentration properties of solutions for the above problem when $$\varepsilon \rightarrow 0$$ ε → 0 .