Infinitely many solutions for a Schrödinger–Kirchhoff equations with concave and convex nonlinearities

Abstract

This paper is concerned with the following Schrödinger–Kirchhoff equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δu + V ( x ) u = k ( x ) | u | q − 2 u − h ( x ) | u | p − 2 u , u ∈ H 1 ( R 3 ) , where a and b are positive constants, 1 < q < 2 < p < + ∞ . Under some suitable assumptions on V ( x ) , k ( x ) and h ( x ) , we obtain the existence of multiple solutions for this problem via a variant of Clark's theorem.