Infinitely many solutions for super-quadratic Kirchhoff-type equations with sign-changing potential

This paper is concerned with the following Kirchhoff type equation This paper is concerned with the following Kirchhoff type equation {-(a + b integral(R3) vertical bar del u vertical bar(2) dx) Delta u + V(x)u = f (x, u), x is an element of R-3, u(x) -> 0, vertical bar x vertical bar -> infinity, where a > 0, b >= 0, V is an element of C(R-3, R), f is an element of C(R-3 x R, R), V(x) and f(x, u)u are both allowed to be sign-changing. Using a weaker assumption lim(vertical bar t vertical bar ->infinity) integral(t)(0) f(x, s)ds/vertical bar t vertical bar(3) = infinity uniformly in x is an element of R-3, instead of the common assumption lim(vertical bar t vertical bar ->infinity) integral(t)(0) f(x, s)ds/vertical bar t vertical bar(4) = infinity uniformly in x is an element of R-3, we establish the existence of infinitely many high energy solutions of the above problem, where some new tricks are introduced to overcome the competing effect of nonlocal term. Our result unifies both asymptotically cubic or super-cubic cases, which generalizes and improves the existing ones. (C) 2016 Elsevier Ltd. All rights reserved.