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

This paper is concerned with the following Kirchhoff type equation −a+b∫R3|∇u|2dx△u+V(x)u=f(x,u),x∈R3,u(x)→0,|x|→∞,where a>0,b≥0, V∈C(R3,R), f∈C(R3×R,R), V(x) and f(x,u)u are both allowed to be sign-changing. Using a weaker assumption lim|t|→∞∫0tf(x,s)ds|t|3=∞ uniformly in x∈R3, instead of the common assumption lim|t|→∞∫0tf(x,s)ds|t|4=∞ uniformly in x∈R3, 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.