Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RNRN

In the present paper, the following Schrödinger–Kirchhoff-type problem: equation(1.1)−(a+b∫RN|∇u|2dx)Δu+V(x)u=f(x,u),inRN is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem.