Existence and Multiplicity of Nontrivial Solutions for Nonlinear Schrödinger Equations with Unbounded Potentials

Jianhua Chen, Xianjiu Huang, Bitao Cheng, Huxiao Luo

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations −∆u + V(x)u = K(x) f (u), x ∈ R N , u ∈ D 1,2 (R N), where N ≥ 3, V(x) and K(x) are both unbounded potential functions and f is a function with a super-quadratic growth. Firstly, we prove the existence of infinitely many solutions with compact embedding and by means of symmetric mountain pass theorem. Moreover, we prove the existence of nontrivial solutions without compact embedding in weighted Sobolev spaces and by means of mountain pass theorem. Our results extend and generalize some existing results.