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

Abstract

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations
$$
\left\{
\begin{array}{ll}
-\Delta u+V(x)u= K(x)f(u),\,\,x\in\mathbb{R}^N,\\
u\in D^{1,2}(\R^N),
\end{array}
\right.
$$
where $N\geq 3$, $V(x)$ 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.