Solutions for Schrödinger equations with variable separated type nonlinear terms

In this paper, we consider the following semilinear Schrödinger equation: $ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = a(x)g(u)&{\mbox{for}}\; x\in \mathbb{R}^{N} ,\\ u(x)\rightarrow0&{\mbox{as}}\; |x|\rightarrow \infty , \end{array} \right. \end{eqnarray*} $ where $ a(x) > 0 $ for all $ \mathbb{R}^{N} $. Under some different superlinear conditions on $ g(u) $, we obtain the existence of solutions for the above problem. In order to regain the compactness of the Sobolev embedding, a competing condition between $ a(x) $ and $ V(x) $ is introduced.