NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION
NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION
Abstract We consider the semilinear Schrödinger equation $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-\triangle u+V(x)u=f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\end{array}\right.\end{eqnarray}$$ where $f(x,u)$ is asymptotically linear with respect to $u$ , $V(x)$ is 1-periodic in each of $x_{1},x_{2},\dots ,x_{N}$ and $\sup [{\it\sigma}(-\triangle +V)\cap (-\infty ,0)]<0<\inf [{\it\sigma}(-\triangle +V)\cap (0,\infty )]$ . We develop a direct approach to find ground …