Generalized Nehari manifold and semilinear Schr\"odinger equation with weak monotonicity condition on the nonlinear term

We study the Schr"odinger equations $-\mathrm{\Delta }u+V(x)u=f(x,u)$ in ${\mathbb{R}}^N$ and $-\mathrm{\Delta }u-\lambda u=f(x,u)$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ is nondecreasing. In ${\mathbb{R}}^N$ we also assume that $V$ and $f$ are periodic in $x_1,\dots ,x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those in \cite{sw1} where $u\mapsto f(x,u)\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.