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

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