Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term

Abstract:

We study the Schrödinger 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 of subcritical 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 that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802–3822], where $u\mapsto f(x,u)/|u|$ was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis.