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

TL;DR

This paper establishes the existence of ground state and infinitely many solutions for certain Schrödinger equations with nonlinearities that are weakly monotonic, extending previous results by relaxing the strict monotonicity condition.

Contribution

It introduces a generalized Nehari manifold approach to handle weak monotonicity in nonlinear terms, broadening the class of Schrödinger equations that can be analyzed.

Findings

Existence of ground state solutions in periodic and bounded domains.

Infinitely many solutions when the nonlinearity is odd in u.

Extension of previous results to weaker monotonicity conditions.

Abstract

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 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 in \cite{sw1} 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.