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Generalized Nehari manifold and semilinear Schrödinger equation with weak monotonicity condition on the nonlinear term
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study the Schr\"odinger equations $-\Delta u + V(x)u = f(x,u)$ in $\rn$ and $-\Delta u - \lambda u = f(x,u)$ in a bounded domain $\Omega\subset\rn$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)/|u|$ is nondecreasing. In $\rn$ 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.

Keyword [en]
Generalized Nehari manifold, Schrödinger equation, strongly indefinite functional, Clarke's subdifferential
National Category
Natural Sciences
Research subject
URN: urn:nbn:se:su:diva-134102OAI: diva2:1001159
Available from: 2016-09-30 Created: 2016-09-30 Last updated: 2016-09-30

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