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Ground state solutions for some indefinite variational problems
Stockholm University, Faculty of Science, Department of Mathematics. matematik.
Institut für Mathematik, Goethe-Universität Frankfurt, Germany.
2009 (English)In: Journal of Functional Analysis, ISSN 0022-1236, Vol. 257, no 12, 21 p.3802-3822 p.Article in journal (Refereed) Published
Abstract [en]

We consider the nonlinear stationary Schr\"odinger equation

$-\Delta u + V(x)u = f(x,u)$ in $\rn$. Here $f$ is

a superlinear, subcritical nonlinearity, and we mainly study the case

where both $V$ and $f$ are periodic in $x$ and $0$ belongs to a

spectral gap of $-\Delta +V$. Inspired by previous work of Li et

al. \cite{lwz} and Pankov \cite{pa}, we develop an

approach to find ground state solutions, i.e., nontrivial solutions with least

possible energy. The approach is based on a direct and surprisingly

simple reduction of the indefinite variational problem to a definite

one and gives rise to a new minimax characterization of the

corresponding critical value. Our method works for merely

continuous nonlinearities $f$ which are allowed to have weaker

asymptotic growth than usually assumed. For odd $f$, we obtain infinitely many geometrically distinct solutions.

The approach also yields new existence and multiplicity results for

the Dirichlet problem for the same type of equations in a bounded


Place, publisher, year, edition, pages
Amsterdam: Elsevier , 2009. Vol. 257, no 12, 21 p.3802-3822 p.
Keyword [en]
Schrödingerequation; Strongly indefinite functional; Minimax principle; Ground state
National Category
Mathematical Analysis
Research subject
URN: urn:nbn:se:su:diva-20670DOI: 10.1016/j.jfa.2009.09.013ISI: 000271728900003OAI: diva2:187196
Available from: 2007-11-28 Created: 2007-11-28 Last updated: 2010-01-04Bibliographically approved

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Szulkin, Andrzej
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ReferencesLink to record
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