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Weak Linking Theorems and Schrödinger Equations with Critical Sobolev Exponent

Published online by Cambridge University Press:  15 September 2003

Martin Schechter  and
Wenming Zou
Martin Schechter
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697-3875, USA; mschecht@math.uci.edu.
Wenming Zou
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; wzou@math.tsinghua.edu.cn.
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Abstract

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V(x)u=K(x)|u|^{2^\ast-2}u+g(x, u), u\in W^{1,2}({\bf R}^N)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x, u)u\leq c|u|^{2^\ast}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

Keywords

LinkingSchrödinger equationscritical Sobolev exponent.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 601 - 619
Copyright
© EDP Sciences, SMAI, 2003

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