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Transactions of the American Mathematical Society

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A perturbation method in critical point theory and applications

Authors: Abbas Bahri and Henri Berestycki
Journal: Trans. Amer. Math. Soc. 267 (1981), 1-32
MSC: Primary 35J65; Secondary 58E05
MathSciNet review: 621969
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Abstract: This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type $ - \Delta u = {\left\vert u \right\vert^{p - 1}}u + h(x)$ in $ \Omega ,\,u = 0$ on $ \partial \Omega $. Here, $ \Omega \subset {{\mathbf{R}}^N}$ is smooth and bounded, and $ h \in {L^2}(\Omega )$ is given. We show that there exists $ {p_N} > 1$ such that for any $ p \in (1,\,{p_N})$ and any $ h \in {L^2}(\Omega )$, the preceding equation possesses infinitely many distinct solutions.

The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.

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Keywords: Nonlinear elliptic partial differential equation, critical point, perturbation, contractible set, level set, even and noneven functional, multiplicity
Article copyright: © Copyright 1981 American Mathematical Society