A perturbation method in critical point theory and applications

Bahri, Abbas; Berestycki, Henri

doi:10.1090/S0002-9947-1981-0621969-9

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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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