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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Existence of a nontrivial solution to a strongly indefinite semilinear equation


Authors: B. Buffoni, L. Jeanjean and C. A. Stuart
Journal: Proc. Amer. Math. Soc. 119 (1993), 179-186
MSC: Primary 35J60; Secondary 35Q99, 45K05, 47H15, 47N20
MathSciNet review: 1145940
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Abstract: Under general hypotheses, we prove the existence of a nontrivial solution for the equation $ Lu = N(u)$, where $ u$ belongs to a Hilbert space $ H$, $ L$ is an invertible continuous selfadjoint operator, and $ N$ is superlinear. We are particularly interested in the case where $ L$ is strongly indefinite and $ N$ is not compact. We apply the result to the Choquard-Pekar equation

$\displaystyle - \Delta u(x) + p(x)u(x) = u(x)\int_{{\mathbb{R}^3}} {\frac{{{u^2}(y)}} {{\vert x - y\vert}}dy,\qquad u \in {H^1}({\mathbb{R}^3}),\quad u \ne 0,} $

where $ p \in {L^\infty }({\mathbb{R}^3})$ is a periodic function.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1145940-X
PII: S 0002-9939(1993)1145940-X
Keywords: Semilinear equation, Choquard-Pekar equation, strongly indefinite operator, lack of compactness
Article copyright: © Copyright 1993 American Mathematical Society