Existence of a nontrivial solution to a strongly indefinite semilinear equation
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- by B. Buffoni, L. Jeanjean and C. A. Stuart PDF
- Proc. Amer. Math. Soc. 119 (1993), 179-186 Request permission
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 \[ - \Delta u(x) + p(x)u(x) = u(x)\int _{{\mathbb {R}^3}} {\frac {{{u^2}(y)}} {{|x - y|}}dy,\qquad u \in {H^1}({\mathbb {R}^3}),\quad u \ne 0,} \] where $p \in {L^\infty }({\mathbb {R}^3})$ is a periodic function.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 119 (1993), 179-186
- MSC: Primary 35J60; Secondary 35Q99, 45K05, 47H15, 47N20
- DOI: https://doi.org/10.1090/S0002-9939-1993-1145940-X
- MathSciNet review: 1145940