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Existence of a nontrivial solution to a strongly indefinite semilinear equation. (English) Zbl 0789.35052

An existence result for the nonlinear equation \(Lu=N(u)\) in a Hilbert space \(H\) is proved in this paper. Here \(L\) is an invertible continuous selfadjoint linear operator and \(N\) is a nonlinear operator with “superquadratic growth”. The problem corresponds to a strongly indefinite equation. The proof uses a Lyapunov-Schmidt reduction and then a version of the Mountain Pass theorem without Palais-Smale condition due to Brezis-Nirenberg. This theorem can be applied to problems with non- compact linear part where “linking” theorems do not work. An application is given to the Choquard-Pekar equation \[ -\Delta u+p(x)u= u(x)\int_{\mathbb{R}^ 3} {{u^ 2(y)} \over {| x-y|}}dy, \] with \(p\in L^ \infty(\mathbb{R}^ 3)\) periodic.

MSC:

35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
35J10 Schrödinger operator, Schrödinger equation
47J05 Equations involving nonlinear operators (general)
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