## 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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### References:

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