An elliptic resonance problem with multiple solutions.(English)Zbl 1155.35358

Let $$\Omega$$ be a bounded open domain with smooth boundary. In this paper it is studied the semilinear elliptic equation $$-\Delta u=f(x,u)$$ in $$\Omega$$, under the Dirichlet boundary condition $$u=0$$ on $$\partial\Omega$$. The mapping $$f$$ is a smooth nonlinearity such that $$\lim_{| t| \rightarrow\infty}f(x,t)/t=\lambda_k$$ for any $$x\in\Omega$$, where $$\lambda_k$$ is the $$k$$th eigenvalue of $$(-\Delta)$$ in $$H^1_0(\Omega )$$. The main results of the present paper establish sufficient conditions for the existence of one or several solutions to the above resonant problem. The proofs combine techniques involving the Morse critical point theory, critical groups computation, and minimax methods (mountain pass and local linking). The reviewer believes that the techniques developed in this paper can be extended for the study of other classes of nonlinear boundary value problems, including quasilinear elliptic equations or equations involving singular potentials.

MSC:

 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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