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Nontrivial solutions of elliptic semilinear equations at resonance. (English) Zbl 0958.35051

The authors consider the following Dirichlet problem \(-\Delta u = \lambda_m +f(x,u)\) in a bounded domain \(\Omega\) with smooth boundary, where \(\lambda _m\) is an eigenvalue of the Laplacian operator in \(\Omega\) with Dirichlet boundary data. They treat the doubly resonant case, both at infinity and zero, \(\lim_{t\to 0}f(x,t)/t= \lim_{t\to \infty}f(x,t)/t=0\). They use critical groups computations to get their existence results.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
49K27 Optimality conditions for problems in abstract spaces
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