## 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

### Keywords:

double resonance; critical groups
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