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On the existence of a nontrivial solution to nonlinear problems at resonance. (English) Zbl 0684.35038
Consider the semilinear elliptic boundary value problem $-\Delta u=\lambda_ ku+g(u)\quad in\quad \Omega;\quad u=0\quad on\quad \partial \Omega,$ where $$\Omega \subset {\mathbb{R}}^ n$$ is a bounded domain with a smooth boundary, $$\lambda_ 0<\lambda_ 1\leq..$$. denote the eigenvalues of -$$\Delta$$ on $$\Omega$$ under the above boundary conditions, and $$g\in C'({\mathbb{R}})$$ satisfies $$g(0)=0$$, g(s)$$\to 0$$ for $$s\to \pm \infty$$ and $$\int^{\infty}_{-\infty}g(s)ds=0.$$
The authors establish the existence of a nontrivial solution, provided that either $$g'(0)>\lambda_ n-\lambda_ k,$$ where $$\lambda_ n$$ is the smallest eigenvalue greater than $$\lambda_ k$$, or $$g'(0)\neq 0$$ and $$g'(0)\int^{0}_{-\infty}g(s)ds\geq 0.$$
They use a variational approach and Morse type arguments in their proof.
Reviewer: G.Hetzer

##### 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 35P99 Spectral theory and eigenvalue problems for partial differential equations
##### Keywords:
existence of a nontrivial solution
Full Text:
##### References:
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