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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

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
Full Text: DOI
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