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Existence of three solutions for a class of elliptic eigenvalue problems. (English) Zbl 0970.35089
The author considers the problem $-\Delta u = \lambda(f(u)+\mu g(u))$ in $\Omega$, $u|_{\partial\Omega}=0$, $\Omega\subset\Bbb R^n$, $f$ and $g$ are continuous functions. It is shown, under suitable conditions, the existence of at least three solutions for some $\lambda>0$, if $|\mu|$ is small enough. The abstract scheme which is corresponding to this problem is a novelty: it is not reducible to the Landesman--Lazer or Ambrosetti-- Rabinowitz methods.

35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
47J30Variational methods (nonlinear operator equations)
58E99Variational problems in infinite-dimensional spaces
Full Text: DOI
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