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Three solutions of a quasilinear elliptic problem near resonance. (English) Zbl 0958.35052
Using critical point theory, the authors prove a multiplicity result near the first eigenvalue $\lambda_1$ for quasilinear elliptic problems of the form $$ -\Delta_pu-\lambda_1|u|^{p-2}u+\varepsilon |u|^{p-2}u=f(x,u)+h(x) $$ with Dirichlet conditions. It is proved that, for sufficiently small $\varepsilon >0$, the problem above has at least three solutions when $f$ and $h$ satisfy a Landesman-Lazer condition.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
58E30Variational principles on infinite-dimensional spaces
35A15Variational methods (PDE)
35A05General existence and uniqueness theorems (PDE) (MSC2000)
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References:
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