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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:
 35J65 Nonlinear boundary value problems for linear elliptic equations 58E30 Variational principles in infinite-dimensional spaces 35A15 Variational methods applied to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
$$p$$-Laplacian; critical point theory
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##### References:
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