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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)
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References:
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