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An asymmetric superlinear elliptic problem at resonance. (English) Zbl 1303.35013

Summary: In this article, we study the existence and multiplicity of solutions to the problem \[ \begin{cases} - \varDelta u = g(x, u)\quad & \text{in } \varOmega ;\\ u = 0,\quad & \text{on } \partial \varOmega, \end{cases} \] where \(\varOmega\) is a bounded domain in \(\mathbb R^N(N \geqslant 2)\) with smooth boundary, and \(g:\overline{\Omega} \times \mathbb R\to\mathbb R\) is a differentiable function. We will assume that \(g(x, s)\) has a resonant behavior for large negative values of \(s\) and that a Landesman-Lazer type condition is satisfied. We also assume that \(g(x, s)\) is superlinear, but subcritical, for large positive values of \(s\). We prove the existence and multiplicity of solutions for problem (1.1) by using minimax methods and infinite-dimensional Morse theory.

MSC:

35J20 Variational methods for second-order elliptic equations
49J35 Existence of solutions for minimax problems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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