Opuscula Math. 32, no. 3 (2012), 473-486
http://dx.doi.org/10.7494/OpMath.2012.32.3.473

Opuscula Mathematica

On the existence of three solutions for quasilinear elliptic problem

Paweł Goncerz

Abstract. We consider a quasilinear elliptic problem of the type $$-\Delta_p u = \lambda (f(u)+\mu g(u))$$ in $$\Omega$$, $$u|_{\partial \Omega} =0$$, where $$\Omega \in \mathbb{R}^N$$ is an open and bounded set, $$f$$, $$g$$ are continuous real functions on $$\mathbb{R}$$ and $$\lambda , \mu \in \mathbb{R}$$. We prove the existence of at least three solutions for this problem using the so called three critical points theorem due to Ricceri.

Keywords: critical point, elliptic problem, minimax inequality, $$p$$-Laplacian, three critical points theorem, weak solution.

Mathematics Subject Classification: 35J20, 35J25, 35J92, 58E05.

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• Paweł Goncerz
• Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland
• Revised: 2011-09-17.
• Accepted: 2011-09-19.