Opuscula Math. 32, no. 3 (2012), 473-486

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
  • Received: 2011-06-27.
  • Revised: 2011-09-17.
  • Accepted: 2011-09-19.
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Cite this article as:
Paweł Goncerz, On the existence of three solutions for quasilinear elliptic problem, Opuscula Math. 32, no. 3 (2012), 473-486, http://dx.doi.org/10.7494/OpMath.2012.32.3.473

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