Opuscula Math. 36, no. 2 (2016), 253-264

Opuscula Mathematica

Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

Lingju Kong

Abstract. We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a \(p(x)\)-biharmonic operator \[\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\in C(\overline{\Omega})\), \(\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)\) is the \(p(x)\)-biharmonic operator, and \(\lambda\gt 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^{*}\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda\gt\lambda^{*}\). Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\).

Keywords: critical points, \(p(x)\)-biharmonic operator, weak solutions, mountain pass lemma.

Mathematics Subject Classification: 35J66, 35J40, 35J92, 47J10.

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  • Lingju Kong
  • Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
  • Communicated by Marius Ghergu.
  • Received: 2015-05-14.
  • Accepted: 2015-08-14.
  • Published online: 2015-12-18.
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Cite this article as:
Lingju Kong, Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators, Opuscula Math. 36, no. 2 (2016), 253-264, http://dx.doi.org/10.7494/OpMath.2016.36.2.253

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