Existence of Multiple Solutions for p-biharmonic Problems with Critical Sobolev Exponent

Abstract

In this paper, by using the concentration-compactness principle and a version of symmetry mountain pass theorem, we establish the existence and multiplicity of solutions to the following p-biharmonic problem with critical nonlinearity:

$$\left\{{\matrix{{\Delta _p^2u = f({x,u}) + \mu {{\vert u \vert}^{{p^*} - 2}}u}\;&\;{{in}\;\Omega,} \cr {u={{\partial u} \over {\partial v}} = 0}\;&\;{{\text{on}}\;\partial \Omega,}}} \right.$$

where Ω is a bounded domain in ℝ^N (N ≥ 3) with smooth boundary, \(\Delta_{p}^{2}u=\Delta({\vert \Delta u \vert}^{p-2} \Delta u ), 1 < p < {N \over 2}, \ p^{*}={N_{p}\over N-2p},\ {\partial u \over \partial \nu}\) is the outer normal derivative, μ is a positive parameter and f: Ω × ℝ → ℝ is a Carathéodory function.