Acta Mathematica Scientia 2009,29B(4):903–918 http://actams.wipm.ac.cn MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT ∗ Dedicated to Professor Wu Wenjun on the occasion of his 90th birthday Li Gongbao ( ) Zhang Guo ( ) College of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China E-mail: ligb@mail.ccnu.edu.cn Abstract In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent: −△pu −△q u = |u| p ⋆ -2 u + µ|u| r-2 u in Ω, u| ∂Ω =0, where Ω ⊂ R N is a bounded domain, N>p, p ⋆ = Np N-p is the critical Sobolev exponent and µ> 0. We prove that if 1 <r<q<p<N , then there is a µ0 > 0, such that for any µ ∈ (0,µ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem. Key words p&q-Laplacian; multiplicity of solutions; critical exponent 2000 MR Subject Classification 35J60; 35B33 1 Introduction In this paper, we study the existence of multiple solutions to the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent    −△ p u −△ q u = |u| p ⋆ −2 u + µ|u| r−2 u in Ω, u| ∂Ω =0, (1.1) where Ω is a bounded domain in R N , µ> 0, 1 <r<q<p<N , and p ⋆ = Np/N − p is the critical Sobolev exponent. If p = q = 2, (1.1) can be reduced to    −△u = |u| 2 ⋆ −2 u + µ|u| r−2 u in Ω, u| ∂Ω =0, (1.2) * Received December 31, 2008. Partially supported by NSFC (10571069 and 10631030), and the Lab of Mathematical Sciences, CCNU, Hubei Province, China