Multiplicity of solutions for critical singular problems

In this work we deal with the class of critical singular quasilinear elliptic problems in R-N of the form -div(\x\(-ap)\del u\(p-2)del u) = alpha\x\(-bq)\u\(q-2)u+beta\x\(-dr) k\u (r-2)u x is an element of R-N. (P) where -1< p < N, a < N/p, a <= b < a + 1, alpha and beta are positive parameters, q = q(a, b) =_ Np/[N - p(a + 1 - b)] and d is an element of R. Moreover, 1 < r < p* = Np/(N - p) and 0 <= k is an element of L-r(d-b)(q/(q-r)) (R-N). Multiplicity results are established by combining a version of the concentration-compactness lemma due to Lions with the Krasnoselski genus and the symmetric mountain-pass theorem due to Rabinowitz. (C) 2005 Elsevier Ltd. All rights reserved.