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Abstract. We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain Í2 c R -div(|VK|p~2VK) = \u\p"~2u + k\u\q~2u, A > 0, where p' is the critical Sobolev exponent, and u\sn = 0. By using critical point methods we obtain the existence of solutions in the following cases: If p < q < p* , there exists A0 > 0 such that for all A > A0 there exists a nontrivial solution.If max(p, p* -pKp -1)) < q < p* , there exists nontrivial solution for all A>0.If 1 < q < p there exists A, such that, for 0 < A < A, , there exist infinitely many solutions.Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.

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