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Critical singular problems via concentration-compactness lemma. (English) Zbl 1159.35373

Summary: We consider existence and multiplicity results of nontrivial solutions for a class of quasilinear degenerate elliptic equations in N of the form

-div|x| -ap u| p-2 u+λ|x| -(a+1)p |u| p-2 u=|x| -bq |u| q-2 u+f,(P)

where x N , 1<p<N, q=q(a,b)Np/[N-p(a+1-b)], λ is a parameter, 0a<(N-p)/p, aba+1, and f(L b q ( N )) * . We look for solutions of problem (P) in the Sobolev space 𝒟 a 1,p ( N ) and we prove a version of a concentration-compactness lemma due to Lions. Combining this result with the Ekeland’s variational principle and the mountain-pass theorem, we obtain existence and multiplicity results.

35J70Degenerate elliptic equations
35J20Second order elliptic equations, variational methods
35D05Existence of generalized solutions of PDE (MSC2000)