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Zbl 1159.35373
Assunção, Ronaldo B.; Carrião, Paulo Cesar; Miyagaki, Olimpio Hiroshi
Critical singular problems via concentration-compactness lemma. (English)
[J] J. Math. Anal. Appl. 326, No. 1, 137-154 (2007). ISSN 0022-247X

Summary: We consider existence and multiplicity results of nontrivial solutions for a class of quasilinear degenerate elliptic equations in $\Bbb R^N$ of the form $$-\text{div}\big[|x|^{-ap}\nabla u|^{p-2}\nabla u\big]+ \lambda|x|^{-(a+1)p}|u|^{p-2}u= |x|^{-bq}|u|^{q-2}u+f, \tag P$$ where $x\in\Bbb R^N$, $1<p<N$, $q=q(a,b)\equiv Np/[N-p(a+1-b)]$, $\lambda$ is a parameter, $0\le a<(N-p)/p$, $a\le b\le a+1$, and $f\in(L_b^q(\Bbb R^N))^*$. We look for solutions of problem (P) in the Sobolev space ${\cal D}_a^{1,p}(\Bbb R^N)$ 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.

MSC 2000:: *35J70 Elliptic equations of degenerate type
35J20 Second order elliptic equations, variational methods
35D05 Existence of generalized solutions of PDE

Keywords: degenerate quasilinear equation; $p$-Laplacian; variational methods; compactness-concentration

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