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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 ${ℝ}^{N}$ of the form

$-\text{div}\left[{|x|}^{-ap}{\nabla u|}^{p-2}\nabla u\right]+{\lambda |x|}^{-\left(a+1\right)p}{|u|}^{p-2}u={|x|}^{-bq}{|u|}^{q-2}u+f,\phantom{\rule{2.em}{0ex}}\left(\mathrm{P}\right)$

where $x\in {ℝ}^{N}$, $1, $q=q\left(a,b\right)\equiv Np/\left[N-p\left(a+1-b\right)\right]$, $\lambda$ is a parameter, $0\le a<\left(N-p\right)/p$, $a\le b\le a+1$, and $f\in {\left({L}_{b}^{q}\left({ℝ}^{N}\right)\right)}^{*}$. We look for solutions of problem (P) in the Sobolev space ${𝒟}_{a}^{1,p}\left({ℝ}^{N}\right)$ 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:
 35J70 Degenerate elliptic equations 35J20 Second order elliptic equations, variational methods 35D05 Existence of generalized solutions of PDE (MSC2000)