Lower semicontinuity of functionals via the concentration-compactness principle (Q5955585)

It is proved that, if \(\Omega\) is a bounded open subset of \({\mathbb R}^N\) and \(1<p<N\), then the functionals NEWLINE\[NEWLINE{\mathcal H}_\lambda(u)={1\over p}\int_\Omega|\nabla u(x)|^pdx-{\lambda\over p}\int_\Omega{|u(x)|^p\over|x|^p} dx,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE{\mathcal S}_\lambda(u)={1\over p}\int_\Omega|\nabla u(x)|^pdx-{\lambda\over p}\bigg(\int_\Omega |u(x)|^{p^*}dx\bigg)^{p/p^*},NEWLINE\]NEWLINE are weakly lower semicontinuous in \(W^{1,p}_0(\Omega)\), provided \(\lambda\) belongs to the subset of \(\mathbb R\) in which they are coercive.NEWLINENEWLINENEWLINEThe result is exploited to prove some existence results for quasilinear elliptic equations. In particular, it is proved that there exists a weak solution of the equation NEWLINE\[NEWLINE-\Delta_pu={\lambda\over|x|^p}|u|^{p-2}u+f\text{ in }{\mathbb R}^N.NEWLINE\]