Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

Abstract

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: \[ \left \{ \begin {matrix} {-\triangle _{p} u = \lambda |u|^{r-2}u + \mu \textstyle {\frac {|u|^{q-2}}{|x|^{s}}}u \quad \text {in $\Omega $}, {}} {\hphantom {-} u|_{\partial \Omega } = 0, }\hfill \end {matrix}\right . \] where $\lambda$ and $\mu$ are two positive parameters and $\Omega$ is a smooth bounded domain in $\mathbf {R}^n$ containing $0$ in its interior. The variational approach requires that $1 < p < n$, $p\leq q\leq p^{*}(s)\equiv \frac {n-s}{n-p}p$ and $p\leq r\leq p^*\equiv p^*(0)=\frac {np}{n-p}$, which we assume throughout. However, the situations differ widely with $q$ and $r$, and the interesting cases occur either at the critical Sobolev exponent ($r=p^*$) or in the Hardy-critical setting ($s=p=q$) or in the more general Hardy-Sobolev setting when $q=\frac {n-s}{n-p}p$. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case $p=2$, especially those corresponding to singularities (i.e., when $0<s\leq p)$.