A Positive Solution for a Weighted p-Laplace Equation with Hardy–Sobolev’s Critical Exponent

Abstract

Here, considering $-\infty<a<\frac{N-p}{p}$, $a\le e\le a+1$, $d=1+a-e$ and $p^*:=p^*(a,e)=\frac{Np}{N-dp}$, the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials

$$\begin{aligned} -\Delta _{ap}u+V(x)|x|^{-ep^*}|u|^{p-2}u=|x|^{-ep^*}f(u) \end{aligned}$$

in ${\mathbb {R}}^N$ is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class $C^1$. We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in $L^{\infty }({\mathbb {R}}^N).$