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

Here, considering \(-\infty<a<\frac{N-p}{p}\) $-\infty <a<\frac{N-p}{p}$ , \(a\le e\le a+1\) $a\le e\le a+1$ , \(d=1+a-e\) $d=1+a-e$ and \(p^*:=p^*(a,e)=\frac{Np}{N-dp}\) $p^{\ast }:=p^{\ast }(a,e)=\frac{\mathit{Np}}{N-dp}$ , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials \(\begin{aligned} -\Delta _{ap}u+V(x)|x|^{-ep^*}|u|^{p-2}u=|x|^{-ep^*}f(u) \end{aligned}\) $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{\mathit{ap}}u+{V(x)|x|}^{-e{p}^{\ast }}{|u|}^{p-2}u={|x|}^{-e{p}^{\ast }}f(u)\end{array}$$ in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class \(C^1\) $C^1$ . 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 \( L^{\infty }({\mathbb {R}}^N). \) $L^{\infty }({\mathbb{R}}^N).$