Existence of least energy positive and nodal solutions for a class of Caffarelli–Kohn–Nirenberg type problems

In this paper we investigate the quasilinear problem given by the following equation in \(\mathbb {R}^{N}\) ${\mathbb{R}}^N$ \(\begin{aligned} -\hbox {div}\left( |x|^{-ap}|\nabla u|^{p-2}\nabla u\right) +|x|^{-bp^{*}} V(x)|u|^{p-2}u = |x|^{-bp^{*}}K(x)f(u). \end{aligned}\) $$\begin{array}{c}\hfill -\text{div}\left({|x|}^{-ap},{|\mathrm{\nabla }u|}^{p-2},\mathrm{\nabla },u\right)+|x|_{}^{-bp_{}^{\ast }}{V(x)|u|}^{p-2}u=|x|_{}^{-bp_{}^{\ast }}K(x)f(u).\end{array}$$ Here, \(1<p<N\) $1<p<N$ , \(0\le a< \frac{N-p}{p}\) $0\le a<\frac{N-p}{p}$ , \(a<b\le a+1\) $a<b\le a+1$ , \(p =p (a,b)=\frac{pN}{N-dp}\) $p=p(a,b)=\frac{\mathit{pN}}{N-dp}$ and \(d=1+a-b\) $d=1+a-b$ . This equation exhibits singularity not only in the nonlinearity but also in the operator. By imposing suitable assumptions on the functions V, K and f, we establish the existence of least energy positive and nodal solutions using minimization technique on the Nehari manifold.