Nonnegative solutions for indefinite Caffarelli–Kohn–Nirenberg-type problems with subcritical or critical growth

Using variational methods we prove the existence of nonnegative solutions for the following class of quasilinear problems given by: $\begin{array}{rcl}& & -\text{div}(|x{|}^{-\mathrm{\Upsilon }p}|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)+|x{|}^{-b{p}^{\ast }}|u{|}^{p-2}u\\ & =& \lambda |x{|}^{-b{p}^{\ast }}a(x)g(u)+\gamma |x{|}^{-b{p}^{\ast }}|u{|}^{p^{\ast }-2}u\text{in}{\mathbb{R}}^N,\end{array}$ for the subcritical case ( $\gamma =0$ $\gamma =0$ ) and also for the critical case ( $\gamma =1$ $\gamma =1$ ). The functions $a:{\mathbb{R}}^N\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are continuous functions that satisfy some additional conditions, $1<p<N$ $1 < p < N$ , $0\le \mathrm{\Upsilon }<\frac{N-p}{p}$ $0 \leq \Upsilon < \frac{N-p}{p}$ , $\mathrm{\Upsilon }<b\le \mathrm{\Upsilon }+1$ $\Upsilon < b \leq \Upsilon +1$ , $p^{\ast }=p^{\ast }(\mathrm{\Upsilon },b)=\frac{pN}{N-dp}$ $p^{*}=p^{*}(\Upsilon ,b)=\frac{pN}{N -d p}$ with $d=1+\mathrm{\Upsilon }-b$ $d = 1 + \Upsilon - b$ .