On the generalised Brezis-Nirenberg problem

For $p \in (1,N)$ and a domain $\Omega$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -\Delta_p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(\Omega), \end{eqnarray*} where $\Delta_p$ is the $p$-Laplace operator defined as $\Delta_p(u) = \text{div}(|\nabla u|^{p-2} \nabla u),$ $p^{*}= \frac{Np}{N-p}$ is the critical Sobolev exponent and $\mathcal{D}_p(\Omega)$ is the Beppo-Levi space defined as the completion of $\text{C}_c^{\infty}(\Omega)$ with respect to the norm $\|u\|_{\mathcal{D}_p} := \left[ \displaystyle \int_{\Omega} |\nabla u|^p \mathrm{d}x \right]^ \frac{1}{p}.$ In this article, we provide various sufficient conditions on $g$ and $\Omega$ so that the above problem admits a positive solution for certain range of $\mu$. As a consequence, for $N \geq p^2$, if $g$ is such that $g^+ \neq 0$ and the map $u \mapsto \displaystyle \int_{\Omega} |g||u|^p \mathrm{d}x$ is compact on $\mathcal{D}_p(\Omega)$, we show that the problem under consideration has a positive solution for certain range of $\mu$. Further, for $\Omega =\mathbb{R}^N$, we give a necessary condition for the existence of positive solution.