On the generalised Brezis-Nirenberg problem

Abstract

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*} -Δ_p u - μg|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(Ω), \end{eqnarray*} where ${\Delta }_p$ is the $p$-Laplace operator defined as ${\Delta }_p(u)=\text{div}(|\nabla u{|}^{p-2}\nabla u),$ $p^{\ast }=\frac{Np}{N-p}$ 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 $\Vert u{\Vert }_{{\mathcal{D}}_p}:={\left[{\int }_{\Omega }|\nabla u{|}^p\mathrm{d}x\right]}^{\frac{1}{p}}.$ 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 $N\ge p^2$, if $g$ is such that $g^{+}\ne 0$ and the map $u\mapsto {\int }_{\Omega }|g||u{|}^p\mathrm{d}x$ 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.

31 pages