On the generalised Brézis–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:

$$\begin{aligned} -\Delta _p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \text{ in } \mathcal {D}_p(\Omega ), \end{aligned}$$

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 $\Vert u\Vert _{\mathcal {D}_p}:= \left[ \displaystyle \int _{\Omega } |\nabla u|^p \mathrm{d}x\right] ^ \frac{1}{p}.$ 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 $N \ge p^2$, if g is such that $g^+ \ne 0$ 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.