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.