Ground states of critical and supercritical problems of Brezis–Nirenberg type

Abstract

We study the existence of symmetric ground states to the supercritical problem $$\begin{aligned} -\Delta v=\lambda v+\left| v\right| ^{p-2}v\;\text {in}\;\Omega ,\quad v=0\;\text {on}\;\partial \Omega , \end{aligned}$$ in a domain of the form $$\begin{aligned} \Omega =\{(y,z)\in \mathbb {R}^{k+1}\times \mathbb {R}^{N-k-1}:\left( \left| y\right| ,z \right) \in \Theta \}, \end{aligned}$$ where \(\Theta \) is a bounded smooth domain such that \(\overline{\Theta } \subset \left( 0,\infty \right) \times \mathbb {R}^{N-k-1}\), \(1\le k\le N-3\), \(\lambda \in \mathbb {R}\), and \(p=\frac{2(N-k)}{N-k-2}\) is the \((k+1)\)-st critical exponent. We show that symmetric ground states exist for \(\lambda \) in some interval to the left of each symmetric eigenvalue and that no symmetric ground states exist in some interval \((-\infty ,\lambda _{*})\) with \(\lambda _{*}>0\) if \(k\ge 2\). Related to this question is the existence of ground states to the anisotropic critical problem $$\begin{aligned} -\text {div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left| u\right| ^{2^{*}-2}u\;\text {in}\; \Theta ,\quad u=0\;\text {on}\; \partial \Theta , \end{aligned}$$ where a, b, c are positive continuous functions on \(\overline{\Theta }\). We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of \(\lambda \), and obtain a bifurcation result for ground states.