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

TL;DR

This paper investigates the existence and properties of symmetric ground states for supercritical and critical elliptic problems in specific domains, identifying parameter intervals for existence and nonexistence, and providing a minimax characterization and bifurcation analysis.

Contribution

It introduces new existence and nonexistence results for symmetric ground states in supercritical problems and offers a minimax framework and bifurcation analysis for anisotropic critical problems.

Findings

Existence of symmetric ground states for certain λ intervals.

Nonexistence of symmetric ground states below a positive threshold λ_* for k≥2.

A minimax characterization and bifurcation results for ground states.

Abstract

We study the existence of symmetric ground states to the supercritical problem \[ -\Delta v=\lambda v+\left\vert v\right\vert ^{p-2}v\text{ \ in }\Omega,\qquad v=0\text{ on }\partial\Omega, \] in a domain of the form \[ \Omega=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \in\Theta\}, \] where $\Theta$ is a bounded smooth domain such that $\overline{\Theta} \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $\lambda\in\mathbb{R},$ 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 $(-\infty,\lambda_{\ast})$ with $\lambda_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem \[ -\text{div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ \Theta,\qquad u=0\quad\text{on}\ \partial\Theta, \] 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.