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

Abstract

We study the existence of symmetric ground states to the supercritical problem \[ -Δv=λv+\left\vert v\right\vert ^{p-2}v\text{ \ in }Ω,\qquad v=0\text{ on }\partialΩ, \] in a domain of the form \[ Ω=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \inΘ\}, \] 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 $(-\infty ,{\lambda }_{\ast })$ with ${\lambda }_{\ast }>0$ if $k\ge 2.$ Related to this question is the existence of ground states to the anisotropic critical problem \[ -\text{div}(a(x)\nabla u)=λb(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ Θ,\qquad u=0\quad\text{on}\ \partialΘ, \] 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.

References in corpus (2)

• Periodic nonlinear Schrödinger equation with application to photonic crystals
• Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents