Ground states of a prescribed mean curvature equation

Ignacio Guerra
Universidad de Santiago de Chile

Abstract: We study the existence of radial ground-state solutions for the problem

$\displaystyle -{\rm div}\left(\frac{\nabla u}{\sqrt{1+\vert\nabla u\vert^2}}\right)=
u^q,\quad u>0$   in$\displaystyle \quad \mathbb{R}^N$      
$\displaystyle u(x) \to 0$   as$\displaystyle \quad \vert x\vert\to\infty ,$      

$ N\ge 3$, $ q>1$. It is known that this problem has infinitely many ground states when $ q\ge {\frac{N+2}{N-2}}$, while no solutions exist if $ q\le \frac N{N-2}$. A question raised by Ni and Serrin in [Atti Convegni Lincei 77 (1985), 231-257], is whether or not ground state solutions exist for $ \frac N{N-2} <q< {\frac{N+2}{N-2}}.$ In this paper we prove the existence of a large, finite number of ground states with fast decay $ O(\vert x\vert^{2-N})$ as $ \vert x\vert \to +\infty$ provided that $ q$ lies below but close enough to the critical exponent $ \frac{N+2}{N-2}$. These solutions develop a bubble towerprofile as $ q$ approaches the critical exponent.

This is joint work with Manuel del Pino (Universidad de Chile)