Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem N 3, q>1. It is known that this problem has infinitely many ground states when , while no solutions exist if . A question raised by Ni and Serrin in [W.-M. Ni, J. Serrin, Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei 77 (1985) 231–257] is whether or not ground state solutions exist for . In this paper we prove the existence of a large, finite number of ground states with fast decay O(x2−N) as x→+∞ provided that q lies below but close enough to the critical exponent . These solutions develop a bubble-tower profile as q approaches the critical exponent