Ground states of a prescribed mean curvature equation

We study the existence of radial ground state solutions for the problem−div(∇u1+|∇u|2)=uq,u>0inRN,u(x)→0as|x|→∞,N⩾3N⩾3, q>1q>1. It is known that this problem has infinitely many ground states when q⩾N+2N−2, while no solutions exist if q⩽NN−2. 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 NN−2