Ground States and Free Boundary Value Problems for the n-Laplacian in n Dimensional Space

Using a new gradient estimate, we prove several theorems on the existence of radial ground states for the n-Laplace equation div(|∇u|n−2 ∇u)+f(u)=0 in Rn, n>1, and the existence of positive radial solutions for the associated Dirichlet–Neumann free boundary value problem in a ball. We treat exponentially subcritical, critical, and supercritical nonlinearities f(u), which also are allowed to have singularities at zero. Moreover, we show that the local behavior of f at zero affects the existence in a crucial way: this allows us to prove the existence of ground states for a large class of functions f(u) without imposing any restriction on their growth for large u.