Abstract

We study local and global properties of positive solutions of -Delta u = u(p)vertical bar del u vertical bar(q) in a domain Omega of R-N , in the range p + q > 1, p >= 0, 0 <= q < 2. We first prove a local Harnack inequality and nonexistence of positive solutions in R-N when p (N - 2) + q (N - 1) < N. Using a direct Bernstein method, we obtain a first range of values p and q in which u (x) <= c(dist(x , partial derivative Omega))q-2/p+q-1. This holds in particular if p + q < 1 + 4/N-1. Using an integral Bernstein method, we obtain a wider range of values of p and q in which all the global solutions are constants. Our result contains Gidas and Spruck's nonexistence result as a particular case. We also study solutions under the form u (x) = r q-2/p+q-1 omega(sigma). We prove existence, nonexistence, and rigidity of the spherical component co in some range of values of N, p, and q.