Large solutions and gradient bounds for quasilinear elliptic equations

We consider the quasilinear degenerate elliptic equation λu−pu+H(x,Du)=0 in where p is the p-Laplace operator, p > 2, λ ≥ 0 and is a smooth open bounded subset of RN (N ≥ 2). Under suitable structure con- ditions on the function H, we prove local and global gradient bounds for the solutions. We apply these estimates to study the solvability of the Dirichlet problem, and the existence, uniqueness and asymptotic behavior of maximal solutions blowing up at the boundary. The ergodic limit for those maximal solutions is also studied and the existence and uniqueness of a so-called additive eigenvalue is proved in this context