BLOW-UP SOLUTIONS FOR SOME NONLINEAR ELLIPTIC EQUATIONS INVOLVING A FINSLER-LAPLACIAN

In this paper we prove existence results and asymptotic behavior for strong solutions u is an element of W-loc(2,2)(Omega) of the nonlinear elliptic problem (P) {-Delta(H) u + H(del u)(q) +lambda u = f in Omega, u -> +infinity on partial derivative Omega, where H is a suitable norm of R-n, Omega subset of R-n is a bounded domain,Delta(H) is the Finsler Laplacian, 1 < q <= 2, lambda > 0, and f is a suitable function in L-loc(infinity). Furthermore, we are interested in the behavior of the solutions when lambda -> 0(+), studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P). 2010 Mathematics Subject Classification: 35J60, 35J25, 35B44.