Eigenfunctions for singular fully non linear equations in unbounded domains

In this paper we prove some Harnack inequality for fully non linear degenerate elliptic equations, in the two dimensional case, extending the results of Davila Felmer and Quaas in the singular case but in all dimensions. We then apply this result for the existence of an eigenfunction in smooth bounded domain.Comment: 30 pages 2 figure

Some Liouville Theorems for the p-Laplacian

We present several Liouville type results for the $p$-Laplacian in $\R^N$. Suppose that $h$ is a nonnegative regular function such that $$h(x) = a|x|^\gamma\ {\rm for}\ |x|\ {\rm large},\ a>0\ {\rm and}\ \gamma> -p.$$ We obtain the following non -existence result: 1) Suppose that $N>p>1$, and $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a nonnegative weak solution of - {\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \;\;\mbox{in }\; \R^N . Suppose that $p-1< q\leq {(N+\gamma)(p-1)\over N-p}$ then $u\equiv 0$. 2) Let $N\leq p$. If $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a weak solution bounded below of $-{\rm div} (|\nabla u|^{p-2 }\nabla u)\geq 0$ in $\R^N$ then $u$ is constant. 3) Let $N>p$ if $u$ is bounded from below and $-{\rm div} (|\nabla u|^{p-2 }\nabla u)=0$ in $\R^N$ then $u$ is constant. 4)If $-\Delta_p u+h(x) u^q\leq 0,$. If $q> p-1$, then $u\equiv 0$.Comment: 19 page

The Dirichlet problem for singular fully nonlinear operators

In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds.Comment: 10 pages, 0 figure

Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators

The main scope of this article is to define the concept of principal eigenvalue for fully non linear second order operators in bounded domains that are elliptic and homogenous. In particular we prove maximum and comparison principle, Holder and Lipschitz regularity. This leads to the existence of a first eigenvalue and eigenfunction and to the existence of solutions of Dirichlet problems within this class of operators.Comment: 37 pages, 0 figure