Abstract:
We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta_p u=f(u)$ in the case $2

2$ and $f(\cdot)$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-\Delta_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.

Abstract:
We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ \mu\vp^{p}=0, \quad \vp>0\quad\text{in $\Omega$} \vp&=0\quad\text{on $\p\Omega$} posed in a (strictly) convex and smooth domain $\Omega\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0

Abstract:
This paper concerns nonlinear elliptic equations in the half space $mathbb{R}_{+}^{n}:={ x=(x',x_{n})in mathbb{R}^{n}:x_{n}$ greater than 0 $}$, $ngeq 2$, with a nonlinear term satisfying some conditions related to a certain Kato class of functions. We prove some existence results and asymptotic behaviour for positive solutions using a potential theory approach.

Abstract:
We show that a system of quasilinear degenerate elliptic inequalities does not have non-trivial solutions for a certain range of parameters in the system. The proof relies on a suitable choice of the test function in the weak formulation of the inequalities.

Abstract:
We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem $-\Delta_p u=a u^{p-1}-b(x) u^q$, $u|_{\partial \Omega}=0$ as $q \to p-1+0$ and as $q \to \infty$ via a scale argument. Here $\Delta_p$ is the $p$-Laplacian with $1

p-1$. If $p=2$, such problems arise in population dynamics. Our main results generalize the results for $p=2$, but some technical difficulties arising from the nonlinear degenerate operator $-\Delta_p$ are successfully overcome. As a by-product, we can solve a free boundary problem for a nonlinear $p$-Laplacian equation.

Abstract:
We prove existence and non existence results for fully nonlinear degenerate elliptic inequalities, by showing that the classical Keller--Osserman condition on the zero order term is a necessary and sufficient condition for the existence of entire sub solutions.

Abstract:
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems.

Abstract:
We prove the existence of solutions for nonlinear degenerate elliptic boundary-value problems of higher order. Solutions are obtained using pseudo-monotonicity theory in a suitable weighted Sobolev space.

Abstract:
In this paper we prove a existence result for solution to a class of nonlinear degenerate elliptic equation associated with a class of partial differential operators of the form where are functions satisfying suitable hypotheses. Here the operator is not uniformly elliptic, but is assume that the following condition is true where is a weight function.

Abstract:
We obtain nontrivial solutions for semilinear elliptic boundary value problems having resonance both at zero and at infinity, when the nonlinear term has asymptotic limits.