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 Mathematics , 2012, Abstract: We prove a weak comparison principle in narrow unbounded domains for solutions to $-\Delta_p u=f(u)$ in the case $22$ 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.
 Mathematics , 2012, 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 $0  Electronic Journal of Differential Equations , 2005, Abstract: This paper concerns nonlinear elliptic equations in the half space$mathbb{R}_{+}^{n}:={ x=(x',x_{n})in mathbb{R}^{n}:x_{n}$greater than 0$}$,$ngeq 2$, with a nonlinear term satisfying some conditions related to a certain Kato class of functions. We prove some existence results and asymptotic behaviour for positive solutions using a potential theory approach.  Electronic Journal of Differential Equations , 2002, Abstract: We show that a system of quasilinear degenerate elliptic inequalities does not have non-trivial solutions for a certain range of parameters in the system. The proof relies on a suitable choice of the test function in the weak formulation of the inequalities.  Mathematics , 2003, Abstract: We discuss the asymptotic behavior of positive solutions of the quasilinear elliptic problem$-\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$1p-1$. 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.