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Monotonicity of solutions of quasilinear degenerate elliptic equation in half-spaces  [PDF]
Alberto Farina,Luigi Montoro,Berardino Sciunzi
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.
Asymptotic Behavior in Degenerate Parabolic Fully Nonlinear equations and its application to Elliptic Eigenvalue Problems  [PDF]
Soojung Kim,Ki-ahm Lee
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
Existence of positive solutions to nonlinear elliptic problem in the half space
Imed Bachar,Habib Maagli,Malek Zribi
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.
A nonexistence result for a system of quasilinear degenerate elliptic inequalities in a half-space  [cached]
Mokthar Kirane,Eric Nabana
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.
Asymptotic behavior of positive solutions of some quasilinear elliptic problems  [PDF]
Zhongmin Guo,Li Ma
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.
Entire subsolutions of fully nonlinear degenerate elliptic equations  [PDF]
Italo Capuzzo Dolcetta,Fabiana Leoni,Antonio Vitolo
Mathematics , 2015,
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.
Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems  [PDF]
Olivier Ley,Vinh Duc Nguyen
Mathematics , 2014,
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.
Higher order nonlinear degenerate elliptic problems with weak monotonicity  [cached]
Youssef Akdim,Elhoussine Azroul,Mohamed Rhoudaf
Electronic Journal of Differential Equations , 2006,
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.
Existence of solutions for a class of nonlinear degenerate elliptic equations  [cached]
Cavalheiro Albo Carlos
Journal of Inequalities and Applications , 2002,
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.
Semilinear elliptic equations having asymptotic limits at zero and infinity  [cached]
Kanishka Perera,Martin Schechter
Abstract and Applied Analysis , 1999, DOI: 10.1155/s1085337599000159
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.
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