Global Solvability of Dirichlet Problem for Fully Nonlinear Elliptic Systems

2014 ◽  
Vol 35 (7-9) ◽  
pp. 1043-1065 ◽  
Author(s):  
Luisa Fattorusso ◽  
Antonio Tarsia
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Systems ◽  
Global Solvability ◽  
Nonlinear Elliptic Systems ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic

scholarly journals Strong Maximum Principle for Viscosity Solutions of Fully Nonlinear Cooperative Elliptic Systems

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2985
Author(s):  
Georgi Boyadzhiev ◽  
Nikolai Kutev
Keyword(s):  
Maximum Principle ◽  
Viscosity Solutions ◽  
Principal Symbol ◽  
Elliptic Systems ◽  
Strong Maximum Principle ◽  
Nonlinear Elliptic Systems ◽  
Fully Nonlinear ◽  
Nonlinear Elliptic ◽  
Weakly Coupled ◽  
First Time

In this paper, we consider the validity of the strong maximum principle for weakly coupled, degenerate and cooperative elliptic systems in a bounded domain. In particular, we are interested in the viscosity solutions of elliptic systems with fully nonlinear degenerated principal symbol. Applying the method of viscosity solutions, introduced by Crandall, Ishii and Lions in 1992, we prove the validity of strong interior and boundary maximum principle for semi-continuous viscosity sub- and super-solutions of such nonlinear systems. For the first time in the literature, the strong maximum principle is considered for viscosity solutions to nonlinear elliptic systems. As a consequence of the strong interior maximum principle, we derive comparison principle for viscosity sub- and super-solutions in case when on of them is a classical one. The main novelty of this work is the reduction of the smoothness of the solution. In the literature the strong maximum principle is proved for classical C2 or generalized C1 solutions, while we prove it for semi-continuous ones.

POSITIVE SOLUTIONS OF NONLINEAR ELLIPTIC SYSTEMS

1993 ◽  
Vol 03 (06) ◽  
pp. 823-837 ◽  
Author(s):  
A. CAÑADA ◽  
J.L. GÁMEZ
Keyword(s):  
Positive Solutions ◽  
Nonlinear Interaction ◽  
Spatial Dependence ◽  
Global Bifurcation ◽  
Elliptic Systems ◽  
Nonlinear Elliptic Systems ◽  
Decoupling Method ◽  
Nonlinear Elliptic ◽  
Cooperative Models

In this paper we prove the existence of nonnegative and non-trivial solutions of problems of the form [Formula: see text] Our main result improves many previous results of other authors and it may be applied to study the three standard situations: competition, prey-predator and cooperative models. We also cover some other cases which, due essentially to the spatial dependence or to a nonlinear interaction, are not any of these three types. The method of proof combines a decoupling method with a global bifurcation result.

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