scholarly journals Dirichlet and Neumann Problems Related to Nonlinear Elliptic Systems: Solvability, Multiple Solutions, Solutions with Positive Components

2012 ◽  
Vol 2012 ◽  
pp. 1-44 ◽  
Author(s):  
Luisa Toscano ◽  
Speranza Toscano
Keyword(s):  
Multiple Solutions ◽  
Reflexive Banach Space ◽  
Existence Theorems ◽  
Elliptic Systems ◽  
General System ◽  
Nonlinear Elliptic Systems ◽  
Variational Equations ◽  
Neumann Problems ◽  
Nonlinear Elliptic ◽  
Regularity Properties

We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.

scholarly journals A New Result Concerning the Solvability of a Class of General Systems of Variational Equations with Nonmonotone Operators: Applications to Dirichlet and Neumann Nonlinear Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Luisa Toscano ◽  
Speranza Toscano
Keyword(s):  
Wide Class ◽  
Elliptic Systems ◽  
Nonlinear Problems ◽  
Nonlinear Elliptic Systems ◽  
Variational Equations ◽  
Neumann Problems ◽  
Nonlinear Elliptic ◽  
Reflexive Space ◽  
General Systems

A new result of solvability for a wide class of systems of variational equations depending on parameters and governed by nonmonotone operators is found in a Banach real and reflexive space with applications to Dirichlet and Neumann problems related to nonlinear elliptic systems.

A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

2021 ◽  
Author(s):  
Anna Kh. Balci ◽  
Andrea Cianchi ◽  
Lars Diening ◽  
Vladimir Maz’ya
Keyword(s):  
Differential Inequality ◽  
Differential Operators ◽  
Elliptic Systems ◽  
Second Order ◽  
Nonlinear Elliptic Systems ◽  
Nonlinear Elliptic ◽  
Regularity Properties ◽  
Partial Differential Operators ◽  
Global Estimates ◽  
Special Case

AbstractA sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $${\mathbb R^n}$$ R n are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the p-Laplace system, our conclusions broaden the range of the admissible values of the exponent p previously known.

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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