A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth

We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =\lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $$ for \(i=1,\cdots,n\), in a bounded \(C^{1,1}\) doma...

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Bibliographic Details
Published in:arXiv.org 2019-10
Main Authors: Nornberg, Gabrielle, Schiera, Delia, Sirakov, Boyan
Format: Article
Language:English
Subjects:
Boundary conditions
Dirichlet problem
Nonlinear systems
Online Access:Get full text
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Summary:We consider fully nonlinear uniformly elliptic cooperative systems with quadratic growth in the gradient, such as $$ -F_i(x, u_i, Du_i, D^2 u_i)- \langle M_i(x)D u_i, D u_i \rangle =\lambda c_{i1}(x) u_1 + \cdots + \lambda c_{in}(x) u_n +h_i(x), $$ for \(i=1,\cdots,n\), in a bounded \(C^{1,1}\) domain \(\Omega\subset \mathbb{R}^N\) with Dirichlet boundary conditions; here \(n\geq 1\), \(\lambda \in\mathbb{R}\), \(c_{ij},\, h_i \in L^\infty(\Omega)\), \(c_{ij}\geq 0\), \(M_i\) satisfies \(0
ISSN:2331-8422