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

Gabrielle Nornberg; Delia Schiera; Boyan Sirakov

doi:10.3934/dcds.2020128

Article Contents

2020, Volume 40, Issue 6: 3857-3881. Doi: 10.3934/dcds.2020128 open access

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

1.
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-carlense 400, 13566-590 São Carlos - SP, Brazil
2.
Università degli Studi dell'Insubria, Via Valleggio 11, 22100 Como, Italy
3.
Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, 22451-900 Gávea - Rio de Janeiro, Brazil

^* Corresponding author
^* Corresponding author

Dedicated to Professor Wei-Ming Ni with admiration

Received: April 30, 2019

Revised: September 30, 2019

Published: March 2020

G. Nornberg was supported by Fapesp grant 2018/04000-9, São Paulo Research Foundation

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Abstract

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<\mu_1 I\leq M_i\leq \mu_2 I $, and $ F_i $ is an uniformly elliptic Isaacs operator.

We obtain uniform a priori bounds for systems, under a weak coupling hypothesis that seems to be optimal. As an application, we also establish existence and multiplicity results for these systems, including a branch of solutions which is new even in the scalar case.

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Mathematics Subject Classification: Primary: 35J47, 35J60, 35J66; Secondary: 35A01, 35A16, 35P30.

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Figure 1. Illustration of Theorem 2.4

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Figure 2. Illustration of Theorem 2.5 for $ \mu_2 h\gneqq 0 $ small in $ L^p $-norm

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