## Advances in Differential Equations

- Adv. Differential Equations
- Volume 6, Number 1 (2001), 1-20.

### Existence for an elliptic system with nonlinear boundary conditions via fixed-point methods

Julián Fernández Bonder and Julio D. Rossi

#### Abstract

In this paper we prove the existence of nonnegative nontrivial solutions of the system $$\left\{\begin{array}{rcll} \Delta u & = & u & \mbox{in } \Omega,\\ \Delta v & = & v, & \end{array}\right.$$ with nonlinear coupling through the boundary given by $$ \left\{\begin{array}{rcll} \frac{\partial u}{\partial n} & = & f(x,u,v) & \mbox{on } \partial \Omega, \\ \frac{\partial v}{\partial n} & = & g(x,u,v), \end{array}\right. $$ under suitable assumptions on the nonlinear terms $f$ and $g$. For the proof we use a fixed-point argument and the key ingredient is a Liouville type theorem for a system of Laplace equations with nonlinear coupling through the boundary of power type in the half space.

#### Article information

**Source**

Adv. Differential Equations, Volume 6, Number 1 (2001), 1-20.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1357141499

**Mathematical Reviews number (MathSciNet)**

MR1799678

**Zentralblatt MATH identifier**

1223.35171

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35D05 35J25: Boundary value problems for second-order elliptic equations 35J55

#### Citation

Fernández Bonder, Julián; Rossi, Julio D. Existence for an elliptic system with nonlinear boundary conditions via fixed-point methods. Adv. Differential Equations 6 (2001), no. 1, 1--20. https://projecteuclid.org/euclid.ade/1357141499