## Differential and Integral Equations

### Singular elliptic systems with asymptotically linear nonlinearities

D.D. Hai

#### Abstract

We prove the existence and nonexistence of positive solutions for the system \begin{equation*} \left\{ \begin{array}{c} -\Delta u_{1}=\mu _{1}f_{1}(u_{2})\ \text{ in }\Omega , \\ -\Delta u_{2}=\mu _{2}f_{2}(u_{1}) \ \text{ in }\Omega , \\ \ \ \ \ \ \ \ u_{1}=u_{2}=0 \ \ \ \ \text{ on }\partial \Omega , \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n\text{ }}$with smooth boundary $\partial \Omega ,$ $\mu _{i}$ are positive parameters, and $f_{i}:(0,\infty )\rightarrow \mathbb{R}$ are asymptotically linear at $\infty$ and allowed to be singular at $0,$ $i=1,2$.

#### Article information

Source
Differential Integral Equations, Volume 26, Number 7/8 (2013), 837-844.

Dates
First available in Project Euclid: 20 May 2013