## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 6 (2006), 669-680.

### Multiple positive solutions for classes of elliptic systems with combined nonlinear effects

Jaffar Ali, Mythily Ramaswamy, and R. Shivaji

#### Abstract

We study the existence of multiple positive solutions to systems of the form \begin{equation*} \begin{cases} \qquad-{\Delta} u ={\lambda} f(v), & \text{ in }{\Omega},\\ \qquad-{\Delta} v ={\lambda} g(u), & \text{ in }{\Omega},\\ \qquad\quad~~ u=0=v, & \text{ on }{\partial}{\Omega}. \end{cases} \end{equation*} Here ${\Delta}$ is the Laplacian operator, ${\lambda}$ is a positive parameter, ${\Omega}$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary and $f, g$ belong to a class of positive functions that have a combined sublinear effect at $\infty$. Our results also easily extend to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 6 (2006), 669-680.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050357

**Mathematical Reviews number (MathSciNet)**

MR2234718

**Zentralblatt MATH identifier**

1212.35162

**Subjects**

Primary: 35J55

Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Ali, Jaffar; Shivaji, R.; Ramaswamy, Mythily. Multiple positive solutions for classes of elliptic systems with combined nonlinear effects. Differential Integral Equations 19 (2006), no. 6, 669--680. https://projecteuclid.org/euclid.die/1356050357