### Multiple positive solutions for a class of variational systems

#### Abstract

We consider the variational system $-\Delta u = \lambda ( \nabla F)(u)$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded region in $\mathbb R^m$ ($m \geq 1$) with $C^1$ boundary, $\lambda$ is a positive parameter, $u\colon\Omega \rightarrow \mathbb R^N$ ($N > 1$), and $\Delta$ denotes the Laplace operator. Here $F\colon \mathbb R^N \rightarrow \mathbb R$ is a function of class $C^2$. Using variational methods, we show how changes in the sign of $F$ lead to multiple positive solutions.

#### Keywords

Positive solutions; systems; variational method

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