An existence result for a singular-regular anisotropic system

Abstract

In this paper we deal with the following singular-regular anisotropic system

$$\begin{aligned} \left\{ \begin{array}{ll} -Lu=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =p\dfrac{v^{q}}{u^{1-p}} &{}\quad in~\Omega , \\ -Lv=-\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}v\right| ^{p_{i}-2}\partial _{i}v\right] =qv^{q-1}u^{p} &{}\quad in~\Omega , \\ u>0\text { and }v>0 &{}\quad in~\Omega , \\ u=0\text { and }v=0 &{}\quad on~\partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega$ is a bounded regular domain in $\mathbb {R}^{N}$ and $1\le p_{1}\le p_{2}\le \cdots \le p_{N}.$ Under some suitable conditions on the parameters p and q, we obtain existence results by using nondifferentiable variational techniques.