On the existence of positive weak solutions for a class of (p,q)-Laplacian nonlinear elliptic system with sign-changing weights

Abstract

In this paper, we prove the existence of positive weak solution for the nonlinear elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _{p}u=\lambda _{1}~ a(x) f(v)+\mu _{1}~ \alpha (x) h(u),&{} x\in \Omega ,\\ -\Delta _{q}v=\lambda _{2}~ b(x) g(u)+\mu _{2}~ \beta (x) \gamma (v),&{} x\in \Omega ,\\ u=0=v, &{} x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Delta _{s}z=div(|\nabla z|^{s-2}\nabla z)$$ , $$s>1$$ , $$\lambda _1,\lambda _2,\mu _1$$ and $$\mu _2$$ are positive parameters, and $$\Omega $$ is a bounded domain in $$\mathbb {R}^N$$ . Here $$a(x),b(x),\alpha (x)$$ and $$\beta (x)$$ are sign-changing functions that maybe negative near the boundary. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on $$f(0),h(0),g(0)$$ and $$\gamma (0)$$ .