Weak solution by the sub-supersolution method for a nonlocal system involving Lebesgue generalized spaces

Abstract

We consider a system of nonlocal elliptic equations $$ \displaylines{ - \mathcal{A}(x, |v|_{L^{r_1(x)}}) \hbox{div}(a_1(|\nabla u|^{p_1(x)})|\nabla u|^{p_1(x)-2}\nabla u)\cr = f_1(x, u,v) |\nabla v|^{\alpha_1(x)}_{L^{q_1(x)}}+g_1(x, u,v) |\nabla v|^{\gamma_1(x)}_{L^{s_1(x)}},\cr - \mathcal{A}(x, |u|_{L^{r_2(x)}}) \hbox{div}(a_2(|\nabla v|^{p_2(x)})|\nabla u|^{p_2(x)-2}\nabla u)\cr = f_2(x, u,v) |\nabla u|^{\alpha_2(x)}_{L^{q_2(x)}}+g_2(x, u,v)|\nabla u|^{\gamma_2(x)}_{L^{s_2(x)}}, } $$with Dirichlet boundary condition, where \(\Omega\) is a bounded domain in\()\mathbb{R}^N\) \((N >1)\) with \(C^2\) boundary. Using sub-supersolution method,we prove the existence of at least one positive weak solution.Also, we study a generalized logistic equation and a sublinear system.