On the existence results for a class of singular elliptic system involving indefinite weight functions and asymptotically linear growth forcing term

Résumé

In this work, we study the existence of positive solutions to the singular system
$$
\left\{\begin{array}{ll}
-\Delta_{p}u = \lambda a(x)f(v)-u^{-\alpha} & \textrm{ in }\Omega,\\
-\Delta_{p}v = \lambda b(x)g(u)-v^{-\alpha} & \textrm{ in }\Omega,\\
u = v= 0 & \textrm{ on }\partial \Omega,
\end{array}\right.
$$
where $\lambda $ is positive parameter, $\Delta_{p}u=\textrm{div}(|\nabla u|^{p-2} \nabla u)$, $p>1$, $ \Omega \subset R^{n} $ some for $ n >1 $, is a bounded domain with smooth boundary $\partial \Omega $ , $ 0<\alpha< 1 $, and $f,g: [0,\infty] \to\R$ are continuous, nondecreasing functions which are asymptotically $ p $-linear at $\infty$. We prove the existence of a positive solution for a certain range of $\lambda$ using the method of sub-supersolutions.