On the Existence of Positive Solutions for a Class of Infinite Semipositone Systems with Singular Weights

S.H. Rasouli

Abstract

In this paper we consider the existence of positive solutions of infinite
semipositone systems with singular weights of the form
$$
\left\{\begin{array}{ll}
-div(|x|^{-ap}\,|\nabla u|^{p-2}\,\nabla u)  = \lambda \,|x|^{-(a+1)p+c_{1}}\, (f(v)-\frac{1}{u^{\alpha}}), & x\in \Omega,\\
-div(|x|^{-bq}\,|\nabla v|^{q-2}\,\nabla v)  = \lambda \,|x|^{-(b+1)q+c_{2}}\, (g(u)-\frac{1}{v^{\beta}}), & x\in \Omega,\\
u =  0 =v,  & x\in\partial \Omega,
\end{array}\right.
$$
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ with
$0\in \Omega,$ $1<p,q<N,$ $0\leq a<\frac{N-p}{p}$, $0\leq b<\frac{N-q}{q},$ $\alpha, \beta \in (0,1),$ and $c_{1},c_{2},\lambda$ are positive parameters. Here $f,g:(0,\infty)\to (0,\infty)$ are $C^{2}$ functions. Our aim in this paper is to establish the existence of positive solution for $\lambda$ large. We use the method of sub-super solutions to establish our existence result.