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

#### 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.

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.

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