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

S.H. Rasouli


In this paper we consider the existence of positive solutions of infinite
semipositone systems with singular weights of the form
-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,
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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The Thai Journal of Mathematics organized and supported by The Mathematical Association of Thailand and Thailand Research Council and the Center for Promotion of Mathematical Research of Thailand (CEPMART).

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