On the Existence Results for a Class of Singular Elliptic System Involving Indefinite Weight Functions and Asymptotically Linear Growth Forcing Terms

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.


Introduction
In this article, we mainly consider the existence of a positive solution of the following singular elliptic system where λ is a positive parameter, ∆ p u = div(|∇u| p−2 ∇u), p > 1, Ω ⊂ R n some for n > 1, is a bounded domain with smooth boundary ∂Ω, 0 < α < 1, and f, g : [0, ∞] → R are continuous, nondecreasing functions which are asymptotically p-linear at ∞.We prove the existence of a positive solution for a certain range of λ.
We consider problem (1.1) under the following assumptions.

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G.A. Afrouzi, S. Shakeri and N.T. Chung (H 1 ) There exist σ 1 > 0, k 1 > 0 and s 1 > 1 such that and that there exist σ 2 > 0, k 2 > 0 and s 2 > 1 such that The case when F (0) < 0 (and finite) is referred to in the literature as a semipositone problem.Finding a positive solution for a semipositone problem is well known to be challenging (see [2,5]) .Here we consider the more challenging case when lim u→0 + F (u) = −∞, which has received attention very recently and is referred to as an infinite semipositone problem.However, most of these studies have concentrated on the case when the nonlinear function satisfies a sublinear condition at ∞ (see [6,7,9]).We refer to [15,16,17,18,19] for additional results on elliptic problems.The only paper to our knowledge dealing with an infinite semipositone problem with an asymptotically linear nonlinearity is [8], where the author is restricted to the case p = 2. Also here the existence of a positive solution is focused near λ1 σ , where λ 1 is the first eigenvalue of −∆.See also [1,11], where asymptotically linear nonlinearities have been discussed in he case of a nonsingular semipostione problem and an infinite positone problem.Motivated by the above papers, in this note, we are interested in the existence of positive solution for problem (1.1), where a, b are continuous functions in Ω and λ is a positive parameter.Our main goal is to improve the result introduced in [12], in which the authors study the existence of positive solutions for an infinite semipositone problem with the nonlinearity f being not dependent of x.We shall establish our an existence result via the method of sub and supersolutions.
Class of Singular Elliptic System 69 resp.z i (i = 1, 2) satisfy: 3) for all non-negative test functions The following lemma was established by Miyagaki in [14]:

Main result
With the hypotheses introduced in previous section, the main result of this paper is given by the following theorem.Proof: Let µ 1 is the principal eigenvalue of operator −∆ p with Dirichlet boundary condition.By anti-maximum principle (see [10]), there exists ξ = ξ(Ω) > 0 such that the solution z µ of is positive in Ω and is such that ∂zµ ∂ν < 0 on ∂Ω, where ν is outward normal vector at ∂Ω.
Since z µ > 0 in Ω and ∂zµ ∂ν < 0 there exist m > 0, A > 0, and δ > 0 be such that We first construct a supersolution for (1.1).Let where M λ ≫ 1 is a large positive constant and e p is the unique positive solution of −∆ p e p = 1 in Ω, e p = 0 on ∂Ω.