On critical exponent for the existence and stability properties of positive weak solutions for some nonlinear elliptic systems involving the (p, q)-Laplacian and indefinite weight function

Abstract

Listen

Translate article

This paper deals with the existence and stability properties of positive weak solutions to classes of nonlinear systems involving the (p,q)-Laplacian of the form $$ \left\{\begin{array}{ll} -\Delta_{p} u = \lambda \,a(x)\,v^{\alpha}-c, & x\in \Omega,\\ -\Delta_{q} v = \lambda \,b(x)\,u^{\beta}-c, & x\in \Omega,\\ u=0=v, & x\in\partial \Omega, \end{array}\right. $$ where Δp denotes the p-Laplacian operator defined by \(\Delta_{p}z={\rm div}(|\nabla z|^{p-2}\nabla z)\), p > 1, λ and c are positive parameters, Ω is a bounded domain in RN (N ≥ 1) with smooth boundary, α, β > 0 and the weights a(x), b(x) satisfying a(x) ∈ C(Ω), b(x) ∈ C(Ω) and a(x) > a0 > 0, b(x) > b0 > 0, for x ∈ Ω. We first study the existence of positive weak solution by using the method of sub-super solution and then we study the stability properties of positive weak solution.