ON POSITIVE WEAK SOLUTIONS FOR A CLASS OF NONLINEAR SYSTEMS

Abstract

We study the positive weak solutions for the system

\(\left. \begin{aligned} −∆_{P,p}u = λa(x)f(v) \quad in \; Ω, \\ −∆{P,p}v = λb(x)g(u) \quad in \; Ω, \\ u= v= 0 \quad on \;  ∂Ω. \end{aligned} \right\}\)

where λ > 0 is a parameter, ∆_P,p with p > 1 and P = P(x) is a weight function, denotes the weighted p-Laplacian defined by ∆_P,pu ≡ div[P(x)|∇u|^p−2∇u], a(x), b(x) are weight functions and Ω ⊂ ℜ^N is a bounded domain with smooth boundary ∂Ω.  We discuss the existence of positive weak solutions for large λ when

\(\displaystyle \lim_{x \to +\infty} \frac{f^{\frac{1}{p-1}} (M(g(x))^{\frac{1}{p-1}})}{x}\) = 0

In particular, we do not assume any sign-changing conditions on a(x) or b(x).  Our approach depends on the method of sub-supersolutions.