Abstract
In this paper, we investigate the stability of positive weak solution for the singular p-Laplacian nonlinear system -div[vertical bar x vertical bar(-ap)vertical bar del u vertical bar(p - 2) del u] + m(x)vertical bar u vertical bar(p - 2) u = lambda vertical bar x vertical bar(-(a+1)p+c)b(x)f(u) in Omega, Bu on partial derivative Omega, where Omega subset of R-n is a bounded domain with smooth boundary Bu = delta h(x)+(1 - delta)partial derivative u/partial derivative n where delta is an element of here [0, 1] h : partial derivative Omega -> R+ with h = 1 when delta = 1, 0 is an element of Omega, 1 < p < n, 0 <= a < n - p/p, m(x) is a weight function, the continuous function b(x) : Omega -> R satisfies either b(x) > 0 or b(x) < 0 for all x is an element of Omega, lambda is a positive parameter and f : [0, infinity) -> R is a continuous function. We provide a simple proof to establish that every positive solution is unstable under certain conditions.