On existence of positive solutions of p-Laplace equations

Abstract

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δ p u = div(|∇u| p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \\ {0}, n ≥ 3, 1 < p 1, p 2 < n, q 1, q 2 > 0, and K 1(x), K 2(x) are double bounded functions.