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

                                                               −Δpu = K(x) uq/|x|p , u > 0 in Rⁿ \ {0},

where n ≥ 3, 1 < p < n, q > 0, Δ_pu = 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_Rⁿ u = 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 

                                                                   −Δ_p1u = K₁(x) v^q1/|x|^p1 ,
                                                                   −Δ_p2v = K2(x) u^q2/|x|^p2 ,

where u, v > 0 in Rⁿ \ {0}, n ≥ 3, 1 < p1, p2 < n, q1, q2 > 0, and K₁(x),K₂(x) are double bounded functions.