Existence of positive entire solutions of a semilinear p-Laplacian problem with a gradient term

Abstract

In this paper, we study a semilinear p-Laplacian problem −Δpu+h(x)|∇u| q = b(x)g(u), u > 0, x ∈ R N , lim |x|→∞ u(x )= 0, where q ∈ (p −1,p), b, h ∈C α(R N ) for some α ∈ (0,1), h(x) 0, b(x) > 0,∀x ∈ R N , and g ∈ C 1 ((0,∞),(0,∞)) which may be singular at 0. Using a sub-supersolution argument and a perturbed argument, we obtain the existence of entire solutions to the problem. No monotonicity condition is imposed on the functions g(s) and g(s) sp−1 .