Positive continuous solutions for some semilinear elliptic problems in the half space

Abstract

The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K^{infty}({mathbb{R}}_{+}^{d}) that was defined in (Bachar et al. 2002:41, 2002) for d=2 and in (Bachar and Mâagli 9(2):153–192, 2005) for dgeq 3 and adapted to study some nonlinear elliptic problems in the half space. The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants λ and μ to the following system: Delta u=lambda f(x,u,v), Delta v=mu g(x,u,v) in {mathbb{R}}_{+}^{d}, lim _{xrightarrow (xi ,0)}u(x)=a_{1}phi _{1}(xi ), lim _{xrightarrow (xi ,0)}v(x)=a_{2}phi _{2}(xi ) for all xi in {mathbb{R}}^{d-1}, lim _{x_{d} rightarrow infty}frac{u(x)}{x_{d}}=b_{1}, lim _{x_{d} rightarrow infty}frac{v(x)}{x_{d}}=b_{2}, where phi _{1} and phi _{2} are nontrivial nonnegative continuous functions on partial {mathbb{R}}_{+}^{d}= {mathbb{R}}^{d-1}times {0}, a_{1}, a_{2}, b_{1}, b_{2} are nonnegative constants such that (a_{1}+b_{1})(a_{2}+b_{2})>0. The functions f and g are nonnegative and belong to a class of functions containing in particular all functions of the type f(x,u,v)=p(x) u^{alpha}g_{1}(v) and g(x,u,v)=q(x)g_{2}(u)v^{beta} with alpha geq 1, beta geq 1, g_{1}, g_{2} are continuous on [0,infty ), and p,q are nonnegative functions in K^{infty}({mathbb{R}}_{+}^{d}).