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

Abstract

Abstract The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions K ∞ ( R + d ) $K^{\infty}({\mathbb{R}}_{+}^{d})$ that was defined in (Bachar et al. 2002:41, 2002) for d = 2 $d=2$ and in (Bachar and Mâagli 9(2):153–192, 2005) for d ≥ 3 $d\geq 3$ 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: Δ u = λ f ( x , u , v ) $\Delta u=\lambda f(x,u,v)$ , Δ v = μ g ( x , u , v ) $\Delta v=\mu g(x,u,v)$ in R + d ${\mathbb{R}}_{+}^{d}$ , lim x → ( ξ , 0 ) u ( x ) = a 1 ϕ 1 ( ξ ) $\lim _{x\rightarrow (\xi ,0)}u(x)=a_{1}\phi _{1}(\xi )$ , lim x → ( ξ , 0 ) v ( x ) = a 2 ϕ 2 ( ξ ) $\lim _{x\rightarrow (\xi ,0)}v(x)=a_{2}\phi _{2}(\xi )$ for all ξ ∈ R d − 1 $\xi \in {\mathbb{R}}^{d-1}$ , lim x d → ∞ u ( x ) x d = b 1 $\lim _{x_{d} \rightarrow \infty}\frac{u(x)}{x_{d}}=b_{1}$ , lim x d → ∞ v ( x ) x d = b 2 $\lim _{x_{d} \rightarrow \infty}\frac{v(x)}{x_{d}}=b_{2}$ , where ϕ 1 $\phi _{1}$ and ϕ 2 $\phi _{2}$ are nontrivial nonnegative continuous functions on ∂ R + d = R d − 1 × { 0 } $\partial {\mathbb{R}}_{+}^{d}= {\mathbb{R}}^{d-1}\times \{0\}$ , a 1 , a 2 , b 1 , b 2 $a_{1}, a_{2}, b_{1}, b_{2}$ are nonnegative constants such that ( a 1 + b 1 ) ( a 2 + b 2 ) > 0 $(a_{1}+b_{1})(a_{2}+b_{2})>0$ . 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 α g 1 ( v ) $f(x,u,v)=p(x) u^{\alpha}g_{1}(v)$ and g ( x , u , v ) = q ( x ) g 2 ( u ) v β $g(x,u,v)=q(x)g_{2}(u)v^{\beta}$ with α ≥ 1 $\alpha \geq 1$ , β ≥ 1 $\beta \geq 1$ , g 1 $g_{1}$ , g 2 $g_{2}$ are continuous on [ 0 , ∞ ) $[0,\infty )$ , and p , q $p,q$ are nonnegative functions in K ∞ ( R + d ) $K^{\infty}({\mathbb{R}}_{+}^{d})$ .

Keywords

• Green function
• Kato class
• Nonlinear elliptic systems
• Positive solution
• Maximum principle
• Schauder fixed point theorem