Positive bounded solutions for semilinear elliptic systems with indefinite weights in the half-space

Abstract

In this article, we study the existence and nonexistence of positive bounded solutions of the Dirichlet problem $$\displaylines{ -\Delta u=\lambda p(x)f(u,v),\quad \text{in } {\mathbb{R}}_+^n,\cr -\Delta v=\lambda q(x)g(u,v), \quad \text{in } {\mathbb{R}}_+^n,\cr u=v=0\quad \text{on }\partial {\mathbb{R}}_+^n,\cr \lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=0, }$$ where ${\mathbb{R}}_+^n=\{x=(x_1,x_2,\dots, x_n)\in {\mathbb{R}}^n: x_n>0\}$ ($n\geq 3$) is the upper half-space and $\lambda$ is a positive parameter. The potential functions p,q are not necessarily bounded, they may change sign and the functions $f,g:\mathbb{R}^2 \to \mathbb{R}$ are continuous. By applying the Leray-Schauder fixed point theorem, we establish the existence of positive solutions for $\lambda$ sufficiently small when $f(0,0)>0$ and $g(0,0)>0$. Some nonexistence results of positive bounded solutions are also given either if $\lambda$ is sufficiently small or if $\lambda$ is large enough.

Keywords

• Semilinear elliptic system
• Leray-Schauder fixed point theorem
• positive bounded solution
• indefinite potential