Existence, uniqueness and stability of positive solutions for a class of semilinear elliptic systems. (English) Zbl 1295.35218

Cui, Renhao; Li, Ping; Shi, Junping; Wang, Yunwen

Topol. Methods Nonlinear Anal. 42, No. 1, 91-104 (2013).

Let \(\Omega\subset {\mathbb R}^n\) be a bounded domain with smooth boundary and assume that \(f\) and \(g\) are smooth real-valued functions defined on \([0,\infty)\times [0,\infty)\) such that \(f_u(u,v)\geq 0\) and \(g_u(u,v)\geq 0\) for all \((u,v)\in [0,\infty)\times [0,\infty)\).
This paper deals with the qualitative analysis of positive solutions of the cooperative semilinear elliptic system \[ \Delta u+\lambda f(u,v)=0,\quad \Delta v+\lambda g(u,v)=0, \] subject to the Dirichlet boundary condition \(u=v=0\) on \(\partial\Omega\).
Under natural sublinearity assumptions, the first main result of this paper establishes a stability property of the solutions. Next, using the stability result and related bifurcation properties, the authors prove the existence and uniqueness of the positive solution and obtain the precise global bifurcation diagram of the system. An application to the logistic system is provided in the final section of this paper.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

35J47	Second-order elliptic systems
35B65	Smoothness and regularity of solutions to PDEs
35B32	Bifurcations in context of PDEs
35B09	Positive solutions to PDEs
35J61	Semilinear elliptic equations

Keywords:

semilinear elliptic system; stability; bifurcation; logistic system