Uniqueness of positive solutions for cooperative Hamiltonian elliptic systems (Q2816345)

This paper is devoted to the study of the following semilinear elliptic system NEWLINE\[NEWLINE\begin{cases} \Delta u+\lambda f(v)=0,\quad x\in \Omega,\\ \Delta v+\lambda g(u)=0,\quad x\in \Omega,\\ u(x)=v(x)=0, \quad x\in \partial\Omega,\end{cases}\tag{1}NEWLINE\]NEWLINE where \(\lambda > 0\), \(\Omega\) is a bounded smooth domain, and \(f\), \(g\) are continuously differentiable functions defined on \(\mathbb R^+\) satisfying \(f'(v)>0\), for \(v>0\) and \(g'(u)>0\), for \(u>0\) with \(f(0)\geq 0\) and \(g(0)\geq 0\).NEWLINENEWLINEUsing implicit function theorems, bifurcation theory, and ordinary differential equations, the authors prove the uniqueness of positive solutions of the system (1) for the case of sublinear and superlinear nonlinearities.