Existence results for classes of Laplacian systems with sign-changing weight (Q2455437)

The authors deal with the boundary value problem \[ \begin{aligned} -\Delta_x u=\lambda F(x,u,v)&\quad\text{in }\Omega,\tag{1}\\ -\Delta_x v=\lambda H(x,u,v) &\quad\text{in }\Omega,\tag{2}\\ u= v= 0 &\quad\text{on }\partial\Omega,\tag{3} \end{aligned} \] where \(\Delta_x\) is the Laplacian, \(\Omega\) is a bounded domain in \(\mathbb{R}^d\), \(d\geq 1\), with smooth boundary. Under suitable assumptions on \(F\) and \(H\), using sub-super-solutions technique, the authors discuss the existence of positive solutions for (1)--(3). Note that, there is no requirement on sign-conditions for \(F(x,0,0)\) or \(H(x,0,0)\), i.e., they could be negative for some \(x\in\Omega\).