On singular quasi-monotone \((p, q)\)-Laplacian systems
In this paper, the existence of solutions of the \((p,q)\)-Laplacian system NEWLINE\[NEWLINE \begin{cases} -\Delta_p u = f(x,u,v) & \text{in} \;\Omega, \\ -\Delta_q v = g(x,u,v) & \text{in} \;\Omega, \\ u, v >0 & \text{in} \;\Omega, \\ u,v=0 & \text{on} \;\partial\Omega \end{cases}NEWLINE\]NEWLINE are obtained, where \(\Omega\) is a smooth bounded domain in \({\mathbb R}^n\), \(n \geq 1\), \(\Delta_p u = \text{div}(|\nabla u|^{p-2} \nabla u)\) is the \(p\)-Laplacian of \(u\), \(1<p, q<\infty\), and \(f\) and \(g\) are Carathéodory functions on \(\Omega\times(0,\infty)\times(0,\infty)\), i.e. \(f(x,s,t)\) and \(g(x,s,t)\) are measurable in \(x\) for all \((s,t)\) and continuous in \((s,t)\) for all \(x\). The sub- and supersolution method and perturbation arguments are used.
scientific article; zbMATH DE number 6053147