A positive solution for some critical \(p\)-Laplacian systems (Q2018882)

The author studies the quasilinear elliptic system \[ \begin{aligned} -\Delta_p u=a|u|^{p-2}u+bu|u|^{p/2-2}|v|^{p/2}+f(x,u,v)\quad & \text{ in }\Omega, \\ -\Delta_p v=bu|u|^{p/2-2}|v|^{p/2}+c|u|^{p-2}u+g(x,u,v)\quad & \text{ in }\Omega, \end{aligned} \] subject to homogeneous Dirichlet boundary conditions. Here \(\Omega\) is a smooth and bounded domain in \(\mathbb R^N\), \(a,b,c\in L^\infty(\Omega)\). The author proves the existence of a positive solution by variational arguments.