Necessary and sufficient conditions of existence for a system involving the \(p\)-Laplacian (\(1 < p < N\))
The existence of positive solutions for the system \[ -\Delta_pu=\mu f(x,u,v)\quad\text{in } \mathbb{R}^N,\quad -\Delta_pv=\mu g(x,u,v)\quad\text{in } \mathbb{R}^N,\quad \lim_{|x|\to\infty}u (x)=0=\lim_{|x|\to\infty}v (x), \] where \(\Delta_p\) is the \(p\)-Laplacian \((1<p<N)\) is studied. For \(0<\mu\leq \mu_*\) there exists a positive solution and for \(\mu> \mu^*\) the system has no positive solution. In some case \(\mu_*=\mu^*\) because these parameters depend on \(p\), \(N\) and the growth of \(f\) and \(g\). The existence result is related to subsolution and supersolution methods and the nonexistence result is a consequence of Picone's identity.
scientific article; zbMATH DE number 1915524
scientific article