Existence and nonexistence of radial positive solutions of superlinear elliptic systems
The author studies the existence of radial solutions of \[ \begin{cases} -\Delta_p u=a(x)\left\vert u\right\vert^{\alpha-1}u +b(x)\left\vert v\right\vert^{\beta-1}v &\text{on } \Omega,\\ -\Delta_q v=c(x)\left\vert u\right\vert^{\gamma-1}u +d(x)\left\vert v\right\vert^{\delta-1}v &\text{on } \Omega,\end{cases}\tag{1} \] subject to homogeneous Dirichlet conditions. \(\Omega\) is either \({\mathbb R}^N\) or an open ball in \({\mathbb R}^N\) centered at 0, \(\Delta_r\) denotes the \(r\)-Laplacian, and \(a\), \(b\), \(c\), \(d\) are continuous function with positive infima. The author establishes two existence results (positive, radial solution), if \(p-1<\alpha\), \(q-1<\delta\) and \(\Omega\) is bounded, and a nonexistence result for \(\Omega={\mathbb R}^N\) (Liouville Theorem). The latter is obtained by means of a blow-up argument due to \textit{B. Gidas} and \textit{J. Spruck} [Commun. Partial Differ. Equ. 6, 883--901 (1981; Zbl 0462.35041)]. The existence results rely on a priori estimates.
scientific article; zbMATH DE number 1740174