Existence and non-existence of positive solutions of the scalar field system in \(\mathbb{R}^ n\). I (Q1913317)

Let \(\Omega= \mathbb{R}^n\) or \(\Omega=\) Euclidean ball in \(\mathbb{R}^n\) \((n\geq 3)\); the author studies the following boundary value problem \[ \Delta u- v+ v^p= 0,\quad \Delta v- u+ u^q= 0\quad (u, v> 0, x\in \Omega)\tag{1} \] with homogeneous Dirichlet boundary data (replaced by \(u, v\to 0\), as \(|x|\to \infty\) when \(\Omega= \mathbb{R}^n\)), where \(p> 1\), \(q> 1\). It is proved that problem (1) admits a solution if and only if \((p+ 1)^{- 1}+ (q+ 1)^{- 1}> (n- 2)/n\). The existence on a ball and on \(\mathbb{R}^n\) are established by a variational approach and an approximation argument respectively.