Existence of entire large positive solutions of semilinear elliptic systems
The authors study the existence of entire solutions of the elliptic system \[ \Delta u= p(|x|v^\alpha,\quad\Delta v= q(|x|) u^\beta\tag{\(*\)} \] in \(\mathbb{R}^N\), \(N\geq 3\), where \(p\) and \(q\) are assumed to be continuous and circumferentially positive in \(\mathbb{R}^N\). A function \(f\) is, by definition, circumferentially positive in a domain \(\Omega\subseteq \mathbb{R}^N\) if \(f\) is nonnegative, and for every \(x_0\in \Omega\) with \(f(x_0)= 0\) there exists a domain \(\Omega_0\subset \Omega\) such that \(x_0\in \Omega_0\) and \(f(x)> 0\) on \(\partial\Omega_0\). Consider first the sublinear case \(0< \alpha\leq\beta< 1\). Given any values \(a,b\geq 0\), the authors prove the existence of an entire positive radial solution of \((*)\) with \(u(0)= a\), \(v(0)= b\). All such solutions are bounded if \[ \int^\infty_0 tp(t) dt< \infty,\quad \int^\infty_0 tq(t) dt< \infty\tag{\(**\)} \] and are large (i.e. blow up at infinity) if these integrals are unbounded. In the sublinear case \(1<\alpha\leq \beta\) let \(G= \{(a,b)\in \mathbb{R}^+\times \mathbb{R}^+:u(0)= a\), \(v(0)= b\) and \((u,v)\) is an entire solution of \((*)\)\}. Assume, \(p\) and \(q\) satisfy \((**)\). Then there are infinitely many entire positive radial solutions of system \((*)\) and \(G\) is a closed bounded convex subset of \(\mathbb{R}^+\times \mathbb{R}^+\). Furthermore, if \(E(G)\) is the closure of \(\{(a, b)\in \partial G: a>0, b>0\}\) then any entire positive radial solution with \((u(0), v(0))\in E(G)\) is large.
scientific article; zbMATH DE number 1515448