Entire solutions blowing up at infinity for semilinear elliptic systems.
The authors consider the elliptic system in \(\mathbb R^N\), with \(N \geq 3\), \[ \begin{cases} \Delta u(x) = p(x) g(v(x)),\\ \Delta v(x) = q(x) f(u(x)). \end{cases}\tag{1} \] Here \(p\) and \(q\) are locally Hölder continuous, non-negative, radially simmetric functions of domain \(\mathbb R^N\), while \(f\) and \(g\) are locally Hölder continuous, positive, non-decreasing functions of domain \([0, +\infty)\). \(u\) and \(v\) are entire solutions of (1) if they are classical positive solutions of domain \(\mathbb R^N\). They are said to be large if they tend to \(+\infty\), as \(| x| \to +\infty\). Define \[ {\mathcal G} := \{(a,b) \in \mathbb R^+ \times \mathbb R^+ : \exists \text{ an entire radial solution of (1)} \text{ so that } (u(0),v(0)) = (a,b)\}. \] The problem of the structure of \({\mathcal G}\) is studied in great detail, together with the problem of establishing whether the solutions are large or bounded. Concerning this second point, the authors extend previous results by \textit{A. V. Lair} and \textit{A. W. Wood} [J. Differ. Equations 164, No. 2, 380--394 (2000; Zbl 0962.35052)] in the case of pure powers.
scientific article; zbMATH DE number 1975954