Existence and uniqueness of positive solutions to certain differential systems (Q1977686)

The authors study the elliptic system \(-\Delta v =f(x,u)\) in \(\Omega\) and \(-\Delta u = v^\beta\) in \(\Omega\) with Dirichlet boundary data on a bounded smooth domain \(\Omega\) of \({\mathbb R}^N\). When \(f(x,u)=u^\alpha\) where \(\alpha>0\), \(\beta>0\) and \(\alpha \beta >1\), this superlinear case was treated by \textit{P. Clément}, \textit{P. Felmer} and \textit{E. Mitidieri} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 2, 367-393 (1997; Zbl 0902.35051)] using the Mountain Pass Theorem. The authors treat there the sublinear case \(f(x,u)=u^\alpha\) where \(\alpha \beta <1\) and the case of a more general nonlinearity \(f\) assumed to have some sub-\(\beta\) growth. The existence is obtained by the study of the critical points of a general functional \(I\) defined on a Banach space \(B\). The growth conditions supposed on \(f\) and the regularity conditions on \(B\) guarantee that a slight modification of an asymptotic condition on \(f\) introduced in [Nonlinear Anal., Theory Methods Appl. 10, 55-64 (1986; Zbl 0593.35045)] by \textit{H. Brezis} and \textit{L. Oswald} to study a semilinear elliptic problem with a sublinear nonlinearity is enough to get a nontrivial minimum of \(I\). A uniqueness result of positive solutions is also derived when the nonlinearity satisfies a monotonicity condition.