Nonzero and positive solutions of a superlinear elliptic system. (English) Zbl 1090.35525

Zuluaga, Mario

Arch. Math., Brno 37, No. 1, 63-70 (2001).

The author considers an undecoupling elliptic system \[ -\Delta u = \lambda u + \delta v + | u | ^{r-1}, \qquad -\Delta v = \theta u + \gamma v + | v | ^{s-1} \] in \(\Omega \subset \mathbb R^n\), \(n\geq 4\), with zero Dirichlet condition. The exponents are assumed to satisfy \(r > 2\), \( s < 2^* = \frac {2n}{n-2}\). Using Leray-Schauder degree theory and measure theory he shows the existence of positive solutions, under conditions relating the spectrum of \(-\Delta \) and the coefficients of the system. Further, applications to biharmonic equations and the scalar case are given.

Reviewer: Martin Kolář (Brno)

MSC:

35J55	Systems of elliptic equations, boundary value problems (MSC2000)
47H11	Degree theory for nonlinear operators

Keywords:

elliptic system; Leray-Schauder degree; maximum principle