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

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.


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