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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.

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