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A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems. (English) Zbl 0823.35064
The author deals with the existence of positive solutions for the system $$-\Delta u= M(x)u+ N(x,u)$$ in $$\Omega$$, $$u=0$$ on $$\partial \Omega$$, where $$\Omega \subset \mathbb{R}^ N$$, $$N\geq 2$$, is a regular bounded domain, $$u= (u_ 1, u_ 2)$$, $$M(x)= (m_{ij} (x))$$ is a $$2\times 2$$ matrix with $$m_{ij} (x)\geq 0$$, for all $$x\in \Omega$$, $$i\neq j$$ and $$N(x,u)= (f(x,u), g(x,u))$$ when $$f$$ has asymptotic behavior at $$+\infty$$ as $$u^ \sigma_ 1$$ and $$g$$ satisfies some subcritical growth. The proofs are based on a known fixed point theorem in conical shells and the a priori bounds technique.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J50 Variational methods for elliptic systems 47J05 Equations involving nonlinear operators (general)