## A priori bounds and existence of positive solutions for semilinear elliptic systems.(English)Zbl 1359.35053

Summary: We provide a-priori $$L^\infty$$ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions $$v^p$$ and $$u^q$$ for any $$(p,q)$$ lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich-Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system.

### MSC:

 35J47 Second-order elliptic systems 35B45 A priori estimates in context of PDEs 35B09 Positive solutions to PDEs 35J61 Semilinear elliptic equations
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### References:

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