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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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