A priori bounds and existence of positive solutions for semilinear elliptic systems

We provide a-priori L∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions vp and uq 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.