Boundedness, a priori estimates and existence of solutions of elliptic systems with nonlinear boundary conditions. (English) Zbl 1234.35107

Summary: Consider the semilinear elliptic system \(-\Delta u=f(x,u,v)\), \(-\Delta v=g(x,u,v)\), \(x\in\Omega\), complemented by the nonlinear boundary conditions \(\partial_vu=f(y,u,v)\), \(\partial_\nu v=\widetilde g(y,u,v)\), \(y\in\partial\Omega\), where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) and \(\partial_\nu\) denotes the derivative with respect to the outer unit normal \(\nu\). We show that any positive very weak solution of this problem belongs to \(L^\infty\) provided the functions \(f,g,\widetilde f,\widetilde g\) satisfy suitable polynomial growth conditions. In addition, all positive solutions are uniformly bounded provided the right-hand sides are bounded in \(L^1\). We also prove that our growth conditions are optimal. Finally, we show that our results remain true for problems involving nonlocal nonlinearities and we use our a priori estimates to prove the existence of positive solutions.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B33 Critical exponents in context of PDEs
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
35B09 Positive solutions to PDEs