Boundedness, a priori estimates and existence of solutions of elliptic systems with nonlinear boundary conditions

Abstract

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 $\partial_{\nu}u =\tilde f(y,u,v)$, $\partial_{\nu}v =\tilde g(y,u,v)$, $y\in\partial\Omega$, 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 $f,g,\tilde f,\tilde g$ 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.