Elliptic and Parabolic Equations - Abstract

Quittner, Pavol

Boundedness of solutions of superlinear elliptic boundary value problems

We study the impact of boundary conditions on the boundedness of very weak solutions of superlinear elliptic boundary value problems. After presenting recent results on scalar problems, we will consider the model elliptic system $-Delta u=f(x,v)$, $-Delta v+v=g(x,u)$ in a bounded smooth domain $Omega$. Here $f,g$ are nonnegative Carathéodory functions satisfying the growth conditions $fleq C(1+ v ^p)$, $gleq C(1+ u ^q)$, and we complement the system with one of the following three boundary conditions on $partialOmega$: $u=v=0$ (Dirichlet), $partial_nu u=partial_nu v = 0$ (Neumann) and $u=partial_nu v = 0$ (Dirichlet--Neumann). In all three cases we find optimal conditions on $p,q$ guaranteeing that all nonnegative solutions belong to $L^infty(Omega)$ (and satisfy suitable a priori estimates). We also consider related problems in non-smooth domains and problems with nonlinear boundary conditions.