\(H^2\)-solutions for some elliptic equations with nonlinear boundary conditions (Q2918418)

The paper deals with \(-\Delta u+b u=f(x)\) in a bounded domain \(\Omega\subset{\mathbb R}^N\) subject to \({{\partial u}\over{\partial n}} =g(u)-\beta(u)\) on \(\partial\Omega\). A typical example of a radiation function is \(\beta(u)=\left|u\right|^3 u\) and a bounded absorption function \(g\). The authors provide a general framework in which the problem has an \(H^2\) solution. The proof relies on variational methods, a priori bounds for a family of approximate problems, and a limit argument.