\(W^{1,1}(\omega)\) solutions of nonlinear problems with nonhomogeneous Neumann boundary conditions
The paper starts with an introduction summarizing some results on nonlinear problems with Dirichlet boundary conditions. The new result here is, however, some existence result for a Neumann type problem, namely \(-\operatorname{div}(a(x,u,\nabla u))+|u|^{s-1}u=0\) on some domain \(\Omega\subset{\mathbb R}^N\) with the boundary condition \(a(x,u,\nabla u)\cdot\eta=\psi\) on \(\partial\Omega\). Here \(\eta\) is the unit outward normal on \(\partial\Omega\), \(\psi\in L^m(\partial\Omega)\) for some \(m>1\), and \(a:\Omega\times{\mathbb R}\times{\mathbb R}^N\to{\mathbb R}^N\) is a Caratheodory function with the usual \(p\)-growth conditions. The main result is the existence of a \(W^{1,1}\) distributional solution to the problem in the borderline case \(m={N-1\over N(p-1)}\), with \(1<p<2-{1\over N}\) and \(0<s<{1+N(p-1)\over N-1}\). The proof uses approximation by problems with \(L^\infty\) data.
scientific article; zbMATH DE number 6526831
scientific article