On the nonlinear Neumann problem with critical and supercritical nonlinearities
Volume 417 / 2003
Dissertationes Mathematicae 417 (2003), 1-59
MSC: 35B33, 35J65, 35Q55.
DOI: 10.4064/dm417-0-1
Abstract
We investigate the solvability of the Neumann problem (1.1) involving a critical Sobolev exponent. In the first part of this work it is assumed that the coefficients $Q$ and $h$ are at least continuous. Moreover $Q$ is positive on $\,{\overline{\!\mit\Omega}}$ and $\lambda>0$ is a parameter. We examine the common effect of the mean curvature and the shape of the graphs of the coefficients $Q$ and $h$ on the existence of low energy solutions. In the second part of this work we consider the same problem with $Q$ replaced by $-Q$. In this case the problem can be supercritical and the existence results depend on integrability conditions on $Q$ and $h$.
