A uniqueness result for a Neumann problem involving the critical Sobolev exponent
Let \(\Omega\) be a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\). The authors mainly interested in positive solutions of the Neumann problem for the following nonlinear elliptic equation: \[ -d^2\Delta u+u=u^p \quad\text{in }\Omega,\qquad u>0 \quad\text{in }\Omega, \qquad \frac{\partial u}{\partial n}=0 \quad\text{on }\partial\Omega, \tag{1} \] where \(d\) is a positive constant, \(p= \frac{N+2}{N-2}\), \(\Delta\) is the Laplacian and \(\nu\) stands for the unit outer normal to \(\partial\Omega\). In the case of small \(d>0\), the authors show the uniqueness (up to rotation) of the one-bubbling solution which concentrates at a point of the boundary.
scientific article; zbMATH DE number 1944287