On the critical Neumann problem with weight in exterior domains.
Let \(\Omega\subset\mathbb R^n\) be a bounded domain with a smooth boundary \(\partial\Omega\) and set \(\Omega^c=\mathbb R^N\setminus \overline{\Omega}\). Here, the authors study the nonlinear Neumann problem \[ -\Delta u+\lambda u= Q(x)|u|^{2^*-2}u \quad\text{in }\Omega^c,\quad u>0, \qquad \frac{\partial u}{\partial\nu}=0 \quad\text{on }\partial\Omega, \] where the coefficient \(Q\) is continuous and positive on \(\overline{\Omega}^c\), \(\lambda>0\) is a parameter and \(2^*= \frac {2N}{N-2}\), \(N\) is a critical Sobolev exponent. The authors study the common effect of the mean curvature for the boundary \(\partial\Omega\) and the shape of the graph of the coefficient \(Q\) on the existence of least energy solutions.
scientific article; zbMATH DE number 1929513
scientific article