On the nonlinear Neumann problem involving the critical Sobolev exponent and Hardy potential (Q1884583)

The author studies the nonlinear elliptic problem involving the Neumann conditions \[ -\Delta u+\frac{\mu}{|x|^2}u=Q(x)|u|^{2^*-2}u,\text{ in }\Omega,\quad\frac{\partial u}{\partial n}=0,\text{ on } \partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(0\in\Omega\), \(2^*=\frac{2N}{N-2}\) if \(N\geq 3\), \(Q\) is continuous and positive on \(\overline\Omega\), \(\mu\) is a real parameter, \(n\) is an outward normal to the boundary \(\partial\Omega\). The author establishes the existence of solutions through the mountain-pass principle.