An upper bound for the sharp constant in the Sobolev inequality and its application to an interior Dirichlet problem for the homogeneous stationary Schrödinger equation \(\Delta u+q(x)u=0\) (Q2455217)

The author deals with the following Dirichlet problem for the Schrödinger operator \[ \Delta_x u(x)+ q(x)= 0\quad\text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Delta_x\) is the Laplacian, \(q(x)\) is a given function in \(\Omega\), and \(\Omega\) is the arbitrary domain in \(\mathbb{R}^d\) with a boundary \(\partial\Omega\subset C^{1,\mu}\), \(0< \mu\leq 1\). The author obtains sufficient conditions for the nonexistence of nontrivial generalized solutions for (1). To this end, he presents an upper bound for the sharp constant in the corresponding Sobolev inequality.