Schrödinger equations with singular potentials: linear and nonlinear boundary value problems (Q2423417)

This manuscript is concerned with linear and nonlinear Schrödinger equations in \(\mathbb{R}^N\), \(N \geq 3\). Specifically, let \(\Omega \subseteq \mathbb{R}^N\) be a \(C^2\) bounded domain and \(F\) a \(C^2\) submanifold of \(\partial \Omega\) with dimension \(0 \leq k \leq N -2\). The authors then consider Schrödinger equations with the potential \[ V(x) = V_F(x) = -\frac{1}{[\mathrm{dist}(x,F)]^2}. \] In particular, they study the linear problem \(H_\gamma u = 0\) and the nonlinear problem \(H_\gamma u +f(u) = 0\), where \(H_\gamma = -\Delta + \gamma V_F\), \(f\) is an odd, monotonically increasing, continuous function on \(\mathbb R\), and \[ \gamma < C_H(V) := \inf_{\phi \in C_c^1(\Omega)} \frac{\int_\Omega |\nabla \phi(x)|^2 \, \mathrm{d} x}{\int_\Omega V(x)|\phi(x)|^2 \, \mathrm{d}x} \] is the Hardy constant relative to \(V\). The authors extend the notion of normalized boundary trace introduced in [the authors, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 34, No. 1, 69--88 (2017; Zbl 1356.35114)] and use it to study the linear equation. They obtain existence, uniqueness, and stability results.