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Mathematische Annalen

, Volume 374, Issue 1–2, pp 361–394 | Cite as

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

  • Moshe MarcusEmail author
  • Phuoc-Tai Nguyen
Article
  • 146 Downloads

Abstract

Let \(\Omega \subset \mathbb {R}^N\) (\(N \ge 3\)) be a \(C^2\) bounded domain and \(F \subset \partial \Omega \) be a \(C^2\) submanifold with dimension \(0 \le k \le N-2\). Denote \(\delta _F=\mathrm {dist}\,(\cdot ,F)\), \(V=\delta _F^{-2}\) and \(C_H(V)\) the Hardy constant relative to V in \(\Omega \). We study positive solutions of equations (LE) \(-L_{\gamma V} u = 0\) and (NE) \(-L_{\gamma V} u+ f(u) = 0\) in \(\Omega \) where \(L_{\gamma V}=\Delta + \gamma V\), \(\gamma < C_H(V)\) and \(f \in C(\mathbb {R})\) is an odd, monotone increasing function. We extend the notion of normalized boundary trace introduced in Marcus and Nguyen (Ann Inst H. Poincaré (C) Non Linear Anal 34:69–88, 2015) and employ it to investigate the linear equation (LE). Using these results we obtain properties of moderate solutions of (NE). Finally we determine a criterion for subcriticality of points on \(\partial {\Omega }\) relative to f and study b.v.p. for (NE). In particular we establish existence and stability results when the data is concentrated on the set of subcritical points.

Mathematics Subject Classification

35J60 35J75 35J10 35J66 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnionHaifaIsrael
  2. 2.Department of Mathematics and StatisticsMasaryk UniversityBrnoCzech Republic

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