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Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  04 September 2023

Sergio Lancelotti  and
Riccardo Molle Open the ORCID record for Riccardo Molle [Opens in a new window]
Sergio Lancelotti
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi n. 24, 10129 Torino, Italy (sergio.lancelotti@polito.it)
Riccardo Molle
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy (molle@mat.uniroma2.it)
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Abstract

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation

$$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$
where $\Omega =\mathbb {R}^N$ or $\mathbb {R}^N\setminus \Omega$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (Dρ), or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (Dρ) is fulfilled. There are no assumptions on the size of $\mathbb {R}^N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}^N$.

Keywords

Nonlinear Schrödinger equationsnormalized solutionsexterior domainspositive solutions

MSC classification

Primary: 35J10: Schrödinger operator
Secondary: 35J60: Nonlinear elliptic equations 35J20: Variational methods for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 34
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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