Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain
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Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabi
2
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(3), 394; https://doi.org/10.3390/sym12030394
Received: 19 December 2019 / Revised: 9 February 2020 / Accepted: 20 February 2020 / Published: 4 March 2020
(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)
We study the large-time behavior of solutions to the nonlinear exterior problem under the nonhomegeneous Neumann boundary condition where is the Schrödinger operator, is the open unit ball in , , , , , , is a nontrivial complex valued function, and is the outward unit normal vector on , relative to . Namely, under a certain condition imposed on , we show that if and , where then the considered problem admits no global weak solutions. However, if , then for all , the problem admits no global weak solutions. The proof is based on the test function method introduced by Mitidieri and Pohozaev, and an adequate choice of the test function.
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Keywords:
nonlinear Schrödinger equation; exterior domain; nonhomegeneous Neumann boundary condition; global weak solution
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MDPI and ACS Style
Alqahtani, A.; Jleli, M.; Samet, B.; Vetro, C. Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain. Symmetry 2020, 12, 394. https://doi.org/10.3390/sym12030394
AMA Style
Alqahtani A, Jleli M, Samet B, Vetro C. Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain. Symmetry. 2020; 12(3):394. https://doi.org/10.3390/sym12030394
Chicago/Turabian StyleAlqahtani, Awatif; Jleli, Mohamed; Samet, Bessem; Vetro, Calogero. 2020. "Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain" Symmetry 12, no. 3: 394. https://doi.org/10.3390/sym12030394
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