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Open AccessArticle

# Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain

by 1 and
1
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
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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
We study the large-time behavior of solutions to the nonlinear exterior problem $L u ( t , x ) = κ | u ( t , x ) | p , ( t , x ) ∈ ( 0 , ∞ ) × D c$ under the nonhomegeneous Neumann boundary condition $∂ u ∂ ν ( t , x ) = λ ( x ) , ( t , x ) ∈ ( 0 , ∞ ) × ∂ D ,$ where $L : = i ∂ t + Δ$ is the Schrödinger operator, $D = B ( 0 , 1 )$ is the open unit ball in $R N$ , $N ≥ 2$ , $D c = R N ∖ D$ , $p > 1$ , $κ ∈ C$ , $κ ≠ 0$ , $λ ∈ L 1 ( ∂ D , C )$ is a nontrivial complex valued function, and $∂ ν$ is the outward unit normal vector on $∂ D$ , relative to $D c$ . Namely, under a certain condition imposed on $( κ , λ )$ , we show that if $N ≥ 3$ and $p < p c$ , where $p c = N N − 2 ,$ then the considered problem admits no global weak solutions. However, if $N = 2$ , then for all $p > 1$ , 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. View Full-Text
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 Style

Alqahtani, 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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