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Keywords = nonhomegeneous Neumann boundary condition

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9 pages, 246 KiB  
Article
Nonexistence of Global Weak Solutions for a Nonlinear Schrödinger Equation in an Exterior Domain
by Awatif Alqahtani, Mohamed Jleli, Bessem Samet and Calogero Vetro
Symmetry 2020, 12(3), 394; https://doi.org/10.3390/sym12030394 - 4 Mar 2020
Cited by 1 | Viewed by 2532
Abstract
We study the large-time behavior of solutions to the nonlinear exterior problem [...] Read more.
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. Full article
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