Nonexistence of global weak solutions for a nonlinear Schrodinger equation in an exterior domain

Calogero Vetro, Mohamed Jleli, Bessem Samet, Awatif Alqahtani

Research output: Contribution to journalArticlepeer-review

Abstract

We study the large-time behavior of solutions to the nonlinear exterior problem Lu(t, x) = κ[pipe]u(t, x)[pipe]p, (t, x) ∈ (0, ∞) x Dc under the nonhomegeneous Neumann boundary condition (t, x) = λ(x), (t, x) ∈ (0, ∞) x ∂D, where L:= i∂t + Δ is the Schrodinger operator, D = B(0, 1) is the open unit ball in RN, N ≥ 2, Dc = RND, p > 1, κ ∈ , κ ≠ 0, λ ∈ L1(∂D, ) is a nontrivial complex valued function, and ∂v is the outward unit normal vector on ∂D, relative to Dc. Namely, under a certain condition imposed on (κ, λ), we show that if N ≥ 3 and p < pc, where pc =, 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.
Original languageEnglish
Pages (from-to)1-9
Number of pages9
JournalSymmetry
Volume12
Publication statusPublished - 2020

All Science Journal Classification (ASJC) codes

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • General Mathematics
  • Physics and Astronomy (miscellaneous)

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