Nonexistence of global weak solutions for a nonlinear Schrodinger equation in an exterior domain
- Autori: Alqahtani A.; Jleli M.; Samet B.; Vetro C.
- Anno di pubblicazione: 2020
- Tipologia: Articolo in rivista
- OA Link: http://hdl.handle.net/10447/412554
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.