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Title: Pohozaev-type identities for a pseudo-relativistic schrodinger operator and applications.
Authors: Bueno, Hamilton Prado
Pereira, Gilberto de Assis
Medeiros, Aldo Henrique de Souza
Issue Date: 2019
Citation: BUENO, H. P.; PEREIRA, G. de A.; MEDEIROS, A. H. de S. Pohozaev-type identities for a pseudo-relativistic schrodinger operator and applications. Complex Variables and Elliptic Equations, v. 67, p. 2481-2506, 2022. Disponível em: <>. Acesso em: 06 jul. 2023.
Abstract: In this paper we prove a Pohozaev-type identity for both the prob- lem (−∆ + m2 ) su = f(u) in RN and its harmonic extension to R N+1 + when 0 < s < 1. So, our setting includes the pseudo-relativistic operator √ −∆ + m2 and the results showed here are original, to the best of our knowledge. The identity is first obtained in the extension setting and then “translated” into the original problem. In order to do that, we develop a specific Fourier trans- form theory for the fractionary operator (−∆ + m2 ) s , which lead us to define a weak solution u of the original problem if the identity (S) Z RN (−∆ + m2 ) s/2u(−∆ + m2 ) s/2 vdx = Z RN f(u)vdx is satisfied by all v ∈ Hs (RN ). The obtained Pohozaev-type identity is then applied to prove both a result of nonexistence of solution to the case f(u) = |u| p−2u if p ≥ 2 ∗ s and a result of existence of a ground state, if f is modeled by κu3/(1+u 2 ), for a constant κ. In this last case, we apply the Nehari-Pohozaev manifold introduced by D. Ruiz. Finally, we prove that positive solutions of (−∆ + m2 ) su = f(u) are radially symmetric and decreasing with respect to the origin, if f is modeled by functions like t α, α ∈ (1, 2 ∗ s − 1) or tln t.
ISSN: 1747-6941
Appears in Collections:DEMAT - Artigos publicados em periódicos

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