Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications

Abstract

We prove a Pohozaev-type identity for both the problem $(-\mathrm{\Delta }+m^2{)}^{s}u=f(u)$ in ${\mathbb{R}}^N$ and its harmonic extension to ${\mathbb{R}}_{+}^{N+1}$ when 0<s<1. So, our setting includes the pseudo-relativistic operator $\sqrt{-\mathrm{\Delta }+m^2}$ 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 transform theory for the fractionary operator $(-\mathrm{\Delta }+m^2{)}^s$, which lead us to define a weak solution u to the original problem if the identity (S) ${\int }_{{\mathbb{R}}^N}(-\mathrm{\Delta }+m^2{)}^{s/2}u(-\mathrm{\Delta }+m^2{)}^{s/2}v\mathrm{d}x={\int }_{{\mathbb{R}}^N}f\left(u\right)v\mathrm{d}x$(S) is satisfied by all $v\in H^s\left({\mathbb{R}}^N\right)$. The obtained Pohozaev-type identity is then applied to prove both a result of non-existence of solution to the case $f\left(u\right)=|u{|}^{p-2}u$ if $p\ge 2_s^{\ast }$ and a result of existence of a ground state, if f is modeled by $\kappa u^3/(1+u^2)$, for a constant κ. In this last case, we apply the Nehari–Pohozaev manifold introduced by D. Ruiz. Finally, we inform that positive solutions of $(-\mathrm{\Delta }+m^2{)}^{s}u=f(u)$ are radially symmetric and decreasing with respect to the origin, if f is modeled by functions like $t^{\alpha }$, $\alpha \in (1,2_s^{\ast }-1)$ or $t\ln t$.

Keywords:

• Pseudo-relativistic Schrödinger operator
• Pohozaev-type identity
• non-existence of solutions
• asymptotic linear nonlinearity
• radial symmetry

AMS Subject Classifications:

• 35J20
• 35Q55
• 35R11

Acknowledgments

All the authors thank Giovany Figueiredo, Olimpio Miyagaki and Minbo Yang for useful conversations.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

G. A. Pereira received research grants by PNPD/CAPES/Brazil; Aldo H. S. Medeiros by CNPq/Brazil. All the authors take part in the project 422806/2018-8 by CNPq/Brazil.