Pohozaev identities for a pseudo-relativistic Schr\"odinger operator and applications

Abstract

In this paper we prove a Pohozaev-type identity for both the problem $(-\Delta+m^2)^su=f(u)$ 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 $\sqrt{-\Delta+m^2}$ 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 $(-\Delta+m^2)^s$, which lead us to define a weak solution $u$ of the original problem if the identity \begin{equation}\label{defsola}\int_{\mathbb{R}^N}(-\Delta+m^2)^{s/2}u(-\Delta+m^2)^{s/2}v\dd x=\int_{ \mathbb{R}^N}f(u)v\dd x\tag{S}\end{equation} is satisfied by all $v\in H^{s}(\mathbb{R}^N)$. The obtained Pohozaev-type identity is then applied to prove both a result of nonexistence of solution to the case $f(u)=|u|^{p-2}u$ if $p\geq 2^{*}_s$ and a result of existence of a ground state, if $f$ is modeled by $\kappa u^3/(1+u^2)$, for a constant $\kappa$. In this last case, we apply the Nehari-Pohozaev manifold introduced by D. Ruiz. Finally, we prove that positive solutions of $(-\Delta+m^2)^su=f(u)$ are radially symmetric and decreasing with respect to the origin, if $f$ is modeled by functions like $t^\alpha$, $\alpha\in(1,2^{*}_s-1)$ or $t\ln t$.