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 $t\ln t$.

我们证明了两个问题的 Pohozaev 类型身份$(-\mathrm{\Delta }+{米}^2{)}^s你=F(你)$在${\mathbb{R}}^{ñ}$及其谐波扩展为${\mathbb{R}}_{+}^{ñ+1}$当 0< s <1。因此，我们的设置包括伪相对论算子$\sqrt{-\mathrm{\Delta }+{米}^2}$据我们所知，这里显示的结果是原始的。身份首先在扩展设置中获得，然后“翻译”为原始问题。为了做到这一点，我们为分数算子开发了一个特定的傅里叶变换理论$(-\mathrm{\Delta }+{米}^2{)}^s$，这导致我们为原始问题定义一个弱解u ，如果恒等式(S)${\int }_{{\mathbb{R}}^{ñ}}(-\mathrm{\Delta }+{米}^2{)}^{s/2}你(-\mathrm{\Delta }+{米}^2{)}^{s/2}v\mathrm{d}X={\int }_{{\mathbb{R}}^{ñ}}F\left(你\right)v\mathrm{d}X$(S)所有人都满意$v\in H^s\left({\mathbb{R}}^{ñ}\right)$. 然后应用所获得的 Pohozaev 型恒等式来证明这两个结果都是不存在解决方案的结果$F\left(你\right)=|你{|}^{p-2}你$如果$p\ge 2_s^{*}$以及存在基态的结果，如果f由下式建模$\kappa {你}^3/(1+{你}^2)$, 对于常数κ。在最后一种情况下，我们应用了 D. Ruiz 引入的 Nehari-Pohozaev 流形。最后，我们告知积极的解决方案$(-\mathrm{\Delta }+{米}^2{)}^s你=F(你)$如果f由以下函数建模，则径向对称且相对于原点递减${吨}^{\alpha }$,$\alpha \in (1,2_s^{*}-1)$或者$吨\ln 吨$.