arXiv 1810.075972018-10math.AP

一个伪相对论性薛定谔算子的Pohozaev恒等式及其应用

Pohozaev identities for a pseudo-relativistic Schrödinger operator and applications

H. Bueno, G. A Pereira, A. H. Souza Medeiros

备注：: 35 pages

摘要：

在本文中，我们证明了一个Pohozaev型恒等式的问题$(-Δ+m^2)^su=f(u)$在$\mathbb{R}^N$和它的调和扩展到$\mathbb{R}^{N+1}_+$时$0<s<1$。因此，我们的设置包括伪相对论算子$\sqrt{-Δ+m^2}$，并且这里显示的结果是原创的，据我们所知。恒等式首先在扩展设置中获得，然后“翻译”到原始问题中。为了做到这一点，我们开发了一个特定的傅立叶变换理论的分数算子$(-Δ+m^2)^s$，这导致我们定义一个弱解$u$的原问题，如果身份\begin{equation}\label{defsola}\int_{\mathbb{R}^N}(-Δ+m^2)^{s/2}u(-Δ+m^2)^{s/2}v\dd x=\int_{ \mathbb{R}^N}f(u)v\dd x\tag{S}\end{equation}是由所有$v\in H^{s}(\mathbb{R}^N)$满足。得到的Pohozaev型恒等式，然后应用于证明的结果不存在的解决方案的情况下，$f(u)=|u|^{p-2}u$，如果$p\geq 2^{*}_s$和一个结果存在的基态，如果$f$是由$κu^3/(1+u^2)$，为一个常数$κ$。在最后一种情况下，我们应用D.鲁伊斯最后，我们证明了$(-Δ+m^2)^su=f(u)$的正解是径向对称的，并相对于原点减少，如果$f$是由类似$t^α$，$α\in(1,2^{*}_s-1)$或$t\ln t$的功能。

英文摘要：

In this paper we prove a Pohozaev-type identity for both the problem $(-Δ+m^2)^su=f(u)$ 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 $\sqrt{-Δ+m^2}$ 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 $(-Δ+m^2)^s$, which lead us to define a weak solution $u$ of the original problem if the identity \begin{equation}\label{defsola}\int_{\mathbb{R}^N}(-Δ+m^2)^{s/2}u(-Δ+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 $κu^3/(1+u^2)$, 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 $(-Δ+m^2)^su=f(u)$ are radially symmetric and decreasing with respect to the origin, if $f$ is modeled by functions like $t^α$, $α\in(1,2^{*}_s-1)$ or $t\ln t$.

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