We are concerned with the following Schrödinger–Newton problem: -ε2⁢Δ⁢u+V⁢(x)⁢u=18⁢π⁢ε2⁢(∫ℝ3u2⁢(ξ)|x-ξ|⁢𝑑ξ)⁢u,x∈ℝ3.-\varepsilon^{2}\Delta u+V(x)u=\frac{1}{8\pi\varepsilon^{2}}\Bigg{(}\int_{% \mathbb{R}^{3}}\frac{u^{2}(\xi)}{|x-\xi|}\,d\xi\Bigg{)}u,\quad x\in\mathbb{R}^% {3}. For ε small enough, we prove the non-degeneracy of the positive solution to the above problem, that is, the corresponding linear operator ℒε⁢(η)=-ε2⁢Δ⁢η⁢(x)+V⁢(x)⁢η⁢(x)-18⁢π⁢ε2⁢(∫ℝ3uε2⁢(ξ)|x-ξ|⁢𝑑ξ)⁢η⁢(x)-14⁢π⁢ε2⁢(∫ℝ3uε⁢(ξ)⁢η⁢(ξ)|x-ξ|⁢𝑑ξ)⁢uε⁢(x)\mathcal{L}_{\varepsilon}(\eta)=-\varepsilon^{2}\Delta\eta(x)+V(x)\eta(x)-% \frac{1}{8\pi\varepsilon^{2}}\Bigg{(}\int_{\mathbb{R}^{3}}\frac{u_{\varepsilon% }^{2}(\xi)}{|x-\xi|}\,d\xi\Bigg{)}\eta(x)-\frac{1}{4\pi\varepsilon^{2}}\Bigg{(% }\int_{\mathbb{R}^{3}}\frac{u_{\varepsilon}(\xi)\eta(\xi)}{|x-\xi|}\,d\xi\Bigg% {)}u_{\varepsilon}(x) is non-degenerate, i.e., ℒε⁢(ηε)=0⇒ηε=0{\mathcal{L}_{\varepsilon}(\eta_{\varepsilon})=0\Rightarrow\eta_{\varepsilon}=0} for small ε>0{\varepsilon>0}. The main tools are the local Pohozaev identities and the blow-up analysis. This may be the first non-degeneracy result on the peak solutions to the Schrödinger–Newton system.

中文翻译：

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Schrödinger–Newton系统的峰解的非简并性

我们关注以下薛定ding-牛顿问题：-ε2⁢Δ⁢u+V⁢（x）⁢u=18⁢π⁢ε2⁢（∫⁢3u2⁢（ξ）|x-ξ|⁢𝑑ξ）⁢u， x∈ℝ3.- \ varepsilon ^ {2} \ Delta u + V（x）u = \ frac {1} {8 \ pi \ varepsilon ^ {2}} \ Bigg {（} \ int _ {％\ mathbb {R } ^ {3}} \ frac {u ^ {2}（\ xi）} {| x- \ xi |} \，d \ xi \ Bigg {）} u，\ quad x \ in \ mathbb {R} ^ ％{3}。对于足够小的ε，我们证明了上述问题的正解的非退化性，即对应的线性算子ℒε⁢（η）=-ε2⁢Δ⁢η⁢（x）+V⁢（x）⁢ η⁢（x）-18⁢π⁢ε2⁢（∫ℝ3uε2⁢（ξ）|x-ξ|⁢𝑑ξ）⁢η⁢（x）-14⁢π⁢ε2⁢（∫ℝ3uε⁢（ξ）⁢η⁢ （ξ）|x-ξ|⁢𝑑ξ）⁢uε⁢（x）\ mathcal {L} _ {\ varepsilon}（\ eta）=-\ varepsilon ^ {2} \ Delta \ eta（x）+ V（x ）\ eta（x）-％\ frac {1} {8 \ pi \ varepsilon ^ {2}} \ Bigg {（} \ int _ {\ mathbb {R} ^ {3}} \ frac {u _ {\ varepsilon％ } ^ {2}（\ xi）} {| x- \ xi |} \，d \ xi \ Bigg {）} \ eta（x）-\ frac {1} {4 \ pi \ varepsilon ^ {2}} \ Bigg {（％} \ int _ {\ mathbb {R} ^ {3}} \ frac {u _ {\ varepsilon}（\ xi）\ eta（\ xi）} {| x- \ xi |} \，d \ xi \ Bigg％{}} u _ {\ varepsilon}（x）是简并的，即ℒε⁢（ηε）=0⇒ηε= 0 {\ mathcal {L} _ {\ varepsilon}（\ eta_ { \ varepsilon}）= 0 \ Rightarrow \ eta _ {\ varepsilon} = 0}对于较小的ε> 0 {\ varepsilon> 0}。主要工具是本地Pohozaev身份和爆炸分析。这可能是Schrödinger-Newton系统的峰值解的第一个非简并性结果。