Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System

We are concerned with the following Schrödinger–Newton problem: - ε 2 Δ u + V ( x ) u = 1 8 π ε 2 ( ∫ ℝ 3 u 2 ( ξ ) | x - ξ | 𝑑 ξ ) u , x ∈ ℝ 3 . -\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 ℒ ε ( η ) = - ε 2 Δ η ( x ) + V ( x ) η ( x ) - 1 8 π ε 2 ( ∫ ℝ 3 u ε 2 ( ξ ) | x - ξ | 𝑑 ξ ) η ( x ) - 1 4 π ε 2 ( ∫ ℝ 3 u ε ( ξ ) η ( ξ ) | x - ξ | 𝑑 ξ ) u ε ( 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., ℒ ε ( η ε ) = 0 ⇒ η ε = 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.

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