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- Title
Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System.
- Authors
Guo, Qing; Xie, Huafei
- Abstract
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.
- Subjects
NON-degenerate perturbation theory; MATHEMATICS; DEGENERATE perturbation theory; ALGEBRA; EIGENVALUES
- Publication
Advanced Nonlinear Studies, 2021, Vol 21, Issue 2, p447
- ISSN
1536-1365
- Publication type
Article
- DOI
10.1515/ans-2021-2128