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- W3152700148 abstract "Abstract We are concerned with the following Schrödinger–Newton problem: <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>8</m:mn> <m:mo></m:mo> <m:mi>π</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mfrac> <m:mo></m:mo> <m:mrow> <m:mo maxsize=260% minsize=260%>(</m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:msub> <m:mrow> <m:mpadded width=+1.7pt> <m:mfrac> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> </m:mfrac> </m:mpadded> <m:mo></m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>ξ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize=260% minsize=260%>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo rspace=12.5pt>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> -varepsilon^{2}Delta u+V(x)u=frac{1}{8pivarepsilon^{2}}Bigg{(}int_{% mathbb{R}^{3}}frac{u^{2}(xi)}{|x-xi|},dxiBigg{)}u,quad xinmathbb{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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=script>ℒ</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>η</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo></m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mo></m:mo> <m:mi>η</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>η</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>8</m:mn> <m:mo></m:mo> <m:mi>π</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mfrac> <m:mo></m:mo> <m:mrow> <m:mo maxsize=260% minsize=260%>(</m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:msub> <m:mrow> <m:mpadded width=+1.7pt> <m:mfrac> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> </m:mfrac> </m:mpadded> <m:mo></m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>ξ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize=260% minsize=260%>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>η</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>4</m:mn> <m:mo></m:mo> <m:mi>π</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mfrac> <m:mo></m:mo> <m:mrow> <m:mo maxsize=260% minsize=260%>(</m:mo> <m:mrow> <m:msub> <m:mo largeop=true symmetric=true>∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:msub> <m:mrow> <m:mpadded width=+1.7pt> <m:mfrac> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>η</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> </m:mfrac> </m:mpadded> <m:mo></m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>ξ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize=260% minsize=260%>)</m:mo> </m:mrow> <m:mo></m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> mathcal{L}_{varepsilon}(eta)=-varepsilon^{2}Deltaeta(x)+V(x)eta(x)-% frac{1}{8pivarepsilon^{2}}Bigg{(}int_{mathbb{R}^{3}}frac{u_{varepsilon% }^{2}(xi)}{|x-xi|},dxiBigg{)}eta(x)-frac{1}{4pivarepsilon^{2}}Bigg{(% }int_{mathbb{R}^{3}}frac{u_{varepsilon}(xi)eta(xi)}{|x-xi|},dxiBigg% {)}u_{varepsilon}(x) is non-degenerate, i.e., <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msub> <m:mi mathvariant=script>ℒ</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msub> <m:mi>η</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>⇒</m:mo> <m:msub> <m:mi>η</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {mathcal{L}_{varepsilon}(eta_{varepsilon})=0Rightarroweta_{varepsilon}=0} for small <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>ε</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {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." @default.
- W3152700148 created "2021-04-26" @default.
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- W3152700148 date "2021-04-15" @default.
- W3152700148 modified "2023-09-23" @default.
- W3152700148 title "Non-Degeneracy of Peak Solutions to the Schrödinger–Newton System" @default.
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