<p>We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\Delta u +f(u)=0$</EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{n}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>∈</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mo stretchy="false">(</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$u\in H^{1}(\mathbb{R}^{n})$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>≢</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$u\not \equiv 0$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </math></EquationSource> <EquationSource Format="TEX">$n\ge 3$</EquationSource> </InlineEquation>. For the model nonlinearity <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>u</mi> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> </math></EquationSource> <EquationSource Format="TEX">$f(u)=-u+|u|^{p-1}u$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$1< p<(n+2)/(n-2)$</EquationSource> </InlineEquation>, arising from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$k\ge 1$</EquationSource> </InlineEquation>, the problem has a unique solution <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>=</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$u=u(|x|)$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="MATHML"><math> <mi>x</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$x\in \mathbb{R}^{n}$</EquationSource> </InlineEquation>, up to translation and reflection, that has precisely <InlineEquation ID="IEq14"> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> <EquationSource Format="TEX">$k$</EquationSource> </InlineEquation> zeros for <InlineEquation ID="IEq15"> <EquationSource Format="MATHML"><math> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$|x|>0$</EquationSource> </InlineEquation>.</p>

Uniqueness of bound states to $\Delta u-u+|u|^{p-1}u= 0$ in $\mathbb{R}^{n} $, $n\ge 3$

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2025年10月10日

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Moxun Tang

摘要

We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $Δ u + f (u) = 0$ $\Delta u +f(u)=0$ in $R^{n}$ $\mathbb{R}^{n}$ , $u \in H^{1} (R^{n})$ $u\in H^{1}(\mathbb{R}^{n})$ , $u ≢ 0$ $u\not \equiv 0$ , $n \geq 3$ $n\ge 3$ . For the model nonlinearity $f (u) = - u + | u |^{p - 1} u$ $f(u)=-u+|u|^{p-1}u$ , $1 < p < (n + 2) / (n - 2)$ $1< p<(n+2)/(n-2)$ , arising from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer $k \geq 1$ $k\ge 1$ , the problem has a unique solution $u = u (| x |)$ $u=u(|x|)$ , $x \in R^{n}$ $x\in \mathbb{R}^{n}$ , up to translation and reflection, that has precisely $k$ $k$ zeros for $| x | > 0$ $|x|>0$ .

Uniqueness of bound states to \(\Delta u-u+|u|^{p-1}u= 0\) in \(\mathbb{R}^{n} \), \(n\ge 3\)

摘要