<p>In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents <Equation ID="Equ135"> <EquationSource Format="TEX">$\begin{aligned} -\Delta u=(|x|^{-(n-2)}u^{p-\varepsilon })u^{p-1-\varepsilon }\quad \text{ in }\Omega , u=0\quad \text{ on }~~\partial \Omega , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow /> <mo>∗</mo> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>-</mo> <mi>ε</mi> </mrow> </msup> <mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>ε</mi> </mrow> </msup> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a smooth bounded domain in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n=3,4,5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the standard convolution, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p=\frac{n+2}{n-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {D}^{1,2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(n+2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(\lambda _{i,\varepsilon }, v_{i,\varepsilon })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>ε</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>ε</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to the linearized problem of the above nonlocal equations for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$i=1,\cdots ,n+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.</p>

Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

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2026年6月17日

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Alessandro Cannone,
Silvia Cingolani,
Minbo Yang,
Shunneng Zhao

摘要

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents $\begin{aligned} -\Delta u=(|x|^{-(n-2)}*u^{p-\varepsilon })u^{p-1-\varepsilon }\quad \text{ in }~~\Omega ,~~ u=0\quad \text{ on }~~\partial \Omega , \end{aligned}$ $\begin{matrix} - Δ u = {(| x |}^{- (n - 2)} * u^{p - ε}) u^{p - 1 - ε} in Ω, u = 0 on \partial Ω, \end{matrix}$ where $\Omega $ $Ω$ is a smooth bounded domain in $\mathbb {R}^n$ $R^{n}$ for $n=3,4,5$ $n = 3, 4, 5$ , $*$ $*$ denotes the standard convolution, $\varepsilon >0$ $ε > 0$ is a small parameter and $p=\frac{n+2}{n-2}$ $p = \frac{n + 2}{n - 2}$ is $\mathcal {D}^{1,2}$ $D^{1, 2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$ $(n + 2)$ -eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(\lambda _{i,\varepsilon }, v_{i,\varepsilon })$ $(λ_{i, ε}, v_{i, ε})$ to the linearized problem of the above nonlocal equations for $i=1,\cdots ,n+2$ $i = 1, \dots, n + 2$ . As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.