<p>This paper investigates the qualitative properties of normalized solutions to the upper critical Choquard equation with nonlocal perturbation: <Equation ID="Equ166"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u=(I_\alpha |u|^{\frac{N+\alpha }{N-2}})|u|^{\frac{N+\alpha }{N-2}-2}u+\mu (I_\beta |u|^{p})|u|^{p-2}u,\ x\in \mathbb {R}^N,\\ \displaystyle \int _{\mathbb {R}^N}u^2\textrm{d}x=c, \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> </mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <msup> <mrow> <mrow /> <mo>∗</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </msup> <msup> <mrow> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mi>u</mi> <mo>+</mo> <mi>μ</mi> <mo stretchy="false">(</mo> </mrow> <msub> <mi>I</mi> <mi>β</mi> </msub> <msup> <mrow> <mrow /> <mo>∗</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </msup> <msup> <mrow> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mspace width="4pt" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mrow /> <msub> <mo>∫</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </msub> <msup> <mi>u</mi> <mn>2</mn> </msup> <mtext>d</mtext> <mi>x</mi> <mo>=</mo> <mi>c</mi> <mo>,</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha ,\beta \in (0,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p \in \left( \frac{N+\beta }{N}, \frac{N+\beta }{N-2} \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> </mrow> <mi>N</mi> </mfrac> <mo>,</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> is an unknown Lagrange multiplier, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$I_\alpha ,I_{\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <mo>,</mo> <msub> <mi>I</mi> <mi>β</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> denote the Riesz potentials. For <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the existence of normalized solutions in several regimes, that is, when <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\frac{N+\beta }{N}< p < \frac{N+\beta +2}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> </mrow> <mi>N</mi> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (mass-subcritical), <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p = \frac{N+\beta +2}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (mass-critical), and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\frac{N+\beta +2}{N}< p < \frac{N+\beta }{N-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> <mi>N</mi> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>β</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (mass-supercritical). For <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mu \le 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we derive a non-existence result. Particularly, to obtain sharp energy estimates crucial for restoring compactness, we classify analyses by ranges of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> across different dimensions, developing tailored scaling techniques within each range to control energy levels below the corresponding compactness thresholds. This enables us to resolve open problems in sharp energy estimation for the mass-subcritical regime: we cover full parameter ranges for <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$N=3,4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and extend admissible parameter ranges for <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$N \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, while providing a more comprehensive characterization of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\alpha , \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> to advance related research. Moreover, the framework applies directly to special cases including <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\alpha = \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation> and van der Waals-type potentials (<InlineEquation ID="IEq19"> <EquationSource Format="TEX">$p = \frac{N+\alpha }{N-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mo>+</mo> <mi>α</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\alpha < \beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo><</mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>), improving upon existing literature in these settings. We anticipate that the energy estimation techniques introduced in this paper will be extended to wider classes of nonlocal critical elliptic equations with mass constraint.</p>

Normalized solutions for the upper critical Choquard equation with nonlocal perturbation

期刊论文
发表日期：

2026年4月3日

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Sitong Chen,
Peng Jin,
Vicenţiu D. Rădulescu,
Xin’ao Zhou

摘要

This paper investigates the qualitative properties of normalized solutions to the upper critical Choquard equation with nonlocal perturbation: $\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u=(I_\alpha *|u|^{\frac{N+\alpha }{N-2}})|u|^{\frac{N+\alpha }{N-2}-2}u+\mu (I_\beta *|u|^{p})|u|^{p-2}u,\ x\in \mathbb {R}^N,\\ \displaystyle \int _{\mathbb {R}^N}u^2\textrm{d}x=c, \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} - Δ u + λ u = (I_{α} {* | u |}^{\frac{N + α}{N - 2}} {) | u |}^{\frac{N + α}{N - 2} - 2} u + μ (I_{β} {* | u |}^{p} {) | u |}^{p - 2} u, x \in R^{N}, \\ \int_{R^{N}} u^{2} d x = c, \end{matrix}) \end{matrix}$ where $N \ge 3$ $N \geq 3$ , $\alpha ,\beta \in (0,N)$ $α, β \in (0, N)$ , $p \in \left( \frac{N+\beta }{N}, \frac{N+\beta }{N-2} \right) $ $p \in (\frac{N + β}{N}, \frac{N + β}{N - 2})$ , $\mu \in \mathbb {R}$ $μ \in R$ , $c>0$ $c > 0$ , $\lambda \in \mathbb {R}$ $λ \in R$ is an unknown Lagrange multiplier, and $I_\alpha ,I_{\beta }$ $I_{α}, I_{β}$ denote the Riesz potentials. For $\mu > 0$ $μ > 0$ , we establish the existence of normalized solutions in several regimes, that is, when $\frac{N+\beta }{N}< p < \frac{N+\beta +2}{N}$ $\frac{N + β}{N} < p < \frac{N + β + 2}{N}$ (mass-subcritical), $p = \frac{N+\beta +2}{N}$ $p = \frac{N + β + 2}{N}$ (mass-critical), and $\frac{N+\beta +2}{N}< p < \frac{N+\beta }{N-2}$ $\frac{N + β + 2}{N} < p < \frac{N + β}{N - 2}$ (mass-supercritical). For $\mu \le 0$ $μ \leq 0$ , we derive a non-existence result. Particularly, to obtain sharp energy estimates crucial for restoring compactness, we classify analyses by ranges of $\alpha $ $α$ , $\beta $ $β$ across different dimensions, developing tailored scaling techniques within each range to control energy levels below the corresponding compactness thresholds. This enables us to resolve open problems in sharp energy estimation for the mass-subcritical regime: we cover full parameter ranges for $N=3,4$ $N = 3, 4$ and extend admissible parameter ranges for $N \ge 5$ $N \geq 5$ , while providing a more comprehensive characterization of $\alpha , \beta $ $α, β$ to advance related research. Moreover, the framework applies directly to special cases including $\alpha = \beta $ $α = β$ and van der Waals-type potentials ( $p = \frac{N+\alpha }{N-2}$ $p = \frac{N + α}{N - 2}$ with $\alpha < \beta $ $α < β$ ), improving upon existing literature in these settings. We anticipate that the energy estimation techniques introduced in this paper will be extended to wider classes of nonlocal critical elliptic equations with mass constraint.