<p>This paper is dedicated to studying the Choquard-type equation <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mrow> <mo>{</mo> <mtable columnalign="left"> <mtr> <mtd> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mi>α</mi> </msub> <mo>∗</mo> <msup> <mrow> <mo>|</mo> <mi>u</mi> <mo>|</mo> </mrow> <mi>p</mi> </msup> <mo stretchy="false">)</mo> <msup> <mrow> <mo>|</mo> <mi>u</mi> <mo>|</mo> </mrow> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <msup> <mrow> <mo>|</mo> <mi>u</mi> <mo>|</mo> </mrow> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <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> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> <EquationSource Format="TEX">$ \left \{ \textstyle\begin{array}{l} - \Delta u + V(x)u = ( {{I_{\alpha }} * {{\left | u \right |}^{ p}}} ){ \left | u \right |^{ p - 2}}u + \lambda {\left | u \right |^{q - 2}}u, \\ u \in H^{1} ({\mathbb{R}^{N}}), \end{array}\displaystyle \right . $</EquationSource> </Equation> where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </math></EquationSource> <EquationSource Format="TEX">$N\geq 3$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo>)</mo> </mrow> </math></EquationSource> <EquationSource Format="TEX">$\alpha \in \left ({0,N}\right )$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <msup> <mn>2</mn> <mo>∗</mo> </msup> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$1< q<2^{*}: = \frac{{2N}}{{N - 2}}$</EquationSource> </InlineEquation> (the critical Sobolev exponent), <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <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> </math></EquationSource> <EquationSource Format="TEX">$p=\frac{N+\alpha }{N-2}$</EquationSource> </InlineEquation> is the upper critical Hardy-Littlewood-Sobolev exponent and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>α</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$I_{\alpha }$</EquationSource> </InlineEquation> is the Riesz potential. When the reciprocal of the potential <i>V</i> is integrable, we establish the existence of solutions to the above problem for both sublinear and superlinear cases by means of the variational methods.</p>

Existence of solutions for Choquard equations with an $L^{1}$-integrable reciprocal potential and upper critical growth

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2026年3月17日

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Ting Guo,
Tianle Xia,
Xianhua Tang

摘要

This paper is dedicated to studying the Choquard-type equation ${\begin{cases} - Δ u + V (x) u = (I_{α} * {| u |}^{p}) {| u |}^{p - 2} u + λ {| u |}^{q - 2} u, \\ u \in H^{1} (R^{N}), \end{cases}$ $ \left \{ \textstyle\begin{array}{l} - \Delta u + V(x)u = ( {{I_{\alpha }} * {{\left | u \right |}^{ p}}} ){ \left | u \right |^{ p - 2}}u + \lambda {\left | u \right |^{q - 2}}u, \\ u \in H^{1} ({\mathbb{R}^{N}}), \end{array}\displaystyle \right . $ where $N \geq 3$ $N\geq 3$ , $α \in (0, N)$ $\alpha \in \left ({0,N}\right )$ , $1 < q < 2^{*} : = \frac{2 N}{N - 2}$ $1< q<2^{*}: = \frac{{2N}}{{N - 2}}$ (the critical Sobolev exponent), $p = \frac{N + α}{N - 2}$ $p=\frac{N+\alpha }{N-2}$ is the upper critical Hardy-Littlewood-Sobolev exponent and $I_{α}$ $I_{\alpha }$ is the Riesz potential. When the reciprocal of the potential V is integrable, we establish the existence of solutions to the above problem for both sublinear and superlinear cases by means of the variational methods.

Existence of solutions for Choquard equations with an \(L^{1}\)-integrable reciprocal potential and upper critical growth

摘要