We study the Choquard equation involving mixed local and nonlocal operators <Equation ID="Equ12"> <EquationSource Format="TEX">$ -\Delta u + (-\Delta )^{s}u + V(x)u = \left( \frac{1}{|x|^{\mu }} * F(u)\right) f(u) \quad \text {in } \mathbb {R}^{2}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>μ</mi> </msup> </mfrac> <mrow /> <mo>∗</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="1em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$s\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu \in (0,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F(t)=\int _{0}^{t} f(\tau )\,d\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mi>t</mi> </msubsup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>τ</mi> </mrow> </math></EquationSource> </InlineEquation>, and f has subcritical exponential growth of Trudinger–Moser type. Under suitable assumptions on the potential V and the nonlinearity f, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

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发表日期：

2026年6月22日

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Shaoxiong Chen,
Hichem Hajaiej,
Min Yang,
Zhipeng Yang

摘要

We study the Choquard equation involving mixed local and nonlocal operators $ -\Delta u + (-\Delta )^{s}u + V(x)u = \left( \frac{1}{|x|^{\mu }} * F(u)\right) f(u) \quad \text {in } \mathbb {R}^{2}, $ $- Δ u + {(- Δ)}^{s} u + V (x) u = (\frac{1}{{| x |}^{μ}} * F (u)) f (u) in R^{2},$ where $s\in (0,1)$ $s \in (0, 1)$ , $\mu \in (0,2)$ $μ \in (0, 2)$ , $F(t)=\int _{0}^{t} f(\tau )\,d\tau $ $F (t) = \int_{0}^{t} f (τ) d τ$ , and f has subcritical exponential growth of Trudinger–Moser type. Under suitable assumptions on the potential V and the nonlinearity f, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.