<p>We establish the existence of a ground state solution for the fractional Choquard equation governed by the superposition operator <Equation ID="Equ80"> <EquationSource Format="TEX">$ \int _{[0, 1]} (-\varDelta )^s u\, d\mu (s), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mo>∫</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mspace width="0.166667em" /> <mi>d</mi> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>and in presence of a confining potential. Here, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> denotes a signed measure on the interval of fractional exponents [0, 1]. The presence of the superposition operator asks for the problem to be addressed with a special care. However, the wide generality of this setting allows to provide entirely new existence results in several special cases of interest, including, e.g., the mixed operator one <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-\varDelta + (-\varDelta )^s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>Δ</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi>Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. We point out that the possibility of considering operators “with the wrong sign" is also a complete novelty.</p>

Ground state solution for the Choquard equation under the superposition of operators of mixed fractional order

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2026年3月9日

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Edoardo Proietti Lippi,
Caterina Sportelli

摘要

We establish the existence of a ground state solution for the fractional Choquard equation governed by the superposition operator $ \int _{[0, 1]} (-\varDelta )^s u\, d\mu (s), $ $\int_{[0, 1]} {(- Δ)}^{s} u d μ (s),$ and in presence of a confining potential. Here, $\mu $ $μ$ denotes a signed measure on the interval of fractional exponents [0, 1]. The presence of the superposition operator asks for the problem to be addressed with a special care. However, the wide generality of this setting allows to provide entirely new existence results in several special cases of interest, including, e.g., the mixed operator one $-\varDelta + (-\varDelta )^s$ $- Δ + {(- Δ)}^{s}$ . We point out that the possibility of considering operators “with the wrong sign" is also a complete novelty.