We prove existence and multiplicity results for the fractional Schrödinger–Poisson–Slater equation <Equation ID="Equ90"> <EquationSource Format="TEX">$ (-\Delta )^{s}u+\bigl (I_{\alpha }*u^{2}\bigr )u=f(|x|,u) \quad \text {in }\mathbb {R}^{N}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <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> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <msub> <mi>I</mi> <mi>α</mi> </msub> <mrow /> <mo>∗</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo>,</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> <mi>N</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<s<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>s</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha \in (1,N).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We seek solutions in a fractional Coulomb–Sobolev space by employing new tools in critical point theory, which connect the behavior of the nonlinearity f at zero and at infinity with the scaling properties of the left hand side of the equation. In various regimes for the nonlinearity f,  we establish compactness results for an associated action functional and find multiple solutions as critical points, whose number is sensitive to the interaction of f with a sequence of eigenvalues <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{\lambda _k\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi>λ</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> defined via the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>–cohomological index of Fadell and Rabinowitz. The use of this index, instead of the classical Krasnoselskii genus, is essential for us to use new critical group estimates and scaling-based linking geometries. In this fractional setting, new regularity results and necessary conditions for solutions to exist are also proved.

Fractional Schrödinger-Poisson-Slater Equations in Coulomb-Sobolev Spaces

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2026年3月19日

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Elisandra Gloss,
Carlo Mercuri,
Kanishka Perera,
Bruno Ribeiro

摘要

We prove existence and multiplicity results for the fractional Schrödinger–Poisson–Slater equation $ (-\Delta )^{s}u+\bigl (I_{\alpha }*u^{2}\bigr )u=f(|x|,u) \quad \text {in }\mathbb {R}^{N}, $ ${(- Δ)}^{s} u + (I_{α} * u^{2}) u = f (| x |, u) in R^{N},$ with $0<s<1$ $0 < s < 1$ , $\alpha \in (1,N).$ $α \in (1, N) .$ We seek solutions in a fractional Coulomb–Sobolev space by employing new tools in critical point theory, which connect the behavior of the nonlinearity f at zero and at infinity with the scaling properties of the left hand side of the equation. In various regimes for the nonlinearity f, we establish compactness results for an associated action functional and find multiple solutions as critical points, whose number is sensitive to the interaction of f with a sequence of eigenvalues $\{\lambda _k\}$ ${λ_{k}}$ defined via the $\mathbb {Z}_2$ $Z_{2}$ –cohomological index of Fadell and Rabinowitz. The use of this index, instead of the classical Krasnoselskii genus, is essential for us to use new critical group estimates and scaling-based linking geometries. In this fractional setting, new regularity results and necessary conditions for solutions to exist are also proved.