We investigate non-local eigenvalue problems governed by the fractional <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation>-Laplacian operator. The main results include the identification of two positive constants, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma _0 \le \gamma _1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>γ</mi> <mn>0</mn> </msub> <mo>≤</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\gamma _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is shown to be an eigenvalue, and any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma \le \gamma _0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>≤</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> is proven not to be an eigenvalue. Furthermore, assuming the N-function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> satisfies the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-condition, we prove that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is isolated on the right. This isolation result is a key tool in deriving subsequent Hölder regularity estimates. All findings are developed in the setting of non-reflexive spaces, characterized by the failure of the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-condition for the complementary N-function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\overline{\Phi }$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi mathvariant="normal">Φ</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>.

Non-reflexive fractional Orlicz-Sobolev spaces: study of eigenvalue equation

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发表日期：

2026年1月22日

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Hamza El-Houari

摘要

We investigate non-local eigenvalue problems governed by the fractional $\psi $ $ψ$ -Laplacian operator. The main results include the identification of two positive constants, $\gamma _0 \le \gamma _1$ $γ_{0} \leq γ_{1}$ , where $\gamma _1$ $γ_{1}$ is shown to be an eigenvalue, and any $\gamma \le \gamma _0$ $γ \leq γ_{0}$ is proven not to be an eigenvalue. Furthermore, assuming the N-function $\Phi $ $Φ$ satisfies the $\Delta _2$ $Δ_{2}$ -condition, we prove that $\gamma _1$ $γ_{1}$ is isolated on the right. This isolation result is a key tool in deriving subsequent Hölder regularity estimates. All findings are developed in the setting of non-reflexive spaces, characterized by the failure of the $\Delta _2$ $Δ_{2}$ -condition for the complementary N-function $\overline{\Phi }$ $\bar{Φ}$ .