<p>In this paper, we are concerned with a new class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation>-fractional spaces involving anisotropic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\overrightarrow{\tau }(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-Laplacian operators, abbreviated as (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\phi ,\overrightarrow{\tau }(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>,</mo> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>)-HFDA, for differential equations with a power-like variable reaction term. By utilizing the Mountain Pass Theorem together with Ekeland’s Principle, we proof the existence of precise intervals of positive parameters that admit nontrivial solutions for an eigenvalue problem since <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\tau _{M}^{+}<q^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>τ</mi> <mrow> <mi>M</mi> </mrow> <mo>+</mo> </msubsup> <mo><</mo> <msup> <mi>q</mi> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Our main results are novel and contribute to the literature on problems involving (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\phi ,\overrightarrow{\tau }(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>,</mo> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>)-HFDA. This investigation enhances the understanding of this specific class of problems.</p>

A class of eigenvalue problems for $(\phi ,\overrightarrow{\tau }(\cdot ))$-fractional spaces

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2026年3月3日

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El-Houari Hamza,
Arhrrabi Elhoussain,
J. Vanterler da C. Sousa,
Leandro S. Tavares

摘要

In this paper, we are concerned with a new class of $\phi $ $ϕ$ -fractional spaces involving anisotropic $\overrightarrow{\tau }(\cdot )$ $\vec{τ} (\cdot)$ -Laplacian operators, abbreviated as ( $\phi ,\overrightarrow{\tau }(\cdot )$ $ϕ, \vec{τ} (\cdot)$ )-HFDA, for differential equations with a power-like variable reaction term. By utilizing the Mountain Pass Theorem together with Ekeland’s Principle, we proof the existence of precise intervals of positive parameters that admit nontrivial solutions for an eigenvalue problem since $\tau _{M}^{+}<q^{-}$ $τ_{M}^{+} < q^{-}$ . Our main results are novel and contribute to the literature on problems involving ( $\phi ,\overrightarrow{\tau }(\cdot )$ $ϕ, \vec{τ} (\cdot)$ )-HFDA. This investigation enhances the understanding of this specific class of problems.

A class of eigenvalue problems for \((\phi ,\overrightarrow{\tau }(\cdot ))\)-fractional spaces

摘要