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<p>This paper investigates the existence and uniqueness of weak solutions to a singular nonlinear Kirchhoff-type problem featuring <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-Hilfer fractional derivatives, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-Laplacian operators, and Dirichlet boundary conditions. By employing a critical point approach combined with genus theory and variational methods, we prove the existence of weak solutions in suitable <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-Hilfer fractional derivative spaces. Furthermore, for the case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\tau = 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish that the obtained solution is both unique and positive. Our main results provide novel contributions to the theory of differential equations involving <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-Hilfer fractional derivatives.</p>

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A STUDY OF SINGULAR KIRCHHOFF PROBLEM IN FRACTIONAL \(\Phi \)-HILFER DERIVATIVE SPACES WITH \(\tau \)-LAPLACIAN OPERATOR

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Arhrrabi Elhoussain,
  • Lmou Hamid

摘要

This paper investigates the existence and uniqueness of weak solutions to a singular nonlinear Kirchhoff-type problem featuring \(\Phi \) Φ -Hilfer fractional derivatives, \(\tau \) τ -Laplacian operators, and Dirichlet boundary conditions. By employing a critical point approach combined with genus theory and variational methods, we prove the existence of weak solutions in suitable \(\Phi \) Φ -Hilfer fractional derivative spaces. Furthermore, for the case \(\tau = 2\) τ = 2 , we establish that the obtained solution is both unique and positive. Our main results provide novel contributions to the theory of differential equations involving \(\Phi \) Φ -Hilfer fractional derivatives.