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<p>This paper investigates a class of nonlinear elliptic Dirichlet boundary value problems governed by a logarithmic double-phase operator and involving a convection term dependent on the gradient. Under suitable growth conditions on the convection term, we establish the existence of weak solutions by leveraging the framework of Young measures and the Galerkin approximation method. Our analysis is conducted within the setting of Musielak–Orlicz Sobolev spaces with variable exponent, specifically in the space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathscr {Y}_{0}^{1, \mathscr {H}_{\log }}(\mathscr {D})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="script">Y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi mathvariant="script">H</mi> <mo>log</mo> </msub> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. To the best of our knowledge, this is the first study addressing such problems in this functional framework. Our results provide new insights into the interplay between logarithmic double-phase structures and gradient-dependent convection effects, paving the way for further investigations in this direction.</p>

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Existence results for logarithmic double phase elliptic equations with convection terms in variable exponent Musielak–Orlicz–Sobolev spaces

  • 期刊论文

  • 发表日期：

    2025年7月19日

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  • Hasna Moujani,
  • Abderrazak Kassidi,
  • Ali El Mfadel,
  • M’hamed El Omari

摘要

This paper investigates a class of nonlinear elliptic Dirichlet boundary value problems governed by a logarithmic double-phase operator and involving a convection term dependent on the gradient. Under suitable growth conditions on the convection term, we establish the existence of weak solutions by leveraging the framework of Young measures and the Galerkin approximation method. Our analysis is conducted within the setting of Musielak–Orlicz Sobolev spaces with variable exponent, specifically in the space \(\mathscr {Y}_{0}^{1, \mathscr {H}_{\log }}(\mathscr {D})\) Y 0 1 , H log ( D ) . To the best of our knowledge, this is the first study addressing such problems in this functional framework. Our results provide new insights into the interplay between logarithmic double-phase structures and gradient-dependent convection effects, paving the way for further investigations in this direction.