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<p>In the present paper, we propose the investigation of variable exponent <i>p</i>-Laplace equations with logarithmic growth in divergence form. Under weak regularity assumptions on the boundary of domains and the exponent <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1914_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(p(\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the global gradient bounds in norms for solutions are well-established in a class of generalized function spaces via the presence of fractional maximal operators <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1914_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\({\textbf{M}}_\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">M</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>. This effort can be developed in the theory of energy functionals satisfying certain nonstandard growth conditions, including problems governed by the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1914_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(p(\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Laplacian operator.</p>

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Gradient Regularity for the Solutions to \(p(\cdot )\)-Laplacian Equations with Logarithmic Perturbation

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Minh-Phuong Tran,
  • Thanh-Nhan Nguyen

摘要

In the present paper, we propose the investigation of variable exponent p-Laplace equations with logarithmic growth in divergence form. Under weak regularity assumptions on the boundary of domains and the exponent \(p(\cdot )\) p ( · ) , the global gradient bounds in norms for solutions are well-established in a class of generalized function spaces via the presence of fractional maximal operators \({\textbf{M}}_\alpha \) M α . This effort can be developed in the theory of energy functionals satisfying certain nonstandard growth conditions, including problems governed by the \(p(\cdot )\) p ( · ) -Laplacian operator.