<p>This paper studies a singular anisotropic system of coupled quasilinear elliptic equations. The system features anisotropic diffusion operators with variable exponents <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2072_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>i</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$p_{i}$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2072_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>q</mi> <mi>i</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$q_{i}$</EquationSource> </InlineEquation>, singular terms of the form <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2072_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>v</mi> <mrow> <mo>−</mo> <msub> <mi>γ</mi> <mn>1</mn> </msub> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$v^{-\gamma _{1}}$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2072_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mrow> <mo>−</mo> <msub> <mi>γ</mi> <mn>2</mn> </msub> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$u^{-\gamma _{2}}$</EquationSource> </InlineEquation>, and nonlinear source terms. By employing variational methods and an approximation problem, we prove the existence of positive solutions under suitable conditions on the nonlinearities.</p>

Positive solutions to a coupled singular anisotropic system with nonstandard growth and singular nonlinearities

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2025年6月3日

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Seyedeh Atefeh Fallahshams,
Abdolrahman Razani

摘要

This paper studies a singular anisotropic system of coupled quasilinear elliptic equations. The system features anisotropic diffusion operators with variable exponents $p_{i}$ $p_{i}$ and $q_{i}$ $q_{i}$ , singular terms of the form $v^{- γ_{1}}$ $v^{-\gamma _{1}}$ and $u^{- γ_{2}}$ $u^{-\gamma _{2}}$ , and nonlinear source terms. By employing variational methods and an approximation problem, we prove the existence of positive solutions under suitable conditions on the nonlinearities.