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<p>Here, we consider a problem involves anisotropic diffusion operators with variable exponents <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p_i\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, singular terms of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(|x|^{-\ell p_i}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>ℓ</mi> <msub> <mi>p</mi> <mi>i</mi> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation>, and nonlinearities with subcritical and critical growth. Utilizing variational techniques alongside the Mountain Pass Theorem, we demonstrate the existence of nonnegative weak solutions, provided that appropriate conditions are met regarding the nonlinearity and the weight function. Our results extend previous work on anisotropic problems and offer new perspectives on how solutions behave when singular weights and critical exponents are involved.</p>

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Existence of nonnegative solutions for an anisotropic elliptic equation with singular weights and critical growth

  • 期刊论文

  • 发表日期：

    2025年7月23日

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  • Seyedeh Atefeh Fallahshams,
  • Abdolrahman Razani,
  • Sami Baraket

摘要

Here, we consider a problem involves anisotropic diffusion operators with variable exponents \(p_i\) p i , singular terms of the form \(|x|^{-\ell p_i}\) | x | - ℓ p i , and nonlinearities with subcritical and critical growth. Utilizing variational techniques alongside the Mountain Pass Theorem, we demonstrate the existence of nonnegative weak solutions, provided that appropriate conditions are met regarding the nonlinearity and the weight function. Our results extend previous work on anisotropic problems and offer new perspectives on how solutions behave when singular weights and critical exponents are involved.