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<p>We study a class of parametric quasilinear elliptic problems under homogeneous Neumann boundary condition. The equation is driven by the <i>p</i>-Laplacian with an additive coercive term, and the reaction combines a variable-exponent source with a Carathéodory perturbation that depends nonlinearly on the gradient. We develop a nonvariational sub-supersolution approach based on an auxiliary truncated problem and the surjectivity theorem for pseudomonotone operators. This method ensures the existence of a positive weak solution, its location between explicit positive constant sub- supersolutions, and global <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> regularity up to the boundary.</p>

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Parametric quasilinear elliptic equations under Neumann boundary condition with full gradient dependence

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  • 出版模式：

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  • 发表日期：

    2026年7月18日

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  • Anderson de Araujo,
  • Luiz Faria,
  • Dumitru Motreanu

摘要

We study a class of parametric quasilinear elliptic problems under homogeneous Neumann boundary condition. The equation is driven by the p-Laplacian with an additive coercive term, and the reaction combines a variable-exponent source with a Carathéodory perturbation that depends nonlinearly on the gradient. We develop a nonvariational sub-supersolution approach based on an auxiliary truncated problem and the surjectivity theorem for pseudomonotone operators. This method ensures the existence of a positive weak solution, its location between explicit positive constant sub- supersolutions, and global \(C^{1}\) C 1 regularity up to the boundary.