<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<q<r<p-1<s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>r</mi> <mo><</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>s</mi> </mrow> </math></EquationSource> </InlineEquation> be real exponents. We study a semilinear boundary value problem for the weak <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p$</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-Laplacian on the Sierpiński gasket in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, subject to Dirichlet boundary conditions. The equation involves three power-type nonlinearities and two positive parameters <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>. By employing variational methods, fibering maps, and a Nehari set decomposition, we prove the existence of two distinct nontrivial weak solutions for sufficiently small values of the parameters <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>. The critical point analysis based on the fibering-map structure provides a rigorous framework for establishing existence results in the context of non-smooth fractal domains. This work contributes to the study of nonlinear elliptic equations in fractal settings.</p>

Variational treatment of semilinear elliptic equation involving the p-Laplacian on the Sierpiński gasket

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2026年6月25日

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Abdelkarim Dahmani,
Siham El Habib,
Kaoutar Lamrini Uahabi

摘要

Let $p>1$ $p > 1$ , and let $0<q<r<p-1<s$ $0 < q < r < p - 1 < s$ be real exponents. We study a semilinear boundary value problem for the weak $p$ $p$ -Laplacian on the Sierpiński gasket in $\mathbb {R}^2$ $R^{2}$ , subject to Dirichlet boundary conditions. The equation involves three power-type nonlinearities and two positive parameters $\lambda $ $λ$ and $\gamma $ $γ$ . By employing variational methods, fibering maps, and a Nehari set decomposition, we prove the existence of two distinct nontrivial weak solutions for sufficiently small values of the parameters $\lambda $ $λ$ and $\gamma $ $γ$ . The critical point analysis based on the fibering-map structure provides a rigorous framework for establishing existence results in the context of non-smooth fractal domains. This work contributes to the study of nonlinear elliptic equations in fractal settings.