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<p>The goal of this article is to show the existence of a positive solution to a Dirichlet problem for the elliptic equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Delta _g u+f(u)=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega \subset {{\mathbb {H}}^2 \times {\mathbb {R}}},\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>2</mn> </msup> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a perturbation of the half-space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\({\mathbb {H}}^2\times {\mathbb {R}}^+_0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mn>2</mn> </msup> <mo>×</mo> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>0</mn> <mo>+</mo> </msubsup> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The corresponding solution is a perturbation of the solution to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(u''(z)+f(u(z))=0,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(u(0)=0,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(u(z)\rightarrow + 1,\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(z \rightarrow + \infty .\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>z</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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A DIRICHLET PROBLEM ON PERTURBED HALF-SPACES OF \({{\mathbb {H}}^2 \times {\mathbb {R}}}\) FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

  • 期刊论文

  • 发表日期：

    2026年1月21日

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  • Filippo Morabito

摘要

The goal of this article is to show the existence of a positive solution to a Dirichlet problem for the elliptic equation \(\Delta _g u+f(u)=0\) Δ g u + f ( u ) = 0 on \(\Omega \subset {{\mathbb {H}}^2 \times {\mathbb {R}}},\) Ω ⊂ H 2 × R , where \(\Omega \) Ω is a perturbation of the half-space \({\mathbb {H}}^2\times {\mathbb {R}}^+_0.\) H 2 × R 0 + . The corresponding solution is a perturbation of the solution to \(u''(z)+f(u(z))=0,\) u ′ ′ ( z ) + f ( u ( z ) ) = 0 , such that \(u(0)=0,\) u ( 0 ) = 0 , \(u(z)\rightarrow + 1,\) u ( z ) → + 1 , as \(z \rightarrow + \infty .\) z → + ∞ .