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<p>We find a large solution to a semilinear Dirichlet problem in a bounded <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^{1,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> domain for a non-local operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\phi (-\Delta \vert _{D})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <msub> <mo stretchy="false">|</mo> <mi>D</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the nonlinearity. Additionally, we prove interior higher regularity results for this operator.</p>

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Large solutions for subordinate spectral Laplacian

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Ivan Biočić,
  • Vanja Wagner

摘要

We find a large solution to a semilinear Dirichlet problem in a bounded \(C^{1,1}\) C 1 , 1 domain for a non-local operator \(\phi (-\Delta \vert _{D})\) ϕ ( - Δ | D ) , an extension of the infinitesimal generator of a subordinate killed Brownian motion. The setting covers and extends the case of the spectral fractional Laplacian. The upper bound for the explosion rate of the large solution is obtained and is given in terms of the renewal function, distance to the boundary, and the Keller-Osserman-type transformation of the nonlinearity. Additionally, we prove interior higher regularity results for this operator.