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<p>We derive the existence of solutions for an asymptotically linear equation driven by the spectral fractional Laplacian operator with mixed Dirichlet-Neumann boundary conditions. When the nonlinear term <i>f</i> is odd and a suitable relation between the perturbation parameter, the limit of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f(\cdot ,t)/t\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>t</mi> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t\rightarrow 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.</p>

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Asymptotically linear fractional problems with mixed boundary conditions

  • 期刊论文

  • 发表日期：

    2026年5月19日

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  • Giovanni Molica Bisci,
  • Alejandro Ortega,
  • Luca Vilasi

摘要

We derive the existence of solutions for an asymptotically linear equation driven by the spectral fractional Laplacian operator with mixed Dirichlet-Neumann boundary conditions. When the nonlinear term f is odd and a suitable relation between the perturbation parameter, the limit of \(f(\cdot ,t)/t\) f ( · , t ) / t as \(t\rightarrow 0\) t → 0 and the eigenvalues occurs, we establish also a multiplicity result via the pseudo-index theory related to the genus.