Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations
Abstract We consider the following nonlocal critical problem with mixed Dirichlet–Neumann boundary conditions: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt"> <m:mtr> <m:mtd columnalign="right"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:msubsup> <m:mn>2</m:mn> <m:mi>s</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mtext>in </m:mtext> <m:mi mathvariant="normal">Ω</m:mi> <m:mtext>,</m:mtext> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mi>u</m:mi> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mtext>on </m:mtext> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mi mathvariant="script">𝒟</m:mi> </m:msub> <m:mtext>,</m:mtext> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="right"> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mo></m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mfrac> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi/> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:mtd> <m:mtd/> <m:mtd columnalign="right"> <m:mrow> <m:mtext>on </m:mtext> <m:msub> <m:mi mathvariant="normal">Σ</m:mi> <m:mi mathvariant="script">𝒩</m:mi> </m:msub> <m:mtext>,</m:mtext> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> see text \left\{\begin{aligned} \displaystyle{}(-\Delta)^{s}u&\displaystyle=\lambda u+|% u|^{2_{s}^{*}-2}u&&\displaystyle\phantom{}\text{in ${\Omega}$,}\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\Sigma_{% \mathcal{D}}}$,}\ \displaystyle\frac{\partial u}{\partial\nu}&\displaystyle=0&&\displaystyle% \phantom{}\text{on ${\Sigma_{\mathcal{N}}}$,}\end{aligned}\right. where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> </m:math> {(-\Delta)^{s}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {s\in(\frac{1}{2},1)} , is the spectral fractional Laplacian operator,
Authors
- Luca Vilasi (ORCID: https://orcid.org/0000-0001-6294-6190)
- Giovanni Molica Bisci (ORCID: https://orcid.org/0000-0003-4802-1727)
- Alejandro Ortega
- Eduardo Colorado
Institutions
- University of Messina (IT)
- Universidad Nacional de Educación a Distancia (ES)
- San Raffaele University of Rome (IT)
- Instituto de Física Fundamental (ES)
- Universidad Carlos III de Madrid (ES)
Publication Details
- Journal
- Advances in Calculus of Variations
- Published
- 2026-06-25
- DOI
- https://doi.org/10.1515/acv-2025-0114
- Primary Topic
- Nonlinear Partial Differential Equations
- Type
- article
- Field-Weighted Citation Impact
- 0.00