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Subcritical nonlocal problems with mixed boundary conditions

Dipartimento di Scienze Pure e Applicate (DiSPeA), Universitá degli Studi di Urbino Carlo Bo, Piazza della Repubblica, 13 - 61029 Urbino, Italy

E-mail Address: giovanni.molicabisci@uniurb.it

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Alejandro Ortega

Departamento de Matemáticas Fundamentales, Facultad de Ciencias, Universidad Nacional de Educacion a Distancia, 28040 Madrid, Spain

E-mail Address: alejandro.ortega@mat.uned.es

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, and

Luca Vilasi

Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres, 31 - 98166 Messina, Italy

E-mail Address: lvilasi@unime.it

Corresponding author.

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https://doi.org/10.1142/S166436072350011XCited by:3 (Source: Crossref)

Abstract

By using linking and $\nabla$ -theorems in this paper we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet–Neumann boundary data,

{(−Δ)su=λu+f(x,u)in Ω,u=0on Σ𝒟,∂u∂ν=0on Σ𝒩,<math display="block" altimg="eq-00002.gif"><mrow><mfenced separators="" open="{" close=""><mrow><mtable equalrows="false" columnlines="none" equalcolumns="false" class="array"><mtr><mtd class="array" columnalign="left"><msup><mrow><mo class="MathClass-open" stretchy="false">(</mo><mo stretchy="false" class="MathClass-bin">−</mo><mi mathvariant="normal">Δ</mi><mo class="MathClass-close" stretchy="false">)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo class="MathClass-rel">=</mo><mi>λ</mi><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>f</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-punc">,</mo><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext>in </mtext></mstyle><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><mi>u</mi><mo class="MathClass-rel">=</mo><mn>0</mn><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext>on </mtext></mstyle><msub><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mi mathvariant="cal">𝒟</mi></mrow></msub><mo class="MathClass-punc">,</mo></mtd></mtr><mtr><mtd class="array" columnalign="left"><mfrac><mrow><mi>∂</mi><mi>u</mi></mrow><mrow><mi>∂</mi><mi>ν</mi></mrow></mfrac><mo class="MathClass-rel">=</mo><mn>0</mn><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext>on </mtext></mstyle><msub><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mi mathvariant="cal">𝒩</mi></mrow></msub><mo class="MathClass-punc">,</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math>

where

${(- Δ)}^{s}$ ,

$s \in (1 / 2, 1)$ , is the spectral fractional Laplacian operator,

$Ω \subset ℝ^{N}$ ,

$N > 2 s$ , is a smooth bounded domain,

$λ > 0$ is a real parameter,

$ν$ is the outward normal to

$\partial Ω$ ,

$Σ_{𝒟}$ ,

$Σ_{𝒩}$ are smooth

$(N - 1)$ -dimensional submanifolds of

$\partial Ω$ such that

$Σ_{𝒟} \cup Σ_{𝒩} = \partial Ω$ ,

$Σ_{𝒟} \cap Σ_{𝒩} = \emptyset$ and

$Σ_{𝒟} \cap {\bar{Σ}}_{𝒩} = Γ$ is a smooth

$(N - 2)$ -dimensional submanifold of

$\partial Ω$ .

Communicated by Neil Trudinger

Keywords:

AMSC: 35J20, 35A15, 35S15, 49J35, 35J61

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Vol. 14, No. 01

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History

Received 8 May 2023

Accepted 1 August 2023

Published: 10 August 2023

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Subcritical nonlocal problems with mixed boundary conditions

Abstract

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