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Multiple solutions of a nonlocal system with singular nonlinearities

College of Sciences, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of Saudi Arabia

Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, 31441 Dammam, Saudi Arabia

E-mail Address: mmkratou@iau.edu.sa

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Kamel Saoudi

https://orcid.org/0000-0002-7647-4814

College of Sciences, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of Saudi Arabia

Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, 31441 Dammam, Saudi Arabia

E-mail Address: kmsaoudi@iau.edu.sa

Corresponding author.

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Aisha AlShehri

College of Sciences, Imam Abdulrahman Bin Faisal University, 31441 Dammam, Kingdom of Saudi Arabia

Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, 31441 Dammam, Saudi Arabia

E-mail Address: asalshehri@iau.edu.sa

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https://doi.org/10.1142/S0129167X21500725Cited by:3 (Source: Crossref)

Abstract

In this work, we study the fractional Laplacian equation with singular nonlinearity:

{(−Δ)su=λa(x)|u|q−2u+1−α2−α−βc(x)|u|−α|v|1−βinΩ,(−Δ)sv=μb(x)|v|q−2v+1−β2−α−βc(x)|u|1−α|v|−βin Ω,u=v=0inℝN∖Ω,<math display="block" altimg="eq-00001.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>a</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>q</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo stretchy="false" class="MathClass-bin">+</mo><mfrac><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>α</mi></mrow><mrow><mn>2</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>α</mi><mo stretchy="false" class="MathClass-bin">−</mo><mi>β</mi></mrow></mfrac><mi>c</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mo stretchy="false" class="MathClass-bin">−</mo><mi>α</mi></mrow></msup><mo class="MathClass-rel">|</mo><mi>v</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>β</mi></mrow></msup><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext class="textrm" mathvariant="normal">in</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mtd></mtr><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>v</mi><mo class="MathClass-rel">=</mo><mi>μ</mi><mi>b</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><mi>v</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mi>q</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>2</mn></mrow></msup><mi>v</mi><mo stretchy="false" class="MathClass-bin">+</mo><mfrac><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>β</mi></mrow><mrow><mn>2</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>α</mi><mo stretchy="false" class="MathClass-bin">−</mo><mi>β</mi></mrow></mfrac><mi>c</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><mi>u</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>α</mi></mrow></msup><mo class="MathClass-rel">|</mo><mi>v</mi><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mo stretchy="false" class="MathClass-bin">−</mo><mi>β</mi></mrow></msup><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext class="textrm" mathvariant="normal">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><mi>v</mi><mo class="MathClass-rel">=</mo><mn>0</mn><mspace width="1em" class="quad"></mspace></mtd><mtd class="array" columnalign="left"><mstyle><mtext class="textrm" mathvariant="normal">in</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>N</mi></mrow></msup><mo stretchy="false" class="MathClass-bin">∖</mo><mi mathvariant="normal">Ω</mi><mo class="MathClass-punc">,</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math>

where

$Ω$ is a bounded domain in

$ℝ^{n}$ with smooth boundary

$\partial Ω$ ,

$N > 2 s$ ,

$s \in (0, 1)$ ,

$0 < α < 1, 0 < β < 1,$

$1 < q < 2 < 2_{s}^{*},$

$2_{s}^{*} = \frac{2 N}{N - 2 s}$ is the fractional Sobolev exponent,

$λ, μ$ are two parameters,

$a, b, c \in C (\bar{Ω})$ are nonnegative weight functions, and

${(- Δ)}^{s}$ is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter

$λ$ and

$μ$ .

Communicated by Shih-Hsien Yu

Keywords:

AMSC: 34B15, 37C25, 35R20

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