A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
This article deals with existence of solutions to the following fractional pp-Laplacian system of equations: (−Δp)su=∣u∣ps*−2u+γαps*∣u∣α−2u∣v∣βinΩ,(−Δp)sv=∣v∣ps*−2v+γβps*∣v∣β−2v∣u∣αinΩ,\left\{\begin{array}{l}{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u+\frac{\gamma \alpha }{{p}_{s}^{* }}{|...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-09-01
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| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0103 |
| _version_ | 1797682418540347392 |
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| author | Bhakta Mousomi Perera Kanishka Sk Firoj |
| author_facet | Bhakta Mousomi Perera Kanishka Sk Firoj |
| author_sort | Bhakta Mousomi |
| collection | DOAJ |
| description | This article deals with existence of solutions to the following fractional pp-Laplacian system of equations: (−Δp)su=∣u∣ps*−2u+γαps*∣u∣α−2u∣v∣βinΩ,(−Δp)sv=∣v∣ps*−2v+γβps*∣v∣β−2v∣u∣αinΩ,\left\{\begin{array}{l}{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u+\frac{\gamma \alpha }{{p}_{s}^{* }}{| u| }^{\alpha -2}u{| v| }^{\beta }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\\ {\left(-{\Delta }_{p})}^{s}v={| v| }^{{p}_{s}^{* }-2}v+\frac{\gamma \beta }{{p}_{s}^{* }}{| v| }^{\beta -2}v{| u| }^{\alpha }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\end{array}\right. where s∈(0,1)s\in \left(0,1), p∈(1,∞)p\in \left(1,\infty ) with N>spN\gt sp, α,β>1\alpha ,\beta \gt 1 such that α+β=ps*≔NpN−sp\alpha +\beta ={p}_{s}^{* }:= \frac{Np}{N-sp} and Ω=RN\Omega ={{\mathbb{R}}}^{N} or smooth bounded domains in RN{{\mathbb{R}}}^{N}. When Ω=RN\Omega ={{\mathbb{R}}}^{N} and γ=1\gamma =1, we show that any ground state solution of the aforementioned system has the form (λU,τλV)\left(\lambda U,\tau \lambda V) for certain τ>0\tau \gt 0 and UU and VV are two positive ground state solutions of (−Δp)su=∣u∣ps*−2u{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u in RN{{\mathbb{R}}}^{N}. For all γ>0\gamma \gt 0, we establish existence of a positive radial solution to the aforementioned system in balls. When Ω=RN\Omega ={{\mathbb{R}}}^{N}, we also establish existence of positive radial solutions to the aforementioned system in various ranges of γ\gamma . |
| first_indexed | 2024-03-11T23:59:27Z |
| format | Article |
| id | doaj.art-bb1166c98c4647d993dd88d6a9da9fe1 |
| institution | Directory Open Access Journal |
| issn | 2169-0375 |
| language | English |
| last_indexed | 2024-03-11T23:59:27Z |
| publishDate | 2023-09-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-bb1166c98c4647d993dd88d6a9da9fe12023-09-18T06:31:15ZengDe GruyterAdvanced Nonlinear Studies2169-03752023-09-01231266177310.1515/ans-2023-0103A system of equations involving the fractional p-Laplacian and doubly critical nonlinearitiesBhakta Mousomi0Perera Kanishka1Sk Firoj2Department of Mathematics, Indian Institute of Science Education and Research Pune (IISER-Pune), Dr. Homi Bhabha Road, Pune – 411008, IndiaDepartment of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USADepartment of Mathematics, Indian Institute of Science Education and Research Pune (IISER-Pune), Dr. Homi Bhabha Road, Pune – 411008, IndiaThis article deals with existence of solutions to the following fractional pp-Laplacian system of equations: (−Δp)su=∣u∣ps*−2u+γαps*∣u∣α−2u∣v∣βinΩ,(−Δp)sv=∣v∣ps*−2v+γβps*∣v∣β−2v∣u∣αinΩ,\left\{\begin{array}{l}{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u+\frac{\gamma \alpha }{{p}_{s}^{* }}{| u| }^{\alpha -2}u{| v| }^{\beta }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\\ {\left(-{\Delta }_{p})}^{s}v={| v| }^{{p}_{s}^{* }-2}v+\frac{\gamma \beta }{{p}_{s}^{* }}{| v| }^{\beta -2}v{| u| }^{\alpha }\hspace{0.33em}\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\hspace{1.0em}\end{array}\right. where s∈(0,1)s\in \left(0,1), p∈(1,∞)p\in \left(1,\infty ) with N>spN\gt sp, α,β>1\alpha ,\beta \gt 1 such that α+β=ps*≔NpN−sp\alpha +\beta ={p}_{s}^{* }:= \frac{Np}{N-sp} and Ω=RN\Omega ={{\mathbb{R}}}^{N} or smooth bounded domains in RN{{\mathbb{R}}}^{N}. When Ω=RN\Omega ={{\mathbb{R}}}^{N} and γ=1\gamma =1, we show that any ground state solution of the aforementioned system has the form (λU,τλV)\left(\lambda U,\tau \lambda V) for certain τ>0\tau \gt 0 and UU and VV are two positive ground state solutions of (−Δp)su=∣u∣ps*−2u{\left(-{\Delta }_{p})}^{s}u={| u| }^{{p}_{s}^{* }-2}u in RN{{\mathbb{R}}}^{N}. For all γ>0\gamma \gt 0, we establish existence of a positive radial solution to the aforementioned system in balls. When Ω=RN\Omega ={{\mathbb{R}}}^{N}, we also establish existence of positive radial solutions to the aforementioned system in various ranges of γ\gamma .https://doi.org/10.1515/ans-2023-0103fractional p-laplaciandoubly criticalground stateexistencesystemleast energy solutionnehari manifold35b0935b3335e2035d3035j5045k05 |
| spellingShingle | Bhakta Mousomi Perera Kanishka Sk Firoj A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities Advanced Nonlinear Studies fractional p-laplacian doubly critical ground state existence system least energy solution nehari manifold 35b09 35b33 35e20 35d30 35j50 45k05 |
| title | A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities |
| title_full | A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities |
| title_fullStr | A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities |
| title_full_unstemmed | A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities |
| title_short | A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities |
| title_sort | system of equations involving the fractional p laplacian and doubly critical nonlinearities |
| topic | fractional p-laplacian doubly critical ground state existence system least energy solution nehari manifold 35b09 35b33 35e20 35d30 35j50 45k05 |
| url | https://doi.org/10.1515/ans-2023-0103 |
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