Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p-Laplacian in R3{{\mathbb{R}}}^{3} of the form: εsp(−Δ)psu+V(x)∣u∣p−2u−ϕ∣u∣ps♯−2u=∣u∣ps*−2u+f(u)inR3,εsp(−Δ)sϕ=∣u∣ps♯inR3,\left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{...
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De Gruyter
2025-02-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2024-0063 |
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| author | Liang Shuaishuai Song Yueqiang Shi Shaoyun |
| author_facet | Liang Shuaishuai Song Yueqiang Shi Shaoyun |
| author_sort | Liang Shuaishuai |
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| description | In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p-Laplacian in R3{{\mathbb{R}}}^{3} of the form: εsp(−Δ)psu+V(x)∣u∣p−2u−ϕ∣u∣ps♯−2u=∣u∣ps*−2u+f(u)inR3,εsp(−Δ)sϕ=∣u∣ps♯inR3,\left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{\mathcal{V}}\left(x){| u| }^{p-2}u-\phi {| u| }^{{p}_{s}^{\sharp }-2}u={| u| }^{{p}_{s}^{* }-2}u+f\left(u)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{sp}{\left(-\Delta )}^{s}\phi ={| u| }^{{p}_{s}^{\sharp }}\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε>0\varepsilon \gt 0 is a positive parameter, s∈(34,1)s\in \left(\frac{3}{4},1), (−Δ)ps{\left(-\Delta )}_{p}^{s} is the fractional p-Laplacian operator, p∈(32,3)p\in \left(\frac{3}{2},3), ps*=3p3−sp{p}_{s}^{* }=\frac{3p}{3-sp} is the Sobolev critical exponent, ps♯=p2(3+2s)(3−sp){p}_{s}^{\sharp }=\frac{p}{2}\frac{\left(3+2s)}{\left(3-sp)} is the upper exponent in the sense of the Hardy-Littlewood-Sobolev inequality, V(x):R3→R{\mathcal{V}}\left(x):{{\mathbb{R}}}^{3}\to {\mathbb{R}} symbolizes a continuous potential function with a local minimum, and the continuous function ff possesses subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann category theory, we use the topological arguments to attain the multiplicity and concentration of the positive solutions for the above system. |
| format | Article |
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| institution | Kabale University |
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| spelling | doaj-art-16cf069f25134fd8af4bf75e422158632025-02-02T15:44:38ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-02-011414727410.1515/anona-2024-0063Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3Liang Shuaishuai0Song Yueqiang1Shi Shaoyun2School of Mathematics, Jilin University, Changchun 130022, Jilin, P.R. ChinaCollege of Mathematics, Changchun Normal University, Changchun, 130032, P.R. ChinaSchool of Mathematics, Jilin University, Changchun 130022, Jilin, P.R. ChinaIn this article, we consider the following double critical fractional Schrödinger-Poisson system involving p-Laplacian in R3{{\mathbb{R}}}^{3} of the form: εsp(−Δ)psu+V(x)∣u∣p−2u−ϕ∣u∣ps♯−2u=∣u∣ps*−2u+f(u)inR3,εsp(−Δ)sϕ=∣u∣ps♯inR3,\left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{\mathcal{V}}\left(x){| u| }^{p-2}u-\phi {| u| }^{{p}_{s}^{\sharp }-2}u={| u| }^{{p}_{s}^{* }-2}u+f\left(u)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{sp}{\left(-\Delta )}^{s}\phi ={| u| }^{{p}_{s}^{\sharp }}\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε>0\varepsilon \gt 0 is a positive parameter, s∈(34,1)s\in \left(\frac{3}{4},1), (−Δ)ps{\left(-\Delta )}_{p}^{s} is the fractional p-Laplacian operator, p∈(32,3)p\in \left(\frac{3}{2},3), ps*=3p3−sp{p}_{s}^{* }=\frac{3p}{3-sp} is the Sobolev critical exponent, ps♯=p2(3+2s)(3−sp){p}_{s}^{\sharp }=\frac{p}{2}\frac{\left(3+2s)}{\left(3-sp)} is the upper exponent in the sense of the Hardy-Littlewood-Sobolev inequality, V(x):R3→R{\mathcal{V}}\left(x):{{\mathbb{R}}}^{3}\to {\mathbb{R}} symbolizes a continuous potential function with a local minimum, and the continuous function ff possesses subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann category theory, we use the topological arguments to attain the multiplicity and concentration of the positive solutions for the above system.https://doi.org/10.1515/anona-2024-0063schrödinger-poisson systemdouble criticalfractional p-laplacevariational methodsljusternik-schnirelmann category35b3335j2035j6047j3058e05 |
| spellingShingle | Liang Shuaishuai Song Yueqiang Shi Shaoyun Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 Advances in Nonlinear Analysis schrödinger-poisson system double critical fractional p-laplace variational methods ljusternik-schnirelmann category 35b33 35j20 35j60 47j30 58e05 |
| title | Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 |
| title_full | Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 |
| title_fullStr | Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 |
| title_full_unstemmed | Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 |
| title_short | Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3 |
| title_sort | concentrating solutions for double critical fractional schrodinger poisson system with p laplacian in r3 |
| topic | schrödinger-poisson system double critical fractional p-laplace variational methods ljusternik-schnirelmann category 35b33 35j20 35j60 47j30 58e05 |
| url | https://doi.org/10.1515/anona-2024-0063 |
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