Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
In this article, we study the following fractional Schrödinger-Poisson system: ε2s(−Δ)su+V(x)u+ϕu=f(u)+∣u∣2s*−2u,inR3,ε2t(−Δ)tϕ=u2,inR3,\left\{\begin{array}{ll}{\varepsilon }^{2s}{\left(-\Delta )}^{s}u+V\left(x)u+\phi u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\...
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-04-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2024-0006 |
| _version_ | 1797219187882459136 |
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| author | Feng Shenghao Chen Jianhua Huang Xianjiu |
| author_facet | Feng Shenghao Chen Jianhua Huang Xianjiu |
| author_sort | Feng Shenghao |
| collection | DOAJ |
| description | In this article, we study the following fractional Schrödinger-Poisson system: ε2s(−Δ)su+V(x)u+ϕu=f(u)+∣u∣2s*−2u,inR3,ε2t(−Δ)tϕ=u2,inR3,\left\{\begin{array}{ll}{\varepsilon }^{2s}{\left(-\Delta )}^{s}u+V\left(x)u+\phi u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{2t}{\left(-\Delta )}^{t}\phi ={u}^{2},\hspace{1.0em}& \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 small parameter, 0<s,t<1,2s+2t>30\lt s,t\lt 1,2s+2t\gt 3, and 2s*=63−2s{2}_{s}^{* }=\frac{6}{3-2s} is the critical Sobolev exponent in dimension 3. By assuming that VV is weakly differentiable and f∈C(R,R)f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) satisfies some lower order perturbations, we show that there exists a constant ε0>0{\varepsilon }_{0}\gt 0 such that for all ε∈(0,ε0]\varepsilon \in (0,{\varepsilon }_{0}], the above system has a semiclassical Nehari-Pohozaev-type ground state solution vˆε{\hat{v}}_{\varepsilon }. Moreover, the decay estimate and asymptotic behavior of {vˆε}\left\{{\hat{v}}_{\varepsilon }\right\} are also investigated as ε→0\varepsilon \to 0. Our results generalize and improve the ones in Liu and Zhang and Ambrosio, and some other relevant literatures. |
| first_indexed | 2024-04-24T12:29:40Z |
| format | Article |
| id | doaj.art-754215cd76e44d18b410760f2da026e9 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-24T12:29:40Z |
| publishDate | 2024-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-754215cd76e44d18b410760f2da026e92024-04-08T07:35:21ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-04-0113111910.1515/anona-2024-0006Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutionsFeng Shenghao0Chen Jianhua1Huang Xianjiu2School of Mathematics and Computer Science, Nanchang University, Nanchang, 330031, P. R. ChinaSchool of Mathematics and Computer Science, Nanchang University, Nanchang, 330031, P. R. ChinaSchool of Mathematics and Computer Science, Nanchang University, Nanchang, 330031, P. R. ChinaIn this article, we study the following fractional Schrödinger-Poisson system: ε2s(−Δ)su+V(x)u+ϕu=f(u)+∣u∣2s*−2u,inR3,ε2t(−Δ)tϕ=u2,inR3,\left\{\begin{array}{ll}{\varepsilon }^{2s}{\left(-\Delta )}^{s}u+V\left(x)u+\phi u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{2t}{\left(-\Delta )}^{t}\phi ={u}^{2},\hspace{1.0em}& \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 small parameter, 0<s,t<1,2s+2t>30\lt s,t\lt 1,2s+2t\gt 3, and 2s*=63−2s{2}_{s}^{* }=\frac{6}{3-2s} is the critical Sobolev exponent in dimension 3. By assuming that VV is weakly differentiable and f∈C(R,R)f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) satisfies some lower order perturbations, we show that there exists a constant ε0>0{\varepsilon }_{0}\gt 0 such that for all ε∈(0,ε0]\varepsilon \in (0,{\varepsilon }_{0}], the above system has a semiclassical Nehari-Pohozaev-type ground state solution vˆε{\hat{v}}_{\varepsilon }. Moreover, the decay estimate and asymptotic behavior of {vˆε}\left\{{\hat{v}}_{\varepsilon }\right\} are also investigated as ε→0\varepsilon \to 0. Our results generalize and improve the ones in Liu and Zhang and Ambrosio, and some other relevant literatures.https://doi.org/10.1515/anona-2024-0006fractional schrödinger-poissonnehari-pohozaev manifoldcritical problemground state solutionsemiclassical35r1135a1535b33 |
| spellingShingle | Feng Shenghao Chen Jianhua Huang Xianjiu Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions Advances in Nonlinear Analysis fractional schrödinger-poisson nehari-pohozaev manifold critical problem ground state solution semiclassical 35r11 35a15 35b33 |
| title | Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions |
| title_full | Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions |
| title_fullStr | Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions |
| title_full_unstemmed | Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions |
| title_short | Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions |
| title_sort | critical fractional schrodinger poisson systems with lower perturbations the existence and concentration behavior of ground state solutions |
| topic | fractional schrödinger-poisson nehari-pohozaev manifold critical problem ground state solution semiclassical 35r11 35a15 35b33 |
| url | https://doi.org/10.1515/anona-2024-0006 |
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