Schrodinger-Poisson systems with singular potential and critical exponent
In this article we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr -\Delta \phi =u^2, \quad x\in\mathbb{R}^3, }$$ where V is a singular potential with the parameter $\alpha$ and the nonlinearity f satisfies critical gro...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/130/abstr.html |
| _version_ | 1818434736795680768 |
|---|---|
| author | Senli Liu Haibo Chen Zhaosheng Feng |
| author_facet | Senli Liu Haibo Chen Zhaosheng Feng |
| author_sort | Senli Liu |
| collection | DOAJ |
| description | In this article we study the Schrodinger-Poisson system
$$\displaylines{
-\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr
-\Delta \phi =u^2, \quad x\in\mathbb{R}^3,
}$$
where V is a singular potential with the parameter $\alpha$ and the nonlinearity f
satisfies critical growth. By applying a generalized version of Lions-type theorem and
the Nehari manifold theory, we establish the existence of the nonnegative ground state
solution when $\lambda=0$.
By the perturbation method, we obtain a nontrivial solution to above system when
$\lambda\neq 0$. |
| first_indexed | 2024-12-14T16:41:44Z |
| format | Article |
| id | doaj.art-ee5e7e084ba449feb70d9d04f4036625 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-14T16:41:44Z |
| publishDate | 2020-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-ee5e7e084ba449feb70d9d04f40366252022-12-21T22:54:19ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912020-12-012020130,117Schrodinger-Poisson systems with singular potential and critical exponentSenli Liu0Haibo Chen1Zhaosheng Feng2 Central South Univ., Changsha, Hunan, China Central South Univ., Changsha, Hunan, China Univ. of Texas Rio Grande Valley, Edinburg, TX, USA In this article we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr -\Delta \phi =u^2, \quad x\in\mathbb{R}^3, }$$ where V is a singular potential with the parameter $\alpha$ and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when $\lambda=0$. By the perturbation method, we obtain a nontrivial solution to above system when $\lambda\neq 0$.http://ejde.math.txstate.edu/Volumes/2020/130/abstr.htmlschrodinger-poisson systemlions-type theoremsingular potentialground state solutioncritical exponent |
| spellingShingle | Senli Liu Haibo Chen Zhaosheng Feng Schrodinger-Poisson systems with singular potential and critical exponent Electronic Journal of Differential Equations schrodinger-poisson system lions-type theorem singular potential ground state solution critical exponent |
| title | Schrodinger-Poisson systems with singular potential and critical exponent |
| title_full | Schrodinger-Poisson systems with singular potential and critical exponent |
| title_fullStr | Schrodinger-Poisson systems with singular potential and critical exponent |
| title_full_unstemmed | Schrodinger-Poisson systems with singular potential and critical exponent |
| title_short | Schrodinger-Poisson systems with singular potential and critical exponent |
| title_sort | schrodinger poisson systems with singular potential and critical exponent |
| topic | schrodinger-poisson system lions-type theorem singular potential ground state solution critical exponent |
| url | http://ejde.math.txstate.edu/Volumes/2020/130/abstr.html |
| work_keys_str_mv | AT senliliu schrodingerpoissonsystemswithsingularpotentialandcriticalexponent AT haibochen schrodingerpoissonsystemswithsingularpotentialandcriticalexponent AT zhaoshengfeng schrodingerpoissonsystemswithsingularpotentialandcriticalexponent |