Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian
The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian \begin{align*} (-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\ (-\Delta)^s_qv+b(x)|v|^{q-2}v&...
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| Format: | Article |
| Language: | English |
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University of Szeged
2016-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5164 |
| _version_ | 1797830604112265216 |
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| author | Mingqi Xiang Binlin Zhang Zhe Wei |
| author_facet | Mingqi Xiang Binlin Zhang Zhe Wei |
| author_sort | Mingqi Xiang |
| collection | DOAJ |
| description | The purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian
\begin{align*}
(-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\
(-\Delta)^s_qv+b(x)|v|^{q-2}v&=H_v(x,u,v)\quad \mbox{in } \mathbb{R}^N,
\end{align*}
where $1<q\leq p$, $sp<N$, $(-\Delta )_m^s$ is the fractional $m$-Laplacian, the coefficients $a, b$ are two continuous and positive functions, and $H_u,H_v$ denote the partial derivatives of $H$ with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the existence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition. |
| first_indexed | 2024-04-09T13:38:45Z |
| format | Article |
| id | doaj.art-c8e32f1baa294bb8a27ae24f3c1a944c |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:45Z |
| publishDate | 2016-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-c8e32f1baa294bb8a27ae24f3c1a944c2023-05-09T07:53:06ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752016-11-01201610711510.14232/ejqtde.2016.1.1075164Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-LaplacianMingqi Xiang0Binlin Zhang1Zhe Wei2Civil Aviation University of China, Tianjin, P. R. ChinaHeilongjiang Institute of Technology, Harbin, P. R. ChinaHeilongjiang Institute of Technology, Harbin, P. R. ChinaThe purpose of this paper is to investigate the existence of solutions to the following quasilinear Schr\"{o}dinger type system driven by the fractional $p$-Laplacian \begin{align*} (-\Delta)^{s}_pu+a(x)|u|^{p-2}u&=H_u(x,u,v)\quad \mbox{in } \mathbb{R}^N,\\ (-\Delta)^s_qv+b(x)|v|^{q-2}v&=H_v(x,u,v)\quad \mbox{in } \mathbb{R}^N, \end{align*} where $1<q\leq p$, $sp<N$, $(-\Delta )_m^s$ is the fractional $m$-Laplacian, the coefficients $a, b$ are two continuous and positive functions, and $H_u,H_v$ denote the partial derivatives of $H$ with respect to the second variable and the third variable. By using the mountain pass theorem, we obtain the existence of nontrivial and nonnegative solutions for the above system. The main feature of this paper is that the nonlinearities do not necessarily satisfy the Ambrosetti-Rabinowitz condition.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5164schrödinger systemfractional $p$-laplacianmountain pass theorem |
| spellingShingle | Mingqi Xiang Binlin Zhang Zhe Wei Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian Electronic Journal of Qualitative Theory of Differential Equations schrödinger system fractional $p$-laplacian mountain pass theorem |
| title | Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian |
| title_full | Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian |
| title_fullStr | Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian |
| title_full_unstemmed | Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian |
| title_short | Existence of solutions to a class of quasilinear Schrödinger systems involving the fractional $p$-Laplacian |
| title_sort | existence of solutions to a class of quasilinear schrodinger systems involving the fractional p laplacian |
| topic | schrödinger system fractional $p$-laplacian mountain pass theorem |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5164 |
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