In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian:
$ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $
where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.
Citation: Hui Liang, Yueqiang Song, Baoling Yang. Some results for a supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian[J]. AIMS Mathematics, 2024, 9(5): 13508-13521. doi: 10.3934/math.2024658
In this work, we focus our attention on the existence of nontrivial solutions to the following supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian:
$ \begin{equation*} \begin{cases} -\Delta_{p}u-\Delta_{q}u+\phi|u|^{q-2} u = f\left(x, u\right)+\mu|u|^{s-2} u & \text { in } \Omega, \\ -\Delta \phi = |u|^q & \text { in } \Omega, \\ u = \phi = 0 & \text { on } \partial \Omega, \end{cases} \end{equation*} $
where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, $ \mu > 0, N > 1 $, and $ -\Delta_{{\wp}}\varphi = div(|\nabla\varphi|^{{\wp}-2} \nabla\varphi) $, with $ {\wp}\in \{p, q\} $, is the homogeneous $ {\wp} $-Laplacian. $ 1 < p < q < \frac{q^*}{2} $, $ q^*: = \frac{Nq}{N-q} < s $, and $ q^* $ is the critical exponent to $ q $. The proof is accomplished by the Moser iterative method, the mountain pass theorem, and the truncation technique. Furthermore, the $ (p, q) $-Laplacian and the supercritical term appear simultaneously, which is the main innovation and difficulty of this paper.
| [1] |
A. Ambrosetti, On Schrödinger-Poisson Systems, Milan J. Math., 76 (2008), 257–274. https://doi.org/10.1007/s00032-008-0094-z doi: 10.1007/s00032-008-0094-z
|
| [2] | R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2nd edn. Academic Press, New York, 2003. |
| [3] |
C. O. Alves, G. M. Figueiredo, Multiplicity and concentration of positive solutions for a class of quasilinear problems, Adv. Nonlinear Stud., 11 (2011), 265–295. https://doi.org/10.1515/ans-2011-0203 doi: 10.1515/ans-2011-0203
|
| [4] |
Y. C. An, H. R. Liu, The Schrödinger-Poisson type system involving a critical nonlinearity on the first Heisenberg group, Isr. J. Math., 235 (2020), 385–411. https://doi.org/10.1515/ans-2011-0203 doi: 10.1515/ans-2011-0203
|
| [5] |
A. Ambrosetti, R. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391–404. https://doi.org/10.1142/S021919970800282X doi: 10.1142/S021919970800282X
|
| [6] |
R. Arora, A. Fiscella, T. Mukherjee, P. Winkert, On double phase Kirchhoff problems with singular nonlinearity, Adv. Nonlinear Anal., 12 (2023), 20220312. https://doi.org/10.1515/anona-2022-0312 doi: 10.1515/anona-2022-0312
|
| [7] | H. Brézis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2011. |
| [8] |
R. Benguria, H. Brézis, E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Commun. Math. Phys., 79 (1981), 167–180. https://doi.org/10.1007/BF01942059 doi: 10.1007/BF01942059
|
| [9] |
V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283–293. https://doi.org/10.12775/TMNA.1998.019 doi: 10.12775/TMNA.1998.019
|
| [10] |
D. Cassani, Z. S. Liu, G. Romani, Nonlocal Planar Schrödinger-Poisson Systems in the Fractional Sobolev Limiting Case, J. Differ. Equations, 383 (2024), 214–269. https://doi.org/10.1016/j.jde.2023.11.018 doi: 10.1016/j.jde.2023.11.018
|
| [11] |
S. T. Chen, M. H. Shu, X. H. Tang, L. X. Wen, Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case, J. Differ. Equations, 327 (2022), 448–480. https://doi.org/10.1016/j.jde.2022.04.022 doi: 10.1016/j.jde.2022.04.022
|
| [12] |
Y. Du, J. B. Su, C. Wang, On a quasilinear Schrödinger-Poisson system, J. Math. Anal. Appl., 505 (2022), 125446. https://doi.org/10.1016/j.jmaa.2021.125446 doi: 10.1016/j.jmaa.2021.125446
|
| [13] | Y. Du, J. B. Su, C. Wang, The quasilinear Schrödinger-Poisson system, J. Math. Phys., 64 (2023), 071502. |
| [14] |
L. Gao, Z. Tan, Existence results for fractional Kirchhoff problems with magnetic field and supercritical growth, J. Math. Phys., 64 (2023), 031503. https://doi.org/10.1063/5.0127185 doi: 10.1063/5.0127185
|
| [15] |
G. Z. Gu, X. H. Tang, J. X. Shen, Multiple solutions for fractional Schrödinger-Poisson system with critical or supercritical nonlinearity, Appl. Math. Lett., 111 (2021), 106605. https://doi.org/10.1016/j.aml.2020.106605 doi: 10.1016/j.aml.2020.106605
|
| [16] |
P. L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1987), 33–97. https://doi.org/10.1007/BF01205672 doi: 10.1007/BF01205672
|
| [17] |
E. H. Lieb, Thomas-Fermi and related theories and molecules, Rev. Mod. Phys., 53 (1981), 603–641. https://doi.org/10.1103/RevModPhys.53.603 doi: 10.1103/RevModPhys.53.603
|
| [18] |
W. Li, V. D. Rădulescu, B. L. Zhang, Infinitely many solutions for fractional Kirchhoff-Schrödinger-Poisson systems, J. Math. Phys., 60 (2019), 011506. https://doi.org/10.1063/1.5019677 doi: 10.1063/1.5019677
|
| [19] |
Z. Y. Liu, L. L. Tao, D. L. Zhang, S. H. Liang, Y. Q. Song, Critical nonlocal Schrödinger-Poisson system on the Heisenberg group, Adv. Nonlinear Anal., 11 (2022), 482–502. https://doi.org/10.1515/anona-2021-0203 doi: 10.1515/anona-2021-0203
|
| [20] | P. Markowich, C. Ringhofer, C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. https://doi.org/10.1007/978-3-7091-6961-2_1 |
| [21] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655–674. https://doi.org/10.1016/j.jfa.2006.04.005 doi: 10.1016/j.jfa.2006.04.005
|
| [22] | Y. Q. Song, Y. Y. Huo, D. D. Repovš, On the Schrödinger-Poisson system with $(p, q)$-Laplacian, Appl. Math. Lett., 141 (2023), 108595. |
| [23] | M. Willem, Minimax theorems, Birkhäuser, Boston, 1996. |
| [24] |
J. J. Zhang, J. M. do Ó, M. Squassina, Fractional Schrödinger-Poisson Systems with a General Subcritical or Critical Nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15–30. https://doi.org/10.1515/ans-2015-5024 doi: 10.1515/ans-2015-5024
|
| [25] | X. J. Zhong, C. L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $ \mathbb{R}^{3}$, Nonlinear Anal. Real World Appl., 39 (2018), 166–184. |
Hui Liang, Yueqiang Song, Baoling Yang. Some results for a supercritical Schrödinger-Poisson type system with $ (p, q) $-Laplacian[J]. AIMS Mathematics, 2024, 9(5): 13508-13521. doi: 10.3934/math.2024658
DownLoad: