Article
| Title: | On the existence of nontrivial solutions for modified fractional Schrödinger-Poisson systems via perturbation method (English) |
| Author: | Goli, Atefe |
| Author: | Rasouli, Sayyed Hashem |
| Author: | Khademloo, Somayeh |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 70 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 293-310 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The existence of nontrivial solutions is considered for the fractional Schrödinger-Poisson system with double quasi-linear terms: $$ \begin{cases} (-\Delta )^{s}u+V(x)u+\phi u -{1\over 2}u (-\Delta )^{s}u^{2}=f(x,u), & x\in \mathbb {R}^{3} ,\\ (-\Delta )^{t} \phi = u^{2}, & x\in \mathbb {R}^{3}, \end{cases} $$ where $(-\Delta )^{\alpha }$ is the fractional Laplacian for $\alpha =s$, $t\in (0,1]$ with $s<t$ and $2t+4s>3$. Under assumptions on $V$ and $f$, we prove the existence of positive solutions and negative solutions for the above system by using perturbation method and the mountain pass theorem. (English) |
| Keyword: | fractional-Schrödinger-Poisson |
| Keyword: | quasi-linear term |
| Keyword: | perturbation method |
| Keyword: | variational method |
| MSC: | 35A02 |
| MSC: | 35A15 |
| MSC: | 35B32 |
| DOI: | 10.21136/AM.2025.0232-23 |
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| Date available: | 2025-05-26T12:19:04Z |
| Last updated: | 2025-06-02 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152983 |
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| Reference: | [1] Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on $\Bbb{R}^N$.Commun. Partial Differ. Equations 20 (1995), 1725-1741. Zbl 0837.35043, MR 1349229, 10.1080/03605309508821149 |
| Reference: | [2] Benci, V., Fortunato, D.: An eigenvalue problem for the Schrödinger-Maxwell equations.Topol. Methods Nonlinear Anal. 11 (1998), 283-293. Zbl 0926.35125, MR 1659454, 10.12775/TMNA.1998.019 |
| Reference: | [3] Caffarelli, L. A., Roquejoffre, J.-M., Sire, Y.: Variational problems with free boundaries for the fractional Laplacian.J. Eur. Math. Soc. (JEMS) 12 (2010), 1151-1179. Zbl 1221.35453, MR 2677613, 10.4171/JEMS/226 |
| Reference: | [4] Caffarelli, L. A., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian.Invent. Math. 171 (2008), 425-461. Zbl 1148.35097, MR 2367025, 10.1007/s00222-007-0086-6 |
| Reference: | [5] Caffarelli, L. A., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces.Calc. Var. Partial Differ. Equ. 41 (2011), 203-240. Zbl 1357.49143, MR 2782803, 10.1007/s00526-010-0359-6 |
| Reference: | [6] Costa, D. G.: On a class of elliptic systems in $\Bbb{R}^n$.Electron. J. Differ. Equ. 1994 (1994), Article ID 7, 14 pages. Zbl 0809.35020, MR 1292598 |
| Reference: | [7] Cotsiolis, A., Tavoularis, N. K.: Sharp Sobolev type inequalities for higher fractional derivatives.C. R., Math., Acad. Sci. Paris 335 (2002), 801-804. Zbl 1032.26010, MR 1947703, 10.1016/S1631-073X(02)02576-1 |
| Reference: | [8] D'Aprile, T., Mugnai, D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations.Proc. R. Soc. Edinb., Sect. A, Math. 134 (2004), 893-906. Zbl 1064.35182, MR 2099569, 10.1017/S030821050000353X |
| Reference: | [9] D'Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations.Math. Models Methods Appl. Sci. 25 (2015), 1447-1476. Zbl 1323.35205, MR 3340706, 10.1142/S0218202515500384 |
| Reference: | [10] Nezza, E. Di, Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces.Bull. Sci. Math. 136 (2012), 521-573. Zbl 1252.46023, MR 2944369, 10.1016/j.bulsci.2011.12.004 |
| Reference: | [11] Laskin, N.: Fractional quantum mechanics and Lévy path integrals.Phys. Lett. A 268 (2000), 298-305. Zbl 0948.81595, MR 1755089, 10.1016/S0375-9601(00)00201-2 |
| Reference: | [12] Liu, X.-Q., Liu, J.-Q., Wang, Z.-Q.: Quasilinear elliptic equations via perturbation method.Proc. Am. Math. Soc. 141 (2013), 253-263. Zbl 1267.35096, MR 2988727, 10.1090/S0002-9939-2012-11293-6 |
| Reference: | [13] Nie, J., Wu, X.: Existence and multiplicity of non-trivial solutions for a class of modified Schrödinger-Poisson systems.J. Math. Anal. Appl. 408 (2013), 713-724. Zbl 1308.81082, MR 3085065, 10.1016/j.jmaa.2013.06.011 |
| Reference: | [14] Peng, X., Jia, G.: Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms.Discrete Contin. Dyn. Syst., Ser. B 27 (2022), 2325-2344. Zbl 1486.35209, MR 4399151, 10.3934/dcdsb.2021134 |
| Reference: | [15] Ruiz, D.: The Schrödinger-Poisson equation under the effect of a nonlinear local term.J. Funct. Anal. 237 (2006), 655-674. Zbl 1136.35037, MR 2230354, 10.1016/j.jfa.2006.04.005 |
| Reference: | [16] Salvatore, A.: Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in $\Bbb{R}^3$.Adv. Nonlinear Stud. 6 (2006), 157-169. Zbl 1229.35065, MR 2219833, 10.1515/ans-2006-0203 |
| Reference: | [17] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions.Princeton Mathematical Series 30. Princeton University Press, Princeton (1970). Zbl 0207.13501, MR 0290095, 10.1515/9781400883882 |
| Reference: | [18] Teng, K.: Multiple solutions for a class of fractional Schrödinger equations in $\Bbb{R}^N$.Nonlinear Anal., Real World Appl. 21 (2015), 76-86. Zbl 1302.35415, MR 3261580, 10.1016/j.nonrwa.2014.06.008 |
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