We are concerned with the following elliptic equations
$ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $
where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mathbb{R}_0^+ \to\mathbb{R}^+_0 $ is a Kirchhoff function, $ A : \Bbb R^N \rightarrow \Bbb R^N $ is a magnetic potential and $ V:\Bbb R^{N}\to(0, \infty) $ is continuous potential. The main purpose is to show the existence of infinitely many large- or small- energy solutions to the problem above. The strategy of the proof for these results is to approach the problem variationally by employing the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition.
Citation: Seol Vin Kim, Yun-Ho Kim. Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field[J]. AIMS Mathematics, 2022, 7(4): 6583-6599. doi: 10.3934/math.2022367
We are concerned with the following elliptic equations
$ \begin{equation*} K(|z|^p_{s, {A}})(-\Delta)^s_{p, A}z+ V(x)|z|^{p-2}z = a(x)|z|^{r-2}z+\lambda f(x, |z|)z \quad {\rm{in}} \; \; \mathbb{R}^{N}, \end{equation*} $
where $ (-\Delta)^{s}_{p, A} $ is the fractional magnetic operator, $ K:\mathbb{R}_0^+ \to\mathbb{R}^+_0 $ is a Kirchhoff function, $ A : \Bbb R^N \rightarrow \Bbb R^N $ is a magnetic potential and $ V:\Bbb R^{N}\to(0, \infty) $ is continuous potential. The main purpose is to show the existence of infinitely many large- or small- energy solutions to the problem above. The strategy of the proof for these results is to approach the problem variationally by employing the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition.
| [1] |
A. Aghajani, A. Razani, Detonation waves in a transverse magnetic field, Michigan Math. J., 53 (2005), 647–664. https://doi.org/10.1307/mmj/1133894171 doi: 10.1307/mmj/1133894171
|
| [2] |
C. O. Alves, G. M. Figueiredo, M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Commun. Part. Diff. Eq., 36 (2011), 1565–1586. https://doi.org/10.1080/03605302.2011.593013 doi: 10.1080/03605302.2011.593013
|
| [3] |
C. O. Alves, S. B. Liu, On superlinear $p(x)$-Laplacian equations in $\Bbb R^{N}$, Nonlinear Anal., 73 (2010), 2566–2579. https://doi.org/10.1016/j.na.2010.06.033 doi: 10.1016/j.na.2010.06.033
|
| [4] |
C. O. Alves, R. C. M. Nemer, S. H. M. Soares, Nontrivial solutions for a mixed boundary problem for Schrödinger equations with an external magnetic field, Topol. Methods Nonlinear Anal., 46 (2015), 329–362. https://doi.org/10.12775/TMNA.2015.050 doi: 10.12775/TMNA.2015.050
|
| [5] |
A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519–543. https://doi.org/10.1006/jfan.1994.1078 doi: 10.1006/jfan.1994.1078
|
| [6] |
A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
|
| [7] |
P. d'Avenia, M. Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var., 24 (2018), 1–24. https://doi.org/10.1051/cocv/2016071 doi: 10.1051/cocv/2016071
|
| [8] | J. Bertoin, Lévy processes, Cambridge: Cambridge University Press, 1996. |
| [9] |
C. Brändle, E. Colorado, A. de Pablo, U. Sánchez, A concave–convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinb. A, 143 (2013), 39–71. https://doi.org/10.1017/S0308210511000175 doi: 10.1017/S0308210511000175
|
| [10] | L. A. Caffarelli, Non-local diffusions, drifts and games, In: Nonlinear partial differential equations, Berlin, Heidelberg: Springer, 2012, 37–52. https://doi.org/10.1007/978-3-642-25361-4_3 |
| [11] | G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A., 112 (1979), 332–336. |
| [12] |
M. M. Chaharlang, A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J., 28 (2021), 429–438. https://doi.org/10.1515/gmj-2019-2077 doi: 10.1515/gmj-2019-2077
|
| [13] |
W. Chen, S. Deng, The Nehari manifold for nonlocal elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys., 66 (2015), 1387–1400. https://doi.org/10.1007/s00033-014-0486-6 doi: 10.1007/s00033-014-0486-6
|
| [14] |
Q.-H. Choi, T. Jung, On the fractional $p$-Laplacian problems, J. Inequal. Appl., 2021 (2021), 41. https://doi.org/10.1186/s13660-021-02569-z doi: 10.1186/s13660-021-02569-z
|
| [15] |
S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field, J. Differ. Equations, 188 (2003), 52–79. https://doi.org/10.1016/S0022-0396(02)00058-X doi: 10.1016/S0022-0396(02)00058-X
|
| [16] |
N. Daisuke, The critical problem of Kirchoff type elliptic equations in dimension four, J. Differ. Equations, 257 (2014), 1168–1193. https://doi.org/10.1016/j.jde.2014.05.002 doi: 10.1016/j.jde.2014.05.002
|
| [17] | M. J. Esteban, P.-L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, In: Partial differential equations and the calculus of variations, Boston: Birkhäuser, 1989,401–449. https://doi.org/10.1007/978-1-4615-9828-2_18 |
| [18] | M. Fabian, P. Habala, P. Hajék, V. Montesinos, V. Zizler, Banach space theory, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-7515-7 |
| [19] |
G. M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl., 401 (2013), 706–713. https://doi.org/10.1016/j.jmaa.2012.12.053 doi: 10.1016/j.jmaa.2012.12.053
|
| [20] |
A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. Theor., 94 (2014), 156–170. https://doi.org/10.1016/j.na.2013.08.011 doi: 10.1016/j.na.2013.08.011
|
| [21] |
B. Ge, Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian, Nonlinear Anal. Real, 30 (2016), 236–247. https://doi.org/10.1016/j.nonrwa.2016.01.003 doi: 10.1016/j.nonrwa.2016.01.003
|
| [22] |
G. Gilboa, S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005–1028. https://doi.org/10.1137/070698592 doi: 10.1137/070698592
|
| [23] |
E. J. Hurtado, O. H. Miyagaki, R. S. Rodrigues, Existence and multiplicity of solutions for a class of elliptic equations without Ambrosetti-Rabinowitz type conditions, J. Dyn. Diff. Equat., 30 (2018), 405–432. https://doi.org/10.1007/s10884-016-9542-6 doi: 10.1007/s10884-016-9542-6
|
| [24] | T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, In: Mathematical physics, spectral theory and stochastic analysis, Basel: Birkhaüser, 2013,247–297. https://doi.org/10.1007/978-3-0348-0591-9_5 |
| [25] |
T. Ichinose, H. Tamura, Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field, Commun. Math. Phys., 105 (1986), 239–257. https://doi.org/10.1007/BF01211101 doi: 10.1007/BF01211101
|
| [26] |
J.-M. Kim, Y.-H. Kim, J. Lee, Existence of weak solutions to a class of Schrödinger type equations involving the fractional $p$-Laplacian in $\Bbb R^{N}$, J. Korean Math. Soc., 56 (2019), 1441–1461. https://doi.org/10.4134/JKMS.j180785 doi: 10.4134/JKMS.j180785
|
| [27] |
Y.-H. Kim, Existence and multiplicity of solutions to a class of fractional $p$-Laplacian equations of Schrödinger-type with concave-convex nonlinearities in $\Bbb R^{N}$, Mathematics, 8 (2020), 1792. https://doi.org/10.3390/math8101792 doi: 10.3390/math8101792
|
| [28] |
I. H. Kim, Y.-H. Kim, K. Park, Existence and multiplicity of solutions for Schrödinger-Kirchhoff type problems involving the fractional $ p(\cdot)$-Laplacian in $\Bbb R^N$, Bound. Value Probl., 2020 (2020), 121. https://doi.org/10.1186/s13661-020-01419-z doi: 10.1186/s13661-020-01419-z
|
| [29] | G. Kirchhoff, Mechanik, Leipzig, Germany: Teubner, 1883. |
| [30] |
G. Li, C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of $p$-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal. Theor., 72 (2010), 4602–4613. https://doi.org/10.1016/j.na.2010.02.037 doi: 10.1016/j.na.2010.02.037
|
| [31] |
X. Lin, X. H. Tang, Existence of infinitely many solutions for $p$-Laplacian equations in $\Bbb R^{N}$, Nonlinear Anal. Theor., 92 (2013), 72–81. https://doi.org/10.1016/j.na.2013.06.011 doi: 10.1016/j.na.2013.06.011
|
| [32] |
S. B. Liu, On ground states of superlinear $p$-Laplacian equations in $\Bbb R^{N}$, J. Math. Anal. Appl., 361 (2010), 48–58. https://doi.org/10.1016/j.jmaa.2009.09.016 doi: 10.1016/j.jmaa.2009.09.016
|
| [33] |
E. D. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
|
| [34] |
B. T. K. Oanh, D. N. Phuong, On multiplicity solutions for a non-local fractional $p$-Laplace equation, Complex Var. Elliptic Equ., 65 (2020), 801–822. https://doi.org/10.1080/17476933.2019.1631287 doi: 10.1080/17476933.2019.1631287
|
| [35] |
P. Pucci, M. Q. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\Bbb R^{N}$, Calc. Var., 54 (2015), 2785–2806. https://doi.org/10.1007/s00526-015-0883-5 doi: 10.1007/s00526-015-0883-5
|
| [36] |
P. Pucci, M. Q. Xiang, B. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27–55. https://doi.org/10.1515/anona-2015-0102 doi: 10.1515/anona-2015-0102
|
| [37] | P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Providence, RI: Amer. Math. Soc., 1986. |
| [38] |
R. Servadei, E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67–102. https://doi.org/10.1090/S0002-9947-2014-05884-4 doi: 10.1090/S0002-9947-2014-05884-4
|
| [39] |
M. Squassina, B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math., 354 (2016), 825–831. https://doi.org/10.1016/j.crma.2016.04.013 doi: 10.1016/j.crma.2016.04.013
|
| [40] |
L. Yang, T. An, J. Zuo, Infinitely many high energy solutions for fractional Schrödinger equations with magnetic field, Bound. Value Probl., 2019 (2019), 196. https://doi.org/10.1186/s13661-019-01309-z doi: 10.1186/s13661-019-01309-z
|
| [41] |
F. L. Wang, D. Hu, M. Q. Xiang, Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems, Adv. Nonlinear Anal., 10 (2021), 636–658. https://doi.org/10.1515/anona-2020-0150 doi: 10.1515/anona-2020-0150
|
| [42] | M. Willem, Minimax theorems, Boston: Birkhäuser, 1996. https://doi.org/10.1007/978-1-4612-4146-1 |
| [43] |
T.-F. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in $\Bbb R^N$ involving sign-changing weight, J. Funct. Anal., 258 (2010), 99–131. https://doi.org/10.1016/j.jfa.2009.08.005 doi: 10.1016/j.jfa.2009.08.005
|
| [44] |
M. Q. Xiang, D. Hu, B. Zhang, Y. Wang, Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth, J. Math. Anal. Appl., 501 (2021), 124269. https://doi.org/10.1016/j.jmaa.2020.124269 doi: 10.1016/j.jmaa.2020.124269
|
| [45] |
M. Q. Xiang, P. Pucci, M. Squassina, B. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst., 37 (2017), 1631–1649. https://doi.org/10.3934/dcds.2017067 doi: 10.3934/dcds.2017067
|
| [46] |
M. Q. Xiang, V. D. Rădulescu, B. Zhang, Nonlocal Kirchhoff problems with singular exponential nonlinearity, Appl. Math. Optim., 84 (2020), 915–954. https://doi.org/10.1007/s00245-020-09666-3 doi: 10.1007/s00245-020-09666-3
|
| [47] |
M. Q. Xiang, B. Zhang, Homoclinic solutions for fractional discrete Laplacian equations, Nonlinear Anal., 198 (2020), 111886. https://doi.org/10.1016/j.na.2020.111886 doi: 10.1016/j.na.2020.111886
|
| [48] |
M. Q. Xiang, B. Zhang, M. Ferrara, Multiplicity results for the non-homogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. R. Soc. A, 471 (2015), 20150034. https://doi.org/10.1098/rspa.2015.0034 doi: 10.1098/rspa.2015.0034
|
Seol Vin Kim, Yun-Ho Kim. Existence and multiplicity of solutions for nonlocal Schrödinger–Kirchhoff equations of convex–concave type with the external magnetic field[J]. AIMS Mathematics, 2022, 7(4): 6583-6599. doi: 10.3934/math.2022367
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