# American Institute of Mathematical Sciences

December  2021, 41(12): 5551-5577. doi: 10.3934/dcds.2021088

## Concentration phenomena for magnetic Kirchhoff equations with critical growth

 1 Department of Mathematics, East China University of Science and Technology, Shanghai 200237, People's Republic of China 2 Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków 30-059, Poland 3 Department of Mathematics, University of Craiova, Craiova 200585, Romania

* Corresponding author: Vicenţiu D. Rădulescu (radulescu@inf.ucv.ro)

Received  March 2021 Published  December 2021 Early access  May 2021

Fund Project: Chao Ji is partially supported by Shanghai Natural Science Foundation (20ZR1413900, 18ZR1409100). The research of Vicențiu D. Rǎdulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI/UEFISCDI, project number PCE 137/2021, within PNCDI Ⅲ. The research of Vicențiu D. Rǎdulescu was also supported by the Slovenian Research Agency program P1-0292.

In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth
 \begin{align*} \left\{ \begin{aligned} &-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{align*}
where
 $\epsilon>0$
is a parameter,
 $a, b>0$
are constants,
 $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$
and
 $A: \mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$
are continuous potentials, and
 $f: \mathbb{R}\rightarrow \mathbb{R}$
is a nonlinear term with subcritical growth. Under a local assumption on the potential
 $V$
, combining variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, we establish multiplicity and concentration properties of solutions to the above problem for
 $\varepsilon$
small. A feature of this paper is that the function
 $f$
is assumed to be only continuous, which allows to consider larger classes of nonlinearities in the reaction.
Citation: Chao Ji, Vicenţiu D. Rădulescu. Concentration phenomena for magnetic Kirchhoff equations with critical growth. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5551-5577. doi: 10.3934/dcds.2021088
##### References:
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show all references

##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013.  Google Scholar [2] C. O. Alves and G. M. Figueiredo, Multiple solutions for a semilinear elliptic equation with critical growth and magnetic field, Milan J. Math., 82 (2014), 389-405.  doi: 10.1007/s00032-014-0225-7.  Google Scholar [3] C. O. Alves, G. M. Figueiredo and M. Yang, Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field, Asymptot. Anal., 96 (2016), 135-159.  doi: 10.3233/ASY-151337.  Google Scholar [4] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Rational Mech. Anal., 170 (2003), 277-295.  doi: 10.1007/s00205-003-0274-5.  Google Scholar [5] P. d'Avenia and C. Ji, Multiplicity and concentration results for a magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$, Int. Math. Res. Not., (2020), doi: 10.1093/imrn/rnaa074 doi: 10.1093/imrn/rnaa074.  Google Scholar [6] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar [7] M. J. Esteban and P.-L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in "Partial differential equations and the calculus of variations", Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, 1 (1989), 401–449.  Google Scholar [8] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.  Google Scholar [9] X.M. He and W.M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differential Equations, 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar [10] X. M. He and W. M. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), Art 91, 39 pp. doi: 10.1007/s00526-016-1045-0.  Google Scholar [11] X. M. He and W. M. Zou, Multiplicity and concentrating solutions for a class of fractional Kirchhoff equation, Manuscripta Math., 158 (2018), 159-203.  doi: 10.1007/s00229-018-1017-0.  Google Scholar [12] C. Ji, F. Fang and B. L. Zhang, A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal., 8 (2019), 267-277.  doi: 10.1515/anona-2016-0240.  Google Scholar [13] C. Ji and V. D. Rădulescu, Multi-bump solutions for the nonlinear magnetic Choquard-Schrödinger equation with deepening potential well, preprint. Google Scholar [14] C. Ji and V. D. Rădulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in $\mathbb{R}^{2}$, Manuscripta Math., 164 (2021), 509-542.  doi: 10.1007/s00229-020-01195-1.  Google Scholar [15] C. Ji and V. D. Rădulescu, Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation, Calc. Var. Partial Differential Equations, 59 (2020), Art 115, 28 pp. doi: 10.1007/s00526-020-01772-y.  Google Scholar [16] C. Ji and V. D. Rădulescu, Concentration phenomena for nonlinear magnetic Schrödinger equations with critical growth, Israel J. Math., 241 (2021), 465-500.  doi: 10.1007/s11856-021-2105-5.  Google Scholar [17] C. Ji and V. D. Rădulescu, Multiplicity and concentration of solutions for Kirchhoff equations with magnetic field, Adv. Nonlinear Stud., (2021), in the press. doi: 10.1515/ans-2021-2130.  Google Scholar [18] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar [19] E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.  Google Scholar [20] X. Mingqi, V. D. Rădulescu and B. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21 (2019), 1850004, 36 pp. doi: 10.1142/s0219199718500049.  Google Scholar [21] Y. G. Oh, Existence of semi-classical bound state of nonlinear Schrödinger equations with potential on the class of $(V)_{a}$, Comm. Partial Differential Equations, 13 (1998), 1499-1519.  doi: 10.1080/03605308808820585.  Google Scholar [22] Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131 (1990), 223-253.  doi: 10.1007/BF02161413.  Google Scholar [23] K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221 (2006), 246-255.  doi: 10.1016/j.jde.2005.03.006.  Google Scholar [24] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar [25] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar [26] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010, 597-632.  Google Scholar [27] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar [28] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [29] H. Zhang and F. B. Zhang, Ground states for the nonlinear Kirchhoff type problems, J. Math. Anal. Appl., 423 (2015), 1671-1692.  doi: 10.1016/j.jmaa.2014.10.062.  Google Scholar [30] J. Zhang and W. M. Zou, Multiplicity and concentration behavior of solutions to the critical Kirchhoff-type problem, Z. Angew. Math. Phys., 68 (2017), Paper No. 57, 27 pp. doi: 10.1007/s00033-017-0803-y.  Google Scholar [31] Y. Zhang, X. Tang and V. D. Rădulescu, Small perturbations for nonlinear Schrödinger equations with magnetic potential, Milan J. Math., 88 (2020), 479-506.  doi: 10.1007/s00032-020-00322-7.  Google Scholar
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