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 and Continuous Dynamical Systems, 2021, 41 (12) : 5551-5577. doi: 10.3934/dcds.2021088
References:
[1]

C. O. AlvesG. 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.

[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.

[3]

C. O. AlvesG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

C. JiF. 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.

[13]

C. Ji and V. D. Rădulescu, Multi-bump solutions for the nonlinear magnetic Choquard-Schrödinger equation with deepening potential well, preprint.

[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.

[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.

[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.

[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.

[18]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[19]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[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.

[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.

[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.

[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.

[24]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[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.

[26]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010, 597-632.

[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.

[28]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[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.

[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.

[31]

Y. ZhangX. 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.

show all references

References:
[1]

C. O. AlvesG. 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.

[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.

[3]

C. O. AlvesG. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

C. JiF. 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.

[13]

C. Ji and V. D. Rădulescu, Multi-bump solutions for the nonlinear magnetic Choquard-Schrödinger equation with deepening potential well, preprint.

[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.

[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.

[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.

[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.

[18]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

[19]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics 14, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[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.

[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.

[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.

[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.

[24]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[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.

[26]

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010, 597-632.

[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.

[28]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

[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.

[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.

[31]

Y. ZhangX. 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.

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