# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022059
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## Critical gauged Schrödinger equations in $\mathbb{R}^2$ with vanishing potentials

 1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China 2 Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore, Via della Garzetta 48, 25133, Brescia, Italy

*Corresponding author: Marco Squassina

Received  October 2021 Revised  March 2022 Early access April 2022

Fund Project: Marco Squassina is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Minbo Yang was partially supported by NSFC (11971436, 12011530199) and ZJNSF(LZ22A010001, LD19A010001)

We study a class of gauged nonlinear Schrödinger equations in the plane
 $\left\{ \begin{array}{l} -\Delta u+V(|x|) u+\lambda\bigg(\int_{|x|}^\infty \frac{h_u(s)}{s}u^2(s)ds+\frac{h_u^2(|x|)}{|x|^2} \bigg)u\\\qquad \, = K(|x|)f(u)+\mu g(|x|)|u|^{q-2}u, \\ u(x) = u(|x|) \; {\rm{in}}\; \mathbb{R}^2, \\\\ \end{array} \right.$
where
 $h_u(s) = \int_0^s\frac{r}{2}u^2(r)dr$
,
 $\lambda,\mu>0$
are constants,
 $V(|x|)$
and
 $K(|x|)$
are continuous functions vanishing at infinity. Assume that
 $f$
is of critical exponential growth and
 $g(x) = g(|x|)$
satisfies some technical assumptions with
 $1\leq q<2$
, we obtain the existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, with the help of the genus theory, we prove the existence of infinitely many solutions if
 $f$
in addition is odd.
Citation: Liejun Shen, Marco Squassina, Minbo Yang. Critical gauged Schrödinger equations in $\mathbb{R}^2$ with vanishing potentials. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022059
##### References:
 [1] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbb{R}^{2}$ involving critical exponent, Ann. Sc. Norm. Super. Pisa, 17 (1990), 481-504. [2] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 2010 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194. [3] F. S. B. Albuquerque, J. L. Carvalho, G. M. Figueiredo and E. S. Medeiros, On a planar non-autonomous Schrödinger-Poisson system involving exponential critical growth, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 40, 30 pp. doi: 10.1007/s00526-020-01902-6. [4] F. S. B. Albuquerque, M. C. Ferreira and U. B. Severo, Ground state solutions for a nonlocal equation in $\mathbb{R}^2$ involving vanishing potentials and exponential critical growth, Milan J. Math., 89 (2021), 263-294.  doi: 10.1007/s00032-021-00334-x. [5] C. O. Alves, M. A. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations, 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y. [6] J. G. Azorero and I. P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.1090/S0002-9947-1991-1083144-2. [7] A. Azzollini and A. Pomponio, Positive energy static solutions for the Chern-Simons-Schrödinger system under a large-distance fall-off requirement on the gauge potentials, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 165, 30 pp. doi: 10.1007/s00526-021-02031-4. [8] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. [9] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556. [11] L. Bergé, A. de Bouard and J. C. Saut, Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [12] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024. [13] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004. [14] D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Commun. Partial Differential Equations, 17 (1992), 407-435.  doi: 10.1080/03605309208820848. [15] P. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano, A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonlinear Differential Equations Appl., 22 (2015), 1831-1850.  doi: 10.1007/s00030-015-0346-x. [16] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003. [17] M. de Souza and J. M. do Ó, A sharp Trudinger-Moser type inequality in $\mathbb{R}^2$, Trans. Amer. Math. Soc., 366 (2014), 4513-4549.  doi: 10.1090/S0002-9947-2014-05811-X. [18] Y. Deng, S. Peng and W. Shuai, Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^2$, J. Differential Equations, 264 (2018), 4006-4035.  doi: 10.1016/j.jde.2017.12.003. [19] M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partical Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7. [20] J. M. do Ó, N-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.  doi: 10.1155/S1085337597000419. [21] J. M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074. [22] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^n$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020. [23] G. Dunne, Self-Dual Chern-Simons Theories, Springer, 1995. doi: 10.1007/978-3-540-44777-1. [24] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6. [25] H. Huh, Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.  doi: 10.1088/0951-7715/22/5/003. [26] R. Jackiw and S. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [27] R. Jackiw and S. Pi, Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40.  doi: 10.1143/PTPS.107.1. [28] C. Ji and F. Fang, Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth, J. Math. Anal. Appl., 450 (2017), 578-591.  doi: 10.1016/j.jmaa.2017.01.065. [29] Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 1550074, 20 pp. doi: 10.1142/S0219199715500741. [30] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005. [31] M. A. Krasnoselski$\mathop {\rm{i}}\limits^ \vee$, Topological Methods in the Theory of Nonlinear Integral Equations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. [32] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [33] Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137. [34] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoam, 1 (1985), 145-201.  doi: 10.4171/RMI/6. [35] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [36] S. I. Pohozaev, The Sobolev embedding in the case $pl = n$, In: Proc. Tech. Sci. Conf. on Adv. Sci., Research, 1964–1965; Math. Section, Moscow, (1965), 158–170. [37] A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535. [38] A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2. [39] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, In: CBMS Regional Conference Series in Mathematics, vol. 65, AMS, Providence RI, 1986. doi: 10.1090/cbms/065. [40] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013. [41] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [42] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583.  doi: 10.1142/S021919970700254X. [43] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028. [44] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [45] Y. Yang and X. Zhu, Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two, J. Funct. Anal., 272 (2017), 3347-3374.  doi: 10.1016/j.jfa.2016.12.028.

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##### References:
 [1] Adimurthi and S. L. Yadava, Multiplicity results for semilinear elliptic equations in bounded domain of $\mathbb{R}^{2}$ involving critical exponent, Ann. Sc. Norm. Super. Pisa, 17 (1990), 481-504. [2] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trudinger-Moser inequality in $\mathbb{R}^N$ and its applications, Int. Math. Res. Not. IMRN, 2010 (2010), 2394-2426.  doi: 10.1093/imrn/rnp194. [3] F. S. B. Albuquerque, J. L. Carvalho, G. M. Figueiredo and E. S. Medeiros, On a planar non-autonomous Schrödinger-Poisson system involving exponential critical growth, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 40, 30 pp. doi: 10.1007/s00526-020-01902-6. [4] F. S. B. Albuquerque, M. C. Ferreira and U. B. Severo, Ground state solutions for a nonlocal equation in $\mathbb{R}^2$ involving vanishing potentials and exponential critical growth, Milan J. Math., 89 (2021), 263-294.  doi: 10.1007/s00032-021-00334-x. [5] C. O. Alves, M. A. Souto and M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth, Calc. Var. Partial Differential Equations, 43 (2012), 537-554.  doi: 10.1007/s00526-011-0422-y. [6] J. G. Azorero and I. P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.  doi: 10.1090/S0002-9947-1991-1083144-2. [7] A. Azzollini and A. Pomponio, Positive energy static solutions for the Chern-Simons-Schrödinger system under a large-distance fall-off requirement on the gauge potentials, Calc. Var. Partial Differential Equations, 60 (2021), Paper No. 165, 30 pp. doi: 10.1007/s00526-021-02031-4. [8] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. [9] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅱ. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556. [11] L. Bergé, A. de Bouard and J. C. Saut, Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007. [12] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024. [13] J. Byeon, H. Huh and J. Seok, On standing waves with a vortex point of order N for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004. [14] D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^2$, Commun. Partial Differential Equations, 17 (1992), 407-435.  doi: 10.1080/03605309208820848. [15] P. Cunha, P. d'Avenia, A. Pomponio and G. Siciliano, A multiplicity result for Chern-Simons-Schrödinger equation with a general nonlinearity, Nonlinear Differential Equations Appl., 22 (2015), 1831-1850.  doi: 10.1007/s00030-015-0346-x. [16] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $\mathbb{R}^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.  doi: 10.1007/BF01205003. [17] M. de Souza and J. M. do Ó, A sharp Trudinger-Moser type inequality in $\mathbb{R}^2$, Trans. Amer. Math. Soc., 366 (2014), 4513-4549.  doi: 10.1090/S0002-9947-2014-05811-X. [18] Y. Deng, S. Peng and W. Shuai, Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^2$, J. Differential Equations, 264 (2018), 4006-4035.  doi: 10.1016/j.jde.2017.12.003. [19] M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger-Moser inequalities, Calc. Var. Partical Differential Equations, 55 (2016), Art. 88, 26 pp. doi: 10.1007/s00526-016-1014-7. [20] J. M. do Ó, N-Laplacian equations in $\mathbb{R}^N$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.  doi: 10.1155/S1085337597000419. [21] J. M. do Ó, E. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074. [22] J. M. do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $\mathbb{R}^n$, J. Differential Equations, 246 (2009), 1363-1386.  doi: 10.1016/j.jde.2008.11.020. [23] G. Dunne, Self-Dual Chern-Simons Theories, Springer, 1995. doi: 10.1007/978-3-540-44777-1. [24] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474.  doi: 10.1090/S0273-0979-1979-14595-6. [25] H. Huh, Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.  doi: 10.1088/0951-7715/22/5/003. [26] R. Jackiw and S. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500. [27] R. Jackiw and S. Pi, Self-dual Chern-Simons solitons, Progr. Theoret. Phys. Suppl., 107 (1992), 1-40.  doi: 10.1143/PTPS.107.1. [28] C. Ji and F. Fang, Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth, J. Math. Anal. Appl., 450 (2017), 578-591.  doi: 10.1016/j.jmaa.2017.01.065. [29] Y. Jiang, A. Pomponio and D. Ruiz, Standing waves for a gauged nonlinear Schrödinger equation with a vortex point, Commun. Contemp. Math., 18 (2016), 1550074, 20 pp. doi: 10.1142/S0219199715500741. [30] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005. [31] M. A. Krasnoselski$\mathop {\rm{i}}\limits^ \vee$, Topological Methods in the Theory of Nonlinear Integral Equations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. [32] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $N$-Laplacian type with critical exponential growth in $\mathbb{R}^N$, J. Funct. Anal., 262 (2012), 1132-1165.  doi: 10.1016/j.jfa.2011.10.012. [33] Y. Li and B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^{N}$, Indiana Univ. Math. J., 57 (2008), 451-480.  doi: 10.1512/iumj.2008.57.3137. [34] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoam, 1 (1985), 145-201.  doi: 10.4171/RMI/6. [35] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/1971), 1077-1092.  doi: 10.1512/iumj.1971.20.20101. [36] S. I. Pohozaev, The Sobolev embedding in the case $pl = n$, In: Proc. Tech. Sci. Conf. on Adv. Sci., Research, 1964–1965; Math. Section, Moscow, (1965), 158–170. [37] A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535. [38] A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2. [39] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, In: CBMS Regional Conference Series in Mathematics, vol. 65, AMS, Providence RI, 1986. doi: 10.1090/cbms/065. [40] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^2$, J. Funct. Anal., 219 (2005), 340-367.  doi: 10.1016/j.jfa.2004.06.013. [41] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [42] J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math., 9 (2007), 571-583.  doi: 10.1142/S021919970700254X. [43] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-484.  doi: 10.1512/iumj.1968.17.17028. [44] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [45] Y. Yang and X. Zhu, Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two, J. Funct. Anal., 272 (2017), 3347-3374.  doi: 10.1016/j.jfa.2016.12.028.
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