## Abstract

We are concerned with the following Kirchhoff equation:

where \(a,\, b\) are positive constants, \(V\in \mathcal {C}(\mathbb {R}^2, (0,\infty ))\) is a trapping potential, and *f* has critical exponential growth of Trudinger–Moser type. By developing some new analytical approaches and techniques, we prove the existence of nontrivial solutions and least energy solutions. Without any monotonicity conditions on *f*, we also give the mountain pass characterization of the least energy solution by constructing a fine path. In particular, we remove the common restriction on \(\liminf _{t\rightarrow +\infty }\frac{tf(t)}{e^{\alpha _0 t^2}}\), which is crucial in the literature to overcome the loss of the compactness caused by the critical exponential nonlinearity. Our approach could be extended to other classes of critical exponential growth problems with trapping potentials.

This is a preview of subscription content, access via your institution.

## References

Alves, C.O., Cassani, D., Tarsi, C., Yang, M.B.: Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb{R} ^2\). J. Differential Equations

**261**, 1933–1972 (2016)Adachi, S., Tanaka, K.: Trudinger type inequalities in \(\mathbf{R}^N\) and their best exponents. Proc. Amer. Math. Soc.

**128**, 2051–2057 (2000)Adimurthi, S.L. Yadava.: Multiplicity results for semilinear elliptic equations in a bounded domain of \(\mathbf{R}^2\) involving critical exponents. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)

**17**, 481–504 (1990)Cao, D.M.: Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbf{R}^2\). Comm. Partial Differential Equations

**17**, 407–435 (1992)Cassani, D., Sani, F., Tarsi, C.: Equivalent Moser type inequalities in \(\mathbb{R} ^2\) and the zero mass case. J. Funct. Anal.

**267**, 4236–4263 (2014)Chen, S.T., Tang, X.H.: On the planar Schrödinger equation with indefinite linear part and critical growth nonlinearity. Calc. Var. Partial Differential Equations

**60**(3), Paper No. 95, 27 (2021)Chen, S.T., Tang, X.H., Wei, J.: Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth. Z. Angew. Math. Phys.

**72**, Paper No. 38, 18 (2021)Chen, S.T., Tang, X.H.: Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth. J. Differential Equations

**269**, 9144–9174 (2020)Chen, W., Yu, F.: On a nonhomogeneous Kirchhoff-type elliptic problem with critical exponential in dimension two. Appl. Anal. https://doi.org/10.1080/00036811.2020, 1745778

de Figueiredo, D.G., Miyagaki, O.H., Ruf, B.: Elliptic equations in \( {R}^2\) with nonlinearities in the critical growth range. Calc. Var. Partial Differential Equations

**3**, 139–153 (1995)de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math.

**55**, 135–152 (2002)do Ó, J.M., Mishra, P.K., Zhang, J.: Solutions concentrating around the saddle points of the potential for two-dimensional Schrödinger equations. Z. Angew. Math. Phys.

**70**, Paper No. 64, 26 (2019)Figueiredo, G.M., Severo, U.B.: Ground state solution for a Kirchhoff problem with exponential critical growth. Milan J. Math.

**84**, 23–39 (2016)Fiscella, A., Pucci, P.: \((p,N)\) equations with critical exponential nonlinearities in \({\mathbb{R}}^N\). J. Math. Anal. Appl.

**501**(1), Paper No. 123379, 25 pp (2021)Ibrahim, S., Masmoudi, N., Nakanishi, K.: Trudinger-Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. (JEMS)

**17**, 819–835 (2015)Jeanjean, L., Tanaka, K.: A remark on least energy solutions in in \(\mathbb{R} ^N\). Proc. Amer. Math. Soc.

**131**, 2399–2408 (2003)Lions, J.L.: On some questions in boundary value problems of mathematical physics. North-Holland Math. Stud.

**30**, 284–346 (1978)Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)

Masmoudi, N., Sani, F.: Trudinger-Moser inequalities with the exact growth condition in \(\mathbb{R} ^N\) and applications. Comm. Partial Differential Equations

**40**, 1408–1440 (2015)Miyagaki, O.H., Pucci, P.: Nonlocal Kirchhoff problems with Trudinger-Moser critical nonlinearities. NoDEA Nonlinear Differential Equations Appl.

**26**, 27, 26 pp (2019)Mingqi, X., Rădulescu, V.D., Zhang, B.L.: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity. Calc. Var. Partial Differential Equations

**58**, 57, 27 pp (2019)Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J.

**20**(1970/71), 1077–1092Naimen, D., Tarsi, C.: Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth. Adv. Differential Equations

**22**, 983–1012 (2017)Qin, D.D., Tang, X.H.: On the planar Choquard equation with indefinite potential and critical exponential growth. J. Differential Equations

**285**, 40–98 (2021)Pohozaev, S.I.: A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.)

**96**, 152–168 (1975)Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.

**43**, 270–291 (1992)Tang, X.H., Chen, S.T.: Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. Adv. Nonlinear Anal.

**9**, 413–437 (2020)Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech.

**17**, 473–483 (1967)Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston Inc., Boston, MA (1996)

Zhang, J., do Ó, J.M.: Standing waves for nonlinear Schrödinger equations involving critical growth of Trudinger-Moser type. Z. Angew. Math. Phys.

**66**, 3049–3060 (2015)

## Acknowledgements

This work is partially supported by the National Natural Science Foundation of China (No. 11971485, No. 12001542), Hunan Provincial Natural Science Foundation (No. 2021JJ40703) and Innovation Project of Graduate Students of Central South University (No.1053320213251). The author Lixi Wen acknowledges the financial support of the China Scholarship Council (No. 202006370225) and would like to thank also the Embassy of the People’s Republic of China in Romania. The research of Vicenţiu D. Rădulescu 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## About this article

### Cite this article

Chen, S., Rădulescu, V.D., Tang, X. *et al.* Planar Kirchhoff equations with critical exponential growth and trapping potential.
*Math. Z.* **302**, 1061–1089 (2022). https://doi.org/10.1007/s00209-022-03102-8

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00209-022-03102-8

### Keywords

- Kirchhoff equation
- Critical exponential growth
- Trudinger–Moser inequality
- Trapping potential

### Mathematics Subject Classification

- 35J20
- 35J62
- 35Q55