On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth

Abstract

In this paper, we investigate the existence of solutions to the planar non-autonomous Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V(|x|)u+\gamma \phi K(|x|)u = \lambda Q(|x|)f(u),\ \&x\in {\mathbb {R}}^{2}, \\&\Delta \phi =K(|x|) u^{2}, \ \&x\in {\mathbb {R}}^{2}, \end{aligned} \right. \end{aligned}$$

where \(\gamma ,\lambda \) are positive parameters, VKQ are continuous potentials, which can be unbounded or vanishing at infinity. By assuming that the nonlinearity f(s) has exponential critical growth, we derive the existence of a ground state solution to the system. A key feature of our approach is a new weighted Trudinger–Moser type inequality proved here.

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees for their valuable suggestions and comments which provided insights that helped to improve the paper.

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Correspondence to G. M. Figueiredo.

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J. L. Carvalho was supported by CAPES/BRAZIL.

G. M. Figueiredo was supported by CNPQ, FAPDF, CAPES/BRAZIL.

E. Medeiros was partially suppoted by Grant 2019/2014 Paraíba State Research Foundation (FAPESQ) and Grant 308900/2019-7 CNPq.

Communicated by P. Rabinowitz.

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Albuquerque, F.S., Carvalho, J.L., Figueiredo, G.M. et al. On a planar non-autonomous Schrödinger–Poisson system involving exponential critical growth. Calc. Var. 60, 40 (2021). https://doi.org/10.1007/s00526-020-01902-6

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Mathematics Subject Classification

  • 35J15
  • 35J25
  • 35J60