Existence of solutions for a fractional Choquard-type equation in \(\mathbb {R}\) with critical exponential growth

Abstract

In this paper, we study the following class of fractional Choquard-type equations

$$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$

where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(I_{\mu }\) is the Riesz potential with \(0<\mu <1\), and F is the primitive function of f. We use variational methods and minimax estimates to study the existence of solutions when f has critical exponential growth in the sense of Trudinger–Moser inequality.

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Acknowledgements

The authors would like to express their sincere gratitude to the referees for carefully reading the manuscript and valuable comments and suggestions.

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Correspondence to Rodrigo Clemente.

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Clemente, R., de Albuquerque, J.C. & Barboza, E. Existence of solutions for a fractional Choquard-type equation in \(\mathbb {R}\) with critical exponential growth. Z. Angew. Math. Phys. 72, 16 (2021). https://doi.org/10.1007/s00033-020-01447-w

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Mathematics Subject Classification

  • 35J20
  • 35J60
  • 35R11

Keywords

  • Fractional Choquard-type equation
  • Critical exponential growth
  • Trudinger–Moser inequality