Semiclassical States of Fractional Choquard Equations with Exponential Critical Growth

This paper is concerned with the following one-dimensional fractional Choquard equation epsilon(-Delta)(1/2)u + V(x)u = epsilon(mu)(I-mu * F(u)) f(u), x is an element of R, where (-Delta)(1/2) denotes the 1/2-Laplacian opertor, I-mu is the Riesz potential with 0 < mu < 1, V is a continuous real function satisfying some mild assumptions and f is an element of C(R, R) is a nonlinearity with exponential critical growth. The present paper has three typical features. Firstly, using a weaker assumption on f, we establish the energy inequality to recover the compactness. Second, without the strictly monotone condition and by establishing some new tricks, we obtain the existence of ground state solutions when epsilon is small enough. Finally, we take advantage of some refined analysis techniques to get over the difficulty carried by the nonlocality of the 1/2-Laplacian and prove the concentration of the ground state solutions when epsilon -> 0. Our results extend and complement the results of Alves et al. (J Differ Equ 261:1933-1972, 2016) and Clemente et al. (Z Angew Math Phys 72:16, 2021].