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

One‐dimensional periodic fractional Schrödinger equations with exponential critical growth

“…Compared to the conditions in Giacomoni et al, 23 do Ó et al, 24 and Clemente et al, 25 it is obvious that $$$ \left({f}_4\right) $$$ is weaker, and we give the lower bound $$$ \frac{\pi }{8{\alpha}_0^2}{e}^{\hat{C}+4{V}_{\infty }} $$$ which only depends on $$$ {\alpha}_0 $$$ and the exact value $$$ {V}_{\infty } $$$ and improve the existing assumptions for the works by Giacomoni et al 23 and do Ó et al, 24 which did not give the accurate lower bound. Condition $$$ \left({f}_4\right) $$$ is inspired by 26,27 but the estimating procedure in $$$ \mathbb{R} $$$ is different with the work by Chen et al 26,27 These changes need us to developed the estimating procedure in $$$ \mathbb{R} $$$ to overcome the new difficulties.…”

“…where $$$ \eta $$$ is a sufficiently large positive real number. We also refer to the recent work 25 which used a weaker assumption that $$$ \lim\ {\operatorname{inf}}_{t\to \infty }F(t){e}^{-\pi {t}^2}=\sqrt{\beta_0} $$$, where $$$ {\beta}_0 $$$ just needs to be a positive number. Compared with $$$ \left({f}_{0,4}\right) $$$, these kind of assumptions involved the exponential growth which is independent of any parameters like $$$ {C}_p $$$ in $$$ \left({f}_{0,4}\right) $$$, which in some sense gives a better illustration of the exponential critical growth for these assumptions just depend on the exponential growth velocity $$$ {\alpha}_0 $$$ of ().…”

“…Inspired by the above works, the present work aims to expand and complete those results already found in the existing literature. In particular, in the framework of one‐dimensional fractional Schrödinger equation (), we will generalize the estimating processes 26,27 in $$$ \mathbb{R} $$$, which can be applicable to a wider class of nonlinear terms compared to the work by de Souza and Araújo, 16 Giacomoni et al, 23 do Ó et al, 24 and Clemente et al 25 …”

“…Do Ó et al 24 studied the nonautonomous fractional Hamiltonian system with critical exponential growth, using the assumption that the form of which is similar to (𝑓 ′ 4 ); they 24, Remark 1.2 also pointed out that the condition (𝑓 ′ 4 ) can replaced by a weaker assumption: (𝑓 1 4 ) lim t→∞ t𝑓 (t)e −𝛼 0 t 2 ≥ 𝜂, where 𝜂 is a sufficiently large positive real number. We also refer to the recent work 25 which used a weaker assumption that lim inf t→∞ F(t)e −𝜋t 2 = √ 𝛽 0 , where 𝛽 0 just needs to be a positive number. Compared with (𝑓 0,4 ), these kind of assumptions involved the exponential growth which is independent of any parameters like C p in (𝑓 0,4 ), which in some sense gives a better illustration of the exponential critical growth for these assumptions just depend on the exponential growth velocity 𝛼 0 of (1.6).…”

On some qualitative aspects for doubly nonlocal equations

“…Concerning the fractional Schrödinger equation with Hartree nonlinearity, we mention the papers [23,24] where D'Avenia, Siciliano and Squassina considered the case of pure power nonlinearities and obtained existence and qualitative properties of the solutions (see also [9] for some orbital stability results, [10] for a Strichartz estimates approach, [19] for the unidimensional case and [35] for scattering results). Recently, the existence of a ground state solution to (1) with prescribed mass has been proved in [13] by a Lagrangian approach [12,39,40] and in [15] without L 2 constraint (see also [8] for a non-autonomous case).…”