Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

Huxiao Luo,Chunji Li,Shengjun Li

doi:10.3390/math7020151

Huxiao Luo, Chunji Li + Show 1 more

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https://doi.org/10.3390/math7020151

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Abstract

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

Highlights

IntroductionWe deal with the following nonlocal equation:. where N ≥ 3, 0 < s < 1, 0 < μ < N, V ∈ C (R N , [0, ∞)), Q ∈ C (R N , (0, ∞)), f ∈ C (R, R) and
In this paper, we deal with the following nonlocal equation: ((−∆)s u + V ( x )u = R RN Q(y) F (u(y)) dy | x −y|μQ( x ) f (u), in R N, u ∈ D s,2 (R N ), (1)where N ≥ 3, 0 < s < 1, 0 < μ < N, V ∈ C (R N, [0, ∞)), Q ∈ C (R N, (0, ∞)), f ∈ C (R, R) and
Motivated by the above works, in the first part of this article, we study the ground state solution for

Summary

Introduction

We deal with the following nonlocal equation:. where N ≥ 3, 0 < s < 1, 0 < μ < N, V ∈ C (R N , [0, ∞)), Q ∈ C (R N , (0, ∞)), f ∈ C (R, R) and. In the second part of this article, we consider the following fractional Choquard equation with zero mass case:. Let {vn } be a sequence such that vn * v in E, there exists a constant M2 > 0 such that. Since |un | = |vn |kun k and un /kun k → v a.e. in R N , we have lim |un ( x )| = ∞ for x ∈ {y ∈ R N : v( x ) 6= 0} It follows from (22), (31), ( F4), and Fatou’s n→∞. In view of Lemmas 8 and 9, there exists a bounded sequence {un } ⊂ E such that (31) holds.

Zero Mass Case

Conclusions