Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
On the Existence of Ground State Solutions
for Fractional Schr¨odinger–Poisson Systems
with General Potentials and Super-quadratic
Zu Gao, Xianhua Tang and Sitong Chen
Abstract. In this article, we are concerned with the following fractional
u + V (x)u + φu = f(u)inR
φ = u
where 0 <s≤ t<1, 2s +2t>3, and f ∈ C(R, R).Undermorerelaxed
assumptions on potential V (x)andf(x), we obtain the existence of
ground state solutions for the above problem by adopting some new
tricks. Our results here extend the existing study.
Mathematics Subject Classiﬁcation. 35R11, 58E30, 47F05.
Keywords. Fractional Schr¨odinger–Poisson systems, ground state solu-
tions, Pohozaev identity, variational methods.
In the present paper, we deal with the existence of ground state solutions for
the following fractional Schr¨odinger–Poisson problem:
u + V (x)u + φu = f (u)inR
φ = u
where 0 <s≤ t<1, 2s +2t>3, and (−Δ)
is the fractional Laplacian of
order s. Here, the fractional Laplacian (−Δ)
is deﬁned, up to normalization
factors, by the following singular integral:
u(x) − u(y)
|x − y|
where P.V. is a commonly used abbreviation for “in the principle value sense”
is a dimensional constant that depends on s. Via the Fourier transform