DOI: 10.1515/dema-2025-0243 ISSN: 2391-4661
Some results of perturbed double critical fractional Schrödinger–Poisson systems in
R
3
${\mathbb{R}}^{3}$
Xian He, Sihua Liang Abstract
In this paper, we study the existence and multiplicity solutions for the following perturbed double critical fractional Schrödinger–Poisson Systems in
R
3
${\mathbb{R}}^{3}$
:
ϵ
2
s
−
Δ
s
u
+
V
x
u
−
ϕ
u
2
s
*
−
3
u
=
u
2
s
*
−
2
u
+
g
x
,
u
in
R
3
,
−
Δ
s
ϕ
=
u
2
s
*
−
1
in
R
3
,
$$\begin{cases}{{\epsilon}}^{2s}{\left(-{\Delta}\right)}^{s}u+\mathcal{V}\left(x\right)u-\phi {\left\vert u\right\vert }^{{2}_{s}^{{\ast}}-3}u={\left\vert u\right\vert }^{{2}_{s}^{{\ast}}-2}u+g\left(x,u\right)\hfill & \,\text{in}\,{\mathbb{R}}^{3},\hfill \\ {\left(-{\Delta}\right)}^{s}\phi ={\left\vert u\right\vert }^{{2}_{s}^{{\ast}}-1}\hfill & \,\text{in}\,{\mathbb{R}}^{3},\hfill \end{cases}$$
where
s
∈
1
2
,
3
4
$s\in \left(\frac{1}{2},\frac{3}{4}\right)$
and
2
s
*
=
6
3
−
2
s
${2}_{s}^{{\ast}}=\frac{6}{3-2s}$
is the Sobolev critical exponent, the potential
V
$\mathcal{V}$
and the nonlinearity
g
satisfy suitable assumptions. We first appropriately decompose the corresponding energy functional, and combining with refined estimates, prove the Palais–Smale condition at
c
level (see Lemma 3.6). Then the existence and multiple of solutions for this problem are obtained by the variational method. Moreover, we also prove the asymptotic properties of these solutions when the parameter
ϵ
is sufficiently small. The main features and novelty of this problem lies in the simultaneous appearance of double critical exponents and nonlocal terms, which requires us to control the relationship between the two critical exponents. To some extent, our main conclusions extend and complement some previous results.