Abstract
AbstractIn this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3}|(-\Delta )^{\frac{\alpha }{2}} u|^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{ in }\,\,{\mathbb {R}}^3, \\ u\in H^\alpha ({\mathbb {R}}^3),\quad u>0 &{} \quad \text{ in }\,\, {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$
ε
2
α
a
+
ε
4
α
-
3
b
∫
R
3
|
(
-
Δ
)
α
2
u
|
2
d
x
(
-
Δ
)
α
u
+
V
(
x
)
u
=
H
(
u
-
β
)
f
(
u
)
in
R
3
,
u
∈
H
α
(
R
3
)
,
u
>
0
in
R
3
,
where $$\varepsilon ,\beta >0$$
ε
,
β
>
0
are small parameters, $$\alpha \in (\frac{3}{4},1)$$
α
∈
(
3
4
,
1
)
and a, b are positive constants, $$(-\Delta )^{\alpha }$$
(
-
Δ
)
α
is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.
Funder
Scientific Research fund of Hunan provincial Education Department
National Natural Science Foundation of China
Natural Science Foundation of Hunan Province
Fundamental Research Funds for the Central Universities
Ministry of Research, Innovation and Digitization
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Physics and Astronomy,General Mathematics
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