AbstractIn this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$
a
+
b
[
u
]
s
,
p
p
(
-
Δ
)
p
s
u
=
λ
|
u
|
q
-
2
u
ln
|
u
|
2
+
|
u
|
p
s
∗
-
2
u
in
Ω
,
u
=
0
in
R
N
\
Ω
,
where $$N >sp$$
N
>
s
p
with $$s \in (0, 1)$$
s
∈
(
0
,
1
)
, $$p>1$$
p
>
1
, and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$
[
u
]
s
,
p
p
=
∬
R
2
N
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
N
+
p
s
d
x
d
y
,
$$p_s^*=Np/(N-ps)$$
p
s
∗
=
N
p
/
(
N
-
p
s
)
is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$
Ω
⊂
R
N
$$(N\ge 3)$$
(
N
≥
3
)
is a bounded domain with Lipschitz boundary and $$\lambda $$
λ
is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$
u
b
. Moreover, for any $$\lambda > 0$$
λ
>
0
, we show that the energy of $$u_b$$
u
b
is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$
b
→
0
.
Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems
