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Sign-changing solution for logarithmic elliptic equations with critical exponent
Authors
Tianhao Liu
Wenming Zou
Publication date
16 August 2023
Publisher
View
on
arXiv
Abstract
In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases} \end{equation} Here, the parameters
N
≥
6
N\geq 6
N
≥
6
,
λ
∈
R
\lambda\in \R
λ
∈
R
,
θ
>
0
\theta>0
θ
>
0
and
2
∗
=
2
N
N
−
2
2^*=\frac{2N}{N-2}
2
∗
=
N
−
2
2
N
​
is the Sobolev critical exponent. We prove the existence of sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain
Ω
⊂
R
N
\Omega\subset \mathbb{R}^{N}
Ω
⊂
R
N
. When
Ω
=
B
R
(
0
)
\Omega=B_R(0)
Ω
=
B
R
​
(
0
)
is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic
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oai:arXiv.org:2308.08719
Last time updated on 24/08/2023