Liouville-type Theorem for Fractional Kirchhoff Equations with Weights
- PDF / 308,933 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 220 Views
Liouville-type Theorem for Fractional Kirchhoff Equations with Weights Anh Tuan Duong1
· Duc Hiep Pham2
Received: 14 April 2020 / Revised: 8 August 2020 / Accepted: 31 August 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we prove a Liouville type theorem for stable solutions to fractional Kirchhoff equations with polynomial nonlinearities and weights. Keywords Liouville-type theorems · Stable solutions · Fractional Kirchhoff equations Mathematics Subject Classification 35B53 · 35J60 · 35B35
1 Introduction In this paper, we are interested in the classification of stable solutions of the fractional Kirchhoff problems on the whole space R N a + b(1 − s)
(u(x) − u(y))2 dxdy (−)s u(x) = h(x)|u(x)| p−1 u(x), N +2s R2N |x − y| (1.1) where 0 < s < 1, a > 0, b > 0 and h is a positive continuous function satisfying h(x) ≥ C|x|d for |x| large, d ≥ 0.
(1.2)
Communicated by Amin Esfahani.
B
Anh Tuan Duong [email protected] Duc Hiep Pham [email protected]
1
Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Hanoi, Vietnam
2
VNU-University of Education, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
123
Bulletin of the Iranian Mathematical Society
A typical example of weight function is of type d
h(x) = C(1 + |x|2 ) 2 . The fractional Laplacian (−)s is defined on the Schwartz space of rapidly decaying functions by (−)s u(x) = c N ,s lim
ε→0 R N \B(x,ε)
u(x) − u(ξ ) dξ, |x − ξ | N +2s
where B(x, ε) is the ball of radius ε centered at x ∈ R N and c N ,s =
RN
1 − cos(x1 ) dx |x| N +2s
−1
.
This operator is extended in the distributional sense to a wider space Ls (R ) = u ∈ L 1loc (R N ); N
RN
|u(x)| dx < ∞ . (1 + |x|) N +2s
We refer the readers to the monograph [29] or the paper [12] for elementary properties of the fractional Laplacian. In the limit s ↑ 1, it is known that Eq. (1.1) becomes the equation of Kirchhoff type − a+b
|∇u| dx u(x) = h(x)|u(x)| p−1 u(x). 2
RN
(1.3)
This type of equation was proposed by Kirchhoff in 1883 as a stationary analogue of the Kirchhoff equation on a bounded domain ⊂ R N u tt − a + b |∇u|2 dx u(x) = h(x)|u(x)| p−1 u(x).
The pioneering work in studying Eq. (1.3) is due to Lions [27], in which, a functional analysis approach has been introduced. For recent results concerning (1.3), we refer the readers to the papers [1,6–8,15,20,21,24–26,30,34]. Very recently, the Kirchhoff equations involving fractional Laplacian have been much studied [2–5,9,18,19,22,23,28,31–33,35]. Most of recent results have been concerned with the existence and multiplicity of solutions under various nonlinearities. For instance, in the paper [9], the existence of nontrivial to the non-degenerate fractional Kirchhoff equation with homogeneous Dirichlet boundary condition has been proved via Morse theory. In [5], by exploiting the variational technique, the authors have studied the existence and asymptotic behavior of nonnegative solutions to a clas
Data Loading...
