Abstract
In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<\alpha<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.
Highlights
U : R2 → R2 is a vector valued function
The boundedness and the integrability of solutions are the important conditions which ensure that the Liouville theorem holds
The Pohozaev identity plays a key role in the proof
Summary
They proved the finite energy solution (i.e., u satisfies ∇u ∈ L2(R2)) is bounded (see [4] and [6]). The boundedness and the integrability of solutions are the important conditions which ensure that the Liouville theorem holds. We apply the integral form of the Pohozaev identity (which was used for the Lane–Emden equations in [3], [5] and [12]) to establish a Liouville theorem. Is convergent at each x ∈ Rn. Since u is bounded, by the Hölder inequality, we obtain that for sufficiently large R, there holds. When z is near x, we first take n s ∈ α−1,∞ Combining this with (8), we prove that (7) is absolutely convergent. Integrating by parts yields z · ∇[u(z)(1 − |u(z)|2)]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
