Nonexistence of anti-symmetric solutions for an elliptic system involving fractional Laplacian

ABSTRACT

In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian $\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u(x)=u^{m_1}(x)v^{n_1}(x),& u(x)\ge 0,x\in {\mathbb{R}}_{+}^n,\\ (-\mathrm{\Delta }{)}^{s}v(x)=u^{m_2}(x)v^{n_2}(x),& v(x)\ge 0,x\in {\mathbb{R}}_{+}^n,\\ u(x^{\mathrm{\prime }},-x_n)=-u(x^{\mathrm{\prime }},x_n),& x=(x^{\mathrm{\prime }},x_n)\in {\mathbb{R}}^n,\\ v(x^{\mathrm{\prime }},-x_n)=-v(x^{\mathrm{\prime }},x_n),& x=(x^{\mathrm{\prime }},x_n)\in {\mathbb{R}}^n,\end{array}$ where 0<s<1, $m_i,n_i>0{}_{(i=1,2)},n>2s,{\mathbb{R}}_{+}^n=\{(x^{\mathrm{\prime }},x_n)|x_n>0\}$. We first show that the solutions only depend on $x_n$ variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to $x_n$ variable (for the critical and subcritical cases $m_i+n_i\le \frac{n+2s}{n-2s}{}_{(i=1,2)}$ in the ${\mathcal{L}}_{2s}$ space). Furthermore, when $m_1=n_2=p,n_1=m_2=q$, in the cases $p+q+2s\ge 1$, we obtain a Liouville theorem for the cases $p+q\le \frac{n+2s}{n-2s}$ in the ${\mathcal{L}}_{2s}$ space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain Ω. Using the anti-symmetric property of the solutions, one can extend the space from ${\mathcal{L}}_{2s}$ to ${\mathcal{L}}_{2s+1}$, we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions.

KEYWORDS:

• Fractional Laplacian
• moving planes
• Liouville theorem
• existence of nontrivial solutions
• singularity estimates

AMS SUBJECT CLASSIFICATIONS:

• 35A01
• 35B09
• 35B53
• 35R11
• 35S05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by National Natural Science Foundation of China [grant number 11871278] and the National Natural Science Foundation of China [grant number 11571093].