Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space

In this paper, let α be any real number between 0 and 2, we study the Dirichlet problem for semi-linear elliptic system involving the fractional Laplacian:

$$\left \{\begin {array}{l} (-{\Delta })^{\alpha /2}u(x)=v^{q}(x),\ \ \ x\in \mathbb {R}^{n}_{+},\\ (-{\Delta })^{\alpha /2}v(x)=u^{p}(x),\ \ \ x\in \mathbb {R}^{n}_{+},\\ u(x)=v(x)=0,\ \ \ \ \ \ \ \ \ \ x\notin \mathbb {R}^{n}_{+}. \end {array}\right .\label {elliptic} $$

(1)

We will first establish the equivalence between PDE problem (1) and the corresponding integral equation (IE) system (Lemma 2). Then we use the moving planes method in integral forms to establish our main theorem, a Liouville type theorem for the integral system (Theorem 3). Then we conclude the Liouville type theorem for the above differential system involving the fractional Laplacian (Corollary 4).