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A priori bounds and positive solutions for non-variational fractional elliptic systems. (English) Zbl 1424.35350

In this paper, the following system \[\left\{\begin{array}{ll}(-\Delta)^su=v^p & \text{in} \ \Omega,\\[1ex](-\Delta)^tv=u^q, & \text{in} \ \Omega\\ [1ex] u=v=0 & \text{in} \ \mathbb{R}^n\backslash\Omega\end{array}\right. \tag{1}\] is considered, where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\), \(n\ge 2\), \(0<s\), \(t<1\), \(p\), \(q>0\) and \((-\Delta)^s\) and \((-\Delta)^t\) are the fractional Laplace operator. The Liouville type theorems on \(\mathbb{R}^n\) and \(\mathbb{R}_+^m\) are obtained if \[\left(\frac{2s}{p}+2t\right)\frac{p}{pq-1}\ge n-2s\quad\text{or}\quad\left(\frac{2t}{q}+2s\right)\frac{q}{pq-1}\ge n-2t.\tag{2}\] A priori bounds of positive solutions to (1) are established. Then the existence of positive solutions to (1) is derived under the assumption (2).

MSC:

35R11 Fractional partial differential equations
35B44 Blow-up in context of PDEs
35B45 A priori estimates in context of PDEs
35B51 Comparison principles in context of PDEs
35J57 Boundary value problems for second-order elliptic systems
35J61 Semilinear elliptic equations