On positive viscosity solutions of fractional Lane-Emden systems

Abstract

In this paper we discuss the existence, nonexistence and uniqueness of positive viscosity solution for the following coupled system involving fractional Laplace operator on a smooth bounded domain $\Omega$ in $\mathbb R^n$: \[ \begin{cases} (-\Delta)^{s}u = v^p & \text{\rm in } \Omega, (-\Delta)^{s}v = u^q & \text{\rm in } \Omega, u= v=0 & \text{\rm in } \mathbb R^n\setminus\Omega. \end{cases} \] By means of an appropriate variational framework and a Holder regularity result for suitable weak solutions of the above system, we prove that such a system admits at least one positive viscosity solution for any $0 0$, $pq \neq 1$ and the couple $(p,q)$ is below the critical hyperbole \[ \frac{1}{p + 1} + \frac{1}{q + 1} = \frac{n - 2s}{n} \] whenever $n > 2s$. Moreover, by using the maximum principles for the fractional Laplace operator, we show that uniqueness occurs whenever $pq 1$ associated to the operator $(-\Delta)^{s}$ for $0 < s < 1$.