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 $\mathbb R^n$: \[ \begin{cases} (-\Delta)^{s}u = v^p & \text{in } \Omega,\\ (-\Delta)^{s}v = u^q & \text{in } \Omega,\\ u= v=0 & \text{in } \mathbb R^n\setminus\Omega. \end{cases} \] By means of an appropriate variational framework and a Hölder 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 $0 < s < 1$, provided that $p, q > 0$, $pq \neq 1$ 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 $pq < 1$. Lastly, assuming $\Omega$ is star-shaped, by using a Rellich type variational identity, we prove that no such a solution exists if $(p,q)$ is on or above the critical hyperbole. A crucial point in our proofs is proving, given a critical point $u \in W_{0}^{ s, ({p+1})/{p}}({\Omega}) \cap W^{ 2s, ({p+1})/{p}}(\Omega)$ of a related functional, that there is a function $v$ in an appropriate Sobolev space (Proposition 2.1) so that $(u,v)$ is a weak solution of the above system and a bootstrap argument can be applied successfully in order to establish its H\"{o}lder regularity (Proposition 3.1). The difficulty is caused mainly by the absence of a $L^p$ Calderón-Zygmund theory with $p > 1$ associated to the operator $(-\Delta)^{s}$ for $0 < s < 1$.