Fractional Elliptic Systems with Nonlinearities of Arbitrary Growth

Abstract

In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: \[ \left\{{llll} \mathcal{A}^s u= v^p & {\rm in} \ \ \Omega \mathcal{A}^s v = f(u) & {\rm in} \ \ \Omega u= v=0 & {\rm on} \ \ \partial\Omega \right. \] where $s\in (0, 1)$ and $\mathcal{A}^s$ denote spectral fractional Laplace operators. We assume that $1< p<\frac{2s}{n-2s}$, and the function $f$ is superlinear and with no growth restriction (for example $f(r)=re^r$); thus the system has a nontrivial solution. Another important example is given by $f(r)=r^q$. In this case, we prove that such a system admits at least one positive solution for a certain set of the couple $(p,q)$ below the critical hyperbola \[ \frac{1}{p + 1} + \frac{1}{q + 1} = \frac{n - 2s}{n} \] whenever $n > 2s$. For such weak solutions, we prove an $L^\infty$ estimate of Brezis-Kato type and derive the regularity property of the weak solutions.