Electron. J. Differential Equations, Vol. 2017 (2017), No. 206, pp. 1-20.

Fractional elliptic systems with nonlinearities of arbitrary growth

Edir Junior Ferreira Leite

In this article 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 $\mathbb{R}^n$:
 \mathcal{A}^s u= v^p \quad\text{in }\Omega\cr
 \mathcal{A}^s v = f(u) \quad\text{in }\Omega\cr
 u= v=0 \quad\text{on }\partial\Omega
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

Submitted June 10, 2017. Published September 7, 2017.
Math Subject Classifications: 35J65, 49K20, 35J40.
Key Words: Fractional elliptic systems; critical growth; critical hyperbola.

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Edir Junior Ferreira Leite
Departamento de Matemática
Universidade Federal de Viçosa
CCE, 36570-000, Viçosa, MG, Brazil
email: edirjrleite@ufv.br

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