Fractional elliptic systems with nonlinearities of arbitrary growth
Date
2017-09-07
Authors
Ferreira Leite, Edir Junior
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
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 Ω in ℝn:
Asu = vp in Ω
Asv = ƒ(u) in Ω
u = v = 0 on ∂Ω
where s ∈ (0, 1) and As denote spectral fractional Laplace operators. We assume that 1 < p < 2s/n-2s, and the function ƒ is superlinear and with no growth restriction (for example ƒ(r) = re r; thus the system has a nontrivial solution. Another important example is given by ƒ(r) = rq. 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
1/p+1 + 1/q+1 = n-2s/n
whenever n > 2s. For such weak solutions, we prove an L∞ estimate of Brezis-Kato type and derive the regularity property of the weak solutions.
Description
Keywords
Fractional elliptic systems, Critical growth, Critical hyperbola
Citation
Ferreira Leite, E. J. (2017). Fractional elliptic systems with nonlinearities of arbitrary growth. Electronic Journal of Differential Equations, 2017(206), pp. 1-20.
Rights
Attribution 4.0 International