Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order
Date
2017-04-18
Authors
Jleli, Mohamed
Samet, Bessem
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
We study the nonexistence of nontrivial solutions for the nonlinear elliptic system
Gα, β, θ(up, uq) = vr
Gλ, μ, θ(vs, vt) = um
u, v ≥ 0,
where 0 < α, β, λ, μ ≤ 2, θ ≥ 0, m > q ≥ p ≥ 1, r > t ≥ s ≥ 1, and Gα, β, θ is the fractional operator of mixed orders α, β, defined by
Gα, β, θ(u, v) = (-∆x)α/2u + |x|2 θ (-∆y)β/2v, in ℝN1 x ℝN2.
Here, (-∆x)α/2, 0 < α < 2, is the fractional Laplacian operator of order α/2 with respect to the variable x ∈ ℝN1, and (-∆y)β/2, 0 < β < 2, is the fractional Laplacian operator of order β/2 with respect to the variable y ∈ ℝN2. Via a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.
Description
Keywords
Liouville-type theorem, Nonexistence, Fractional Grushin operator
Citation
Jleli, M., & Samet, B. (2017). Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order. <i>Electronic Journal of Differential Equations, 2017</i>(105), pp. 1-11.
Rights
Attribution 4.0 International