Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order

Abstract

<p>We study the nonexistence of nontrivial solutions for the nonlinear elliptic system</p> <pre>G_{α, β, θ}(u^p, u^q) = v^r<br> G_{λ, μ, θ}(v^s, v^t) = u^m<br> u, v ≥ 0,</pre> <p>where 0 < α, β, λ, μ ≤ 2, θ ≥ 0, m > q ≥ p ≥ 1, r > t ≥ s ≥ 1, and G_{α, β, θ} is the fractional operator of mixed orders α, β, defined by</p> <pre>G_{α, β, θ}(u, v) = (-∆_x)^α/2u + |x|^2θ (-∆_y)^β/2v, in ℝ^N₁ x ℝ^N₂.</pre> <p>Here, (-∆_x)^α/2, 0 < α < 2, is the fractional Laplacian operator of order α/2 with respect to the variable x ∈ ℝ^N₁, and (-∆_y)^β/2, 0 < β < 2, is the fractional Laplacian operator of order β/2 with respect to the variable y ∈ ℝ^N₂. Via a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters.</p>