Liouville theorems for an integral equation of Choquard type

Abstract

We establish sharp Liouville theorems for the integral equation u(x) = R n u p−1 (y) |x − y| n−α R n u p (z) |y − z| n−β dzdy, x ∈ R n , where 0 < α, β < n and p > 1. Our results hold true for positive solutions under appropriate assumptions on p and integrability of the solutions. As a consequence, we derive a Liouville theorem for positive H α 2 (R n) solutions of the higher fractional order Choquard type equation (−∆) α 2 u = 1 |x| n−β * u p u p−1 in R n .

Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order

“…By Proposition 1, we have U λ0 = 0 in R n \ {0}. Then using (26) and (27), we derive V λ0 ≥ 0 in R n \ {0}. Now formula (17) yields…”

“…Since U λ is continuous with respect to λ, we already have U λ0 ≥ 0 in B λ0 \ {0}. From (26) and (27), we have…”

“…Using the integral equation method, equation (2) with α = 2 and c 2 = 0 was recently investigated in [27]. Some classification results for nonnegative H α (R n ) solutions of (2) with 0 < α < n and c 2 = 0 were also established in [12,22,25,26,28,33,42] using the integral equation method.…”

Nonlinear Inequalities with Double Riesz Potentials

“…The nonexistence of positive solutions to double-nonlocal inequality Eq. 1 with p > 1, q > 0 and p + q ≤ N+β N−α was established by different methods in [10,Theorem 1].…”

Nonlinear Inequalities with Double Riesz Potentials

“…In the case α = 2 our results are fully consistent with the results established in [11,Theorem 1] for the 2nd order elliptic inequality (4). The nonexistence of positive solutions to double-nonlocal inequality (1) with p > 1, q > 0 and p + q ≤ N +β N −α was established by different methods in [8,Theorem 1].…”