Uniqueness of entire ground states for the fractional plasma problem

Abstract

We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime that is dominated by aggregation. Mathematics Subject Classification 35K55 • 35R11 • 49K20Communicated by Manuel del Pino.

Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation

“…Fortunately, the integral equation ω = (G s ω − W x 2 ) + in the half plane Π can be reduced to an integral equation in R 4 by a suitable transform. And hence, we can apply the method of moving planes and Theorem 1.1 in [18] to prove the uniqueness of maximizers, which completes the proof of Theorem 1.2. Lemma 3.4.…”

“…Since φ is continuous and the support of (φ(y) − W ) + is compact, one can apply the method of moving planes to deduce that φ is radially symmetric with respect to some point y 0 = (0, y 0 4 ) and hence unique up to translations in y 4 by Theorem 1.1 of [18]. Therefore, there exists a unique function ω s µ ∈ Σ s µ , whose support is a half disk centered at the origin.…”

On the global classical solutions for the generalized SQG equation

“…Ao et al [1] considered the case when f (τ ) = τ p + with 1 < p < 1+s 1−s . Their construction seems to rely in essential way on the p-power form of f , for example, it relies on the existence, uniqueness and asymptotic behavior of the ground state solution to the fractional plasma equation established by Chan et al [7]. We construct the solutions by the variational method, which does not rely on the properties of the limit equation (see section 2 below).…”

Limit profiles for singularly perturbed Choquard equations with local repulsion

“…and the minimum occurs at the optimal λ * = π 2 2 1/5 . Taking into account the uniqueness of the spherically symmetric nonincreasing minimizer for s TF in the case α = 2, which follows from [13, Theorem 1.2] (see also [14,Lemma 5.2]), the function…”