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.

Appendix A: Riesz potential of the weighted Jacobi polynomials

Here we give a brief derivation of the expressions in (7.3) about the Riesz potential of the weighted Jacobi polynomials $(1-|x|^2)^{-s}P_n^{(-s,N/2-1)} (2|x|^2-1)$ restricted on the unit ball. This relation can be established essentially by reversing the sign of s as for the fractional Laplacian of $(1-|x|^2)^{s}P_n^{(s,N/2-1)} (2|x|^2-1)$ in [24, Theorem 3], so that the Riesz potential can be represented as the inverse Mellin transform

$$\begin{aligned} \frac{(-1)^n2^{-2s}\Gamma (1+n-s)}{n!}\frac{1}{2\pi i}\int _{{\mathscr {C}}} \frac{\Gamma (\tau )\Gamma (\frac{N}{2}-s+n-\tau )}{\Gamma (\frac{N}{2}-\tau )\Gamma (1+n+\tau )}|x|^{-2\tau }d\tau , \end{aligned}$$

(A.1)

where ${\mathscr {C}}$ is a contour from $\sigma -i\infty$ to $\sigma +i\infty$ with $0< \sigma < N/2-s+n$. If $|x|<1$, the contour integral is reduced to the sum of residues around the poles of $\Gamma (\tau )$, leading to

$$\begin{aligned}&\frac{(-1)^n2^{-2s}\Gamma (1+n-s)}{n!}\sum _{k=0}^n \frac{(-1)^k}{k!}\frac{\Gamma (N/2-s+n+k)}{\Gamma (N/2+k)\Gamma (1+n-k)}|x|^{2k}\\&\quad = \lambda _n(-1)^n \frac{\Gamma (N/2+n)}{n!\Gamma (N/2)} {}_2F_1(-n,N/2+n-s;N/2;|x|^2) =\lambda _nP_n^{(-s,N/2-1)}(2|x|^2-1) \end{aligned}$$

using the equivalent definition $P_n^{(a,b)}(z) = (-1)^n \frac{\Gamma (1+b+n)}{n!\Gamma (1+b)} {}_2F_1(-n,1+a+b+n;1+b; (1+z)/2)$ for Jacobi polynomials. For $|x|>1$, the contour integral (A.1) is evaluated by summing the residues around the poles of $\Gamma (\frac{N}{2}-s+n-\tau )$, leading to

$$\begin{aligned}&\frac{(-1)^n2^{-2s}\Gamma (1+n-s)}{n!} \sum _{k=0}^\infty \frac{(-1)^k}{k!}\frac{\Gamma (N/2+n+k-s)}{ \Gamma (s-n-k)\Gamma (N/2+2n+1-s+k) }|x|^{-N-2n-2k+2s} \\&\quad = \frac{2^{-2s}\Gamma (1+n-s)\sin \pi s}{n!\pi } |x|^{-N-2n+2s} \sum _{k=0}^\infty \frac{\Gamma (N/2+n+k-s) \Gamma (1+n-s+k)}{ \Gamma (N/2+2n+1-s+k)k! }|x|^{-2k} \\&\quad = \lambda _n \mu _n |x|^{-N-2n+2s}{ }_2F_1\left( 1-s+n, \frac{N}{2}+n-s; 1+2n+\frac{N}{2}-s;|x|^{-2}\right) . \end{aligned}$$