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The Hölder interpolation inequality says that for $u\in C^{l,\alpha }(B_R(x_0))$ and every $\epsilon >0$,More generally, each lower derivative norm can be bounded by an arbitrarily small multiple of the top $C^{l,\alpha }$ seminorm plus a constant multiple of the $C^0$ norm. The powers of $R$ make the inequality invariant under scaling of the ball.

For a radial function $u=u(r)$ on ${\mathbb{R}}^d$, the radial Laplacian isThe problem therefore becomesThe equation has the scaling symmetryChoosing $\lambda =u(0{)}^{-(p-1)/2}$ normalizes any positive solution to $u_{\lambda }(0)=1$.