Precompactness conjecture for critical radial solutions of the Lin–Ni–Takagi problem

Let $B_R\subset\mathbb{R}^N$ be the ball of radius $R$, let $2^*=\frac{2N}{N-2}$, and let $(u_k)_{k\geq 0}$ be radial solutions of the critical Lin–Ni–Takagi problem with exponent $p_k=2^*$ for every $k\geq 0$. Suppose that the sequence is uniformly bounded in the energy norm. Critical precompactness conjecture. Any such sequence $(u_k)_{k\geq 0}$ is precompact in $C^2(\overline{B_R})$. The theorem preceding this conjecture proves precompactness except possibly at explicitly determined exceptional radii in dimensions $N=3$ and $N=6$; the conjecture asserts that precompactness also holds at those remaining radii, completing the exactly critical case.