Classification of radial blow-up at the first critical exponent for the Lin-Ni-Takagi problem in the ball

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R\subset {\mathbb{R}}^N$ for $N\ge 3$: \begin{equation*} \left { \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & $\text{ in }$ B_R, \ \partial_νu_p & = 0 & $\text{ on }$ \partial B_R, \end{aligned} \right. \end{equation*} when $p$ is close to the first critical Sobolev exponent $2^{\ast }=\frac{2N}{N-2}$. We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as $p\to 2^{\ast }$, we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of $p$. We show in particular that, if $p\ge 2^{\ast }$, finite-energy radial solutions are precompact in $C^2(\overline{B_R})$ provided that $N\ge 7$. Sufficient conditions are also given in smaller dimensions if $p=2^{\ast }$. Finally we compare and interpret our results to the bifurcation analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).