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 \(N \ge 3\):

$$\begin{aligned} \left\{ \begin{array}{ll} - \triangle u_p + u_p = |u_p|^{p-2}u_p &{}\quad \text { in } B_R, \\ \partial _\nu u_p = 0 &{}\quad \text { on } \partial B_R, \end{array} \right. \end{aligned}$$

when p is close to the first critical Sobolev exponent \(2^* = \frac{2N}{N-2}\). We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as \(p \rightarrow 2^*\), 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 \(p \ge 2^*\), 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 \(p=2^*\). Finally we compare and interpret our results in light of the bifurcation analysis of Bonheure, Grumiau and Troestler in (Nonlinear Anal 147:236–273, 2016).