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 \(B_R \subset \mathbb {R}^N\) $B_R\subset {\mathbb{R}}^N$ for \(N \ge 3\) $N\ge 3$ : \(\begin{aligned} \left\{ \begin{array}{ll} - \triangle u_p + u_p = |u_p|^{p-2}u_p &{}\quad \text { in } B_R, \\ \partial _\nu u_p = 0 &{}\quad \text { on } \partial B_R, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-▵u_p+u_p={|u_p|}^{p-2}{u}_p\hfill & \text{in}{B}_R,\hfill \\ {\partial }_{\nu }{u}_p=0\hfill & \text{on}\partial B_R,\hfill \end{array}\right)\end{array}$$ when p is close to the first critical Sobolev exponent \(2^* = \frac{2N}{N-2}\) $2^{\ast }=\frac{2N}{N-2}$ . We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as \(p \rightarrow 2^*\) $p\to 2^{\ast }$ , 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 \(p \ge 2^*\) $p\ge 2^{\ast }$ , finite-energy radial solutions are precompact in \(C^2(\overline{B_R})\) $C^2(\overline{B_R})$ provided that \(N\ge 7\) $N\ge 7$ . Sufficient conditions are also given in smaller dimensions if \(p=2^*\) $p=2^{\ast }$ . 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).