Denis Bonheure

Université libre de Bruxelles

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

We investigate the behaviour of radial positive solutions to the Lin-Ni-Takagi problem in the ball \(B_R \subset \mathbb R^N\):

\[\begin{split}\left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R \\ \partial_\nu u_p & = 0 & \textrm{ in } \partial B_R, \end{aligned} \right.\end{split}\]

when \(p \to 2^* = \frac{2N}{N-2}\). We obtain a complete classification of finite energy blowing-up solutions to this problem. In particular, we show that if \(p \geq 2^\ast\), then solutions are compact provided that \(N\geq 7\). We also give an interpretation of our results in view of the bifurcation analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).