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

par Bonheure, Denis ;Casteras, Jean-Baptiste ;Premoselli, Bruno
Référence Mathematische Annalen
Publication Publié, 2024-06-01

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Résumé :

We investigate the behaviour of radial solutions to the Lin–Ni–Takagi problem in the ball BR⊂RN for N≥3: (Formula presented.) when p is close to the first critical Sobolev exponent 2∗=2NN-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→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≥2∗, finite-energy radial solutions are precompact in C2(BR¯) provided that N≥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).