Lane Emden problems: asymptotic behavior of low energy nodal solutions

Massimo Grossi; Christopher Grumiau; Filomena Pacella

doi:10.1016/j.anihpc.2012.06.005

paper

Lane Emden problems: asymptotic behavior of low energy nodal solutions

arXiv:1206.3971 · doi:10.1016/j.anihpc.2012.06.005

Abstract

We study the nodal solutions of the Lane Emden Dirichlet problem $- Δ u = ∣ u ∣^{p - 1} u w i t h D B C o na s m oo t hb o u n d e dd o main$ Ω $in$ \IR^2 $an d w h er e$ p>1 $. W eco n s i d er so l u t i o n s$ u_p $s a t i s f y in g$ p \int_Ω\abs{\nabla u_p}^2\to 16πe\quad\hbox{as}p\rightarrow+\infty\qquad (*) $an d w e a r e in t er es t e d in t h es ha p e an d t h e a sy m pt o t i c b e ha v i or a s$ p\rightarrow+\infty $. F i r s tw e p r o v e t ha t (*) h o l d s f or l e a s t e n er g y n o d a l so l u t i o n s . T h e n w eo b t ain so m ees t ima t es an d t h e a sy m pt o t i c p r o f i l eo f t hi s k in d o f so l u t i o n s . F ina l l y, in so m ec a ses, w e p r o v e t ha t$ pu_p $c anb ec ha r a c t er i z e d a s t h e d i f f er e n ceo f tw o G r ee n^{'} s f u n c t i o n s an d t h e n o d a l l in e in t er sec t s t h e b o u n d a r y o f$ Ω $, f or l a r g e$ p$.

Annales de l'Institut Henri Poincaré, 2012

Lane Emden problems: asymptotic behavior of low energy nodal solutions

Abstract

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