We consider the semilinear Lane–Emden problem (Ep) -Delta u = |u|^(p−1) u in Omega u =0 on ∂Omega where p > 1 and Omega is a smooth bounded symmetric domain of R^2. We show that for families (u_p) of sign-changing symmetric solutions of (Ep) an upper bound on their Morse index implies concentration of the positive and negative part, u^±_p , at the same point, as p →+∞. Then, an asymptotic analysis of u^+_p and u^−_p shows that the asymptotic profile of (u_p), as p →+∞, is that of a tower of two different bubbles.
Morse index and sign-changing bubble towers for Lane–Emden problems / DE MARCHIS, Francesca; Ianni, Isabella; Pacella, Filomena. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - STAMPA. - 195:...(2016), pp. 357-369. [10.1007/s10231-014-0467-6]
Morse index and sign-changing bubble towers for Lane–Emden problems
DE MARCHIS, FRANCESCA;Isabella Ianni;PACELLA, Filomena
2016
Abstract
We consider the semilinear Lane–Emden problem (Ep) -Delta u = |u|^(p−1) u in Omega u =0 on ∂Omega where p > 1 and Omega is a smooth bounded symmetric domain of R^2. We show that for families (u_p) of sign-changing symmetric solutions of (Ep) an upper bound on their Morse index implies concentration of the positive and negative part, u^±_p , at the same point, as p →+∞. Then, an asymptotic analysis of u^+_p and u^−_p shows that the asymptotic profile of (u_p), as p →+∞, is that of a tower of two different bubbles.| File | Dimensione | Formato | |
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