JournalsjemsVol. 17, No. 8pp. 2037–2068

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

  • Francesca De Marchis

    Università di Roma Tor Vergata, Italy
  • Isabella Ianni

    Seconda Università di Napoli, Caserta, Italy
  • Filomena Pacella

    Università di Roma La Sapienza, Italy
Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems cover
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Abstract

We consider the semilinear Lane–Emden problem

\labelproblemAbstract{Δu=up1u\mboxinΩu=0\mboxonΩ(Ep)\label{problemAbstract}\left\{\begin{array}{lr}-\Delta u= |u|^{p-1}u\qquad \mbox{ in }\Omega\\ u=0\qquad\qquad\qquad\mbox{ on }\partial \Omega \end{array}\right.\tag{$\mathcal E_p$}

where p>1p>1 and Ω\Omega is a smooth bounded domain of R2\mathbb R^2. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of \eqref{problemAbstract}, as p+p\to+\infty. Among other results we show, under some symmetry assumptions on Ω\Omega, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p+p\to+\infty, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in R2\mathbb R^2.

Cite this article

Francesca De Marchis, Isabella Ianni, Filomena Pacella, Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems. J. Eur. Math. Soc. 17 (2015), no. 8, pp. 2037–2068

DOI 10.4171/JEMS/549