Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems
Francesca De Marchis, Isabella Ianni, Filomena Pacella · Journal of the European Mathematical Society · 2015
We consider the semilinear Lane–Emden problem \tag{$\mathcal E_p$}\begin{cases}-\Delta u= |u|^{p-1}u\qquad &\text{ in }\Omega\\ u=0\qquad\qquad\qquad&\text{ on }\partial \Omega \end{cases} where p>1 and \Omega is a smooth bounded domain of \mathbb R^2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( \mathcal E_p ), as p\to+\infty . Among other results we show, under some symmetry assumptions on \Omega , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as 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 \mathbb R^2 .