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Special Session 2: Recent advances in nonlinear Schrodinger systems

New solutions for the Lane-Emden problem in planar domains

Isabella Ianni

Sapienza Universita di Roma
Italy

Co-Author(s):

Abstract:

We consider the Lane-Emden problem

$\begin{eqnarray*} && -\Delta u=|u|^{p-1}u \quad \mbox{ in } \Omega, \ &&u=0 \quad \mbox{ on }\partial\Omega, \end{eqnarray*}$ where

$\Omega\subset\mathbb R^2$ is a smooth bounded domain. When the exponent

$p$ is large, the existence and multiplicity of solutions strongly depend on the geometric properties of the domain, which also deeply affect their qualitative behaviour. Remarkably, a wide variety of solutions, both positive and sign-changing, have been found when

$p$ is sufficiently large. In this talk, we focus on this topic and show the existence of new sign-changing solutions that exhibit an unexpected concentration phenomenon as

$p\rightarrow +\infty$ . These results are obtained in collaboration with L. Battaglia and A. Pistoia.

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