| 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. |
|