Lane-Emden systems with Neumann boundary conditions: existence, regularity and qualitative properties of least energy nodal solutions

Hugo Tavares (IST Lisbona)

Abstract. In this talk we deal with the following system of Lane-Emden equations (also known as elliptic Hamiltonian system) with Neumann boundary conditions
\[
-\Delta u = |v|^{q-1} v\quad \text{ in }\Omega,\qquad -\Delta v = |u|^{p-1} u\quad \text{ in }\Omega,\qquad \partial_\nu u=\partial_\nu v=0\quad \text{ on }\partial\Omega,
\]
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$; in this setting, all nontrivial solutions are necessarily sign-changing. In the subcritical case (i.e., $(p,q)$ below the so called critical hyperbola), we show existence of least energy nodal solutions (l.e.n.s.), regularity, as well as a symmetry breaking and partial regularity results in balls and annuli. In the critical case, under some additional assumptions on $p$ and $q$, we also prove the existence of l.e.n.s., as well as a symmetry breaking in annuli. Our results also apply to the scalar associated model
\[
-\Delta u=|u|^{p-2}u \text{ in } \Omega,\quad u_\nu=0 \text{ on } \partial \Omega,
\]
where our approach provides new results as well as alternative proofs of known facts. If time permits, we also provide in this case additional results, illustrating a continuity of l.e.n.s. with respect to $p$, and an existence result in the slightly supercritical regime.

The talk is based on joint works with Alberto Saldaña, Angela Pistoia and Delia Schiera