Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane-Emden equation in dimension two

TL;DR

This paper provides sharp estimates and proves non-degeneracy of low energy nodal solutions for the Lane-Emden equation in two dimensions, confirming a conjecture about the behavior of least energy solutions as the parameter grows large.

Contribution

It establishes sharp bounds and non-degeneracy for low energy solutions and confirms a conjecture on the asymptotic behavior of least energy nodal solutions.

Findings

Sharp estimates for low energy nodal solutions.

Proof of non-degeneracy of these solutions.

Validation of a conjecture on the asymptotic behavior of least energy solutions.

Abstract

We study the Lane-Emden problem \[\begin{cases} -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega, u_p=0 &\text{on}\quad\partial\Omega, \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. We obtain sharp estimates and non-degeneracy of low energy nodal solutions $u_p$ (i.e. nodal solutions satisfying $\lim_{p\to+\infty}p\int_{\Omega}|u_p|^{p+1}dx=16\pi e$). As applications, we prove that the comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$ holds automatically for least energy nodal solutions, which confirms a conjecture raised by (Grossi-Grumiau-Pacella, Ann.I.H. Poincare-AN, 30 (2013), 121-140) and (Grossi-Grumiau-Pacella, J.Math.Pures Appl. 101 (2014), 735-754).