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

Abstract

We study the Lane-Emden problem \[\begin{cases} -Δu_p =|u_p|^{p-1}u_p&\text{in}\quad Ω, u_p=0 &\text{on}\quad\partialΩ, \end{cases}\] 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 $p(\Vert u_p^{+}{\Vert }_{\infty }-\Vert u_p^{-}{\Vert }_{\infty })=O(1)$ 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).