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).
Categories: math.AP