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

We study the Lane-Emden problem $$\{\begin{array}{lll}-\mathrm{\Delta }{u}_p=\mathrm{\mid }{u}_p{\mathrm{\mid }}^{p-1}{u}_p& \text{in}\mathrm{\Omega },u_p=0& \text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^2$ 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 ${\lim }_{p\to +\mathrm{\infty }}p{\int }_{\mathrm{\Omega }}\mathrm{\mid }{u}_p{\mathrm{\mid }}^{p+1}dx=16\pi e$). As applications, we prove that the comparable condition $p(\mathrm{\parallel }{u}_p^{+}{\mathrm{\parallel }}_{\mathrm{\infty }}-\mathrm{\parallel }{u}_p^{-}{\mathrm{\parallel }}_{\mathrm{\infty }})=O(1)$ 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).