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

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