\section{Applications}\label{section 3}

In this section we prove Theorem \ref{thm 4} and Theorem \ref{thm 5}.
First, we give a variational characterisation of the second eigenvalue. For this, let

\[\M\dq \S\cap\left\{u\in\wp :\int_{\Om}|u|^{p-2}u=0 \right\},\]
\[\F_2\dq\left\{h(S^{1}): h \mbox{ is an odd continuous map from }S^{1} \mbox{ into }\S\cap W_{\Ga_0} \right\},\]
\[\G_2\dq\left\{h(S^{1}): h \mbox{ is an odd continuous map from }S^{1} \mbox{ into }\S\cap W_{\Ga_1} \right\}.\]
Then 
\[\mu_2(\Om)=\inf\limits_{u\in\M} J(u),\ \nu_2(\Om)=\inf\limits_{\A\in\F_2}\sup\limits_{u\in\A} J(u),\ \tau_2(\Om)=\inf\limits_{\A\in\G_2}\sup\limits_{u\in\A} J(u). \]
\noi{\bf  Proof of Theorem \ref{thm 4}.} By the translation invariance of $p$-Laplacian, we can take $\Om=B_{R_1}(0)\setminus \overline{B_{R_0}(0)}$ for some $0\leq R_0<R_1<\infty$. 


 Let $u$ be the eigenfunction associated with $\mu_2(\Om)$, then $u$ must change its sign in $\Om$. Suppose the nodal set $\{x\in\Om: u(x)=0\}$ of $u$ is a sphere of radius $r\in(R_0,R_1)$ centred at origin. Then we have
 \[\tau_1(B_r(0)\setminus\overline{B_{R_0}(0)})=\mu_2(\Om);\qquad \nu_1(B_{R_1}(0)\setminus\overline{B_{r}(0)})=\mu_2(\Om).\]
Now for $s\in (0,r-R_0)$, by Theorem \ref{thm 1} and Theorem \ref{thm 2} we have
\[\tau_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})\leq \mu_2(\Om)\mbox{ and } \nu_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})<\mu_2(\Om).\] 
 Let $\phi_1$ and $ \phi_2$ be positive eigenfunctions corresponding to $\tau_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})$ and $\nu_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})$ respectively, with the normalisation $\int|\phi_1|^{p-1}=1=\int|\phi_2|^{p-1}$. Let $\tilde{\phi}$ denotes the extension of $\phi$ to $\Om$ by zero. Then $\phi=\tilde{\phi_1}-\tilde{\phi_2}\in \wp$ with $\int_{\Om}|\phi|^{p-2}\phi=0$, and
	\begin{align*}
	J(\phi)=\int_{\Om}|\nabla \phi|^p&=\int|\nabla \phi_1|^p+\int|\nabla \phi_2|^p\\
	&= \tau_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})\int|\phi_1|^p+\nu_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})\int|\phi_2|^p\\
	&<\mu_2(\Om)\left(\int|\phi_1|^p+\int|\phi_2|^p\right)=\mu_2(\Om)\int_{\Om}|\phi|^p.
	\end{align*}
This contradicts the variational characterisation of $\mu_2(\Om)$.


\noindent {\bf Proof of Theorem \ref{thm 5}.} We give the result only for $\nu_2(\Om)$, a similar proof holds for $\tau_2(\Om)$. By the translation invariance of $p$-Laplacian, we can take $\Om=B_{R_1}(0)\setminus \overline{B_{R_0}(0)}$ for some $0< R_0<R_1<\infty$.  Let $u$ be an eigenfunction associated with $\nu_2(\Om)$.
	Suppose the nodal set $\{x\in\Om:u(x)=0\}$ of $u$ is a sphere of radius $r\in(R_0,R_1)$ centred at the origin. Then we have 
 	
	\[\la_1(B_r(0)\setminus\overline{B_{R_0}(0)})=\nu_2(\Om);\qquad  \tau_1(B_{R_1}(0)\setminus\overline{B_{r}(0)})=\nu_2(\Om).\] Thus, for $s\in (0,r-R_0)$,  by Theorem 1.1 of \cite{Anoop} and Theorem \ref{thm 2} we have 
	
	\[\la_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})< \nu_2(\Om) \;\mbox{ and }\; \tau_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})<\nu_2(\Om).\] 
	Let $\phi_1$ and $ \phi_2$ be the positive eigenfunctions corresponding to the eigenvalues $\la_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})$ and $\tau_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})$ respectively, with the normalisation $\int|\phi_1|^{p}=1=\int|\phi_2|^{p}$. Now consider the set $\A\dq \{a\tilde{\phi_1}+b\tilde{\phi_2}: |a|^p+|b|^p=1\}$. Then $\A\in\F_2$ and for $\phi\in\A$ we have
	\begin{align*}
	J(\phi)=\int_{\Om}|\nabla \phi|^p&=\int|\nabla \phi_1|^p+\int|\nabla \phi_2|^p\\
	&= \la_1(B_r(se_1)\setminus\overline{B_{R_0}(0)})|a|^p+\tau_1(B_{R_1}(0)\setminus\overline{B_r(se_1)})|b|^p\\
	&<\nu_2(\Om)\left(|a|^p+|b|^p\right)=\nu_2(\Om).
	\end{align*}
A contradiction to the variational characterisation of $\nu_2(\Om)$. 
\qed
{"config": "arxiv", "file": "1806.11491/applications.tex"}