We begin by defining a list of essential mathematics tools extracted from the article, then we state the main result. Let \((M,d)\) be a metric space equipped with the distance \(d\), \(B_\delta(u)\) be an open ball of center \(u\in M\) and of radius \(\delta>0\). For \(f\) a real-valued function on \(M\), we denote by \(|df|(u)\) the supremum of the \(\sigma\)'s in \(\mathbb R_+\) such that there is \(\mathcal{H}\), a \(M\)-valued continuous map on \(B_\delta(u)\times [0,\delta]\) satisfying \(d(\mathcal{H}(u,t),t)\leq t\) and \(f(\mathcal{H}(u,t))\leq f(v)-\sigma t\), for every \(v\in B_\delta(u)\) and \(t\in [0,\delta]\). Let \(u\in M\) and \(c\in\mathbb R\) such that \(f(u)=c\), if \(|df(u)|=0\), then we say that \(u\) is a critical point of \(f\) and \(c\) is a critical value of \(f\). We say that \(f\) satisfies the Palais-Smale condition at level \(c\) if for any sequence \((u_k)_{k\in\mathbb N}\) in \(M\) satisfying \(\lim_{k\to\infty}f(u_k)=c\) and \(\lim_{k\to\infty}(|df|(u_k))=0\) has a convergent subsequent in \(M\). We say that \(f\) satisfies the Cerami-Palais-Smale condition if for every sequence \((u_k)_{k\in\mathbb N}\) in \(M\) satisfying \(\lim_{k\to\infty}f(u_k)=c\) and \(\lim_{k\in\infty}((1+d(u_k,\hat{u}))|df|(u_k))=0\) has a convergent subsequent in \(M\) where \(\hat{u}\) belongs to \(M\). Let \(\psi\) be an isometry on \(M\), we say that \(f\) is \(\psi\)-invariant if \(f(\psi(u))=f(u)\) for \(u\in M\). We say that \(A\), a subset of \(M\), is \(\psi\)-invariant, when \(\psi(A)\subset A\) and Krasnoselskii-type genera are defined by \(\gamma(A)=\min\{k\geq 1:\text{there is a } \psi-\text{invariant and continuous map }\varphi:A\to\mathbb R^k\setminus\{0\}\}\) and \(\overline{\gamma}(A)=\sup\{k\geq 1:\text{there is a } \psi-\text{invariant and continuous map }\varphi:\mathbb R^k\setminus\{0\}\to A\}\). Let \(X\) be a real Banach space and \(f,g\) be real-valued positive even continuous functions on \(X\) and which are homogeneous of degree \(p>0\). We say that \(u\in X\) is an eigenvector if \(g(u)\neq 0\) and \(u\) is a critical point of \(f_{\restriction{M_u}}\) such that \(M_u=\{v\in X:g(v)=g(u)\}\) and \(\lambda=\frac{f(u)}{g(u)}\) is the eigenvalue associated to \(u\). We define \(\underline{\lambda }_k=\inf\{\max_Af:A\text{ is a compact and symmetric subset of }M\text{ with }\gamma(A)\geq k\}\) and \(\overline{\lambda }_k=\inf\{\max_Af:A\text{ is a compact and symmetric subset of }M\text{ with }\overline{\gamma}(A)\geq k\}\). The mountain pass eigenvalue associated to \(u\in M\) is defined by \(\lambda_{mp}(u)=\inf_{\varphi\in \Phi}\max_{t\in[-1,1]}f(\varphi(t))\) such that \(\Phi\) is a nonempty set of continuous maps \(\varphi:[-1,1]\to M\) such that \(\varphi(-1)=-u\) and \(\varphi(1)=u\). Consequently, the main result states that \(\lambda_{mp}(u)=\underline{\lambda}_2=\overline{\lambda}_2\) whenever \(u\) is an eigenvector with eigenvalue \(\lambda_1\), see Theorem 4.4. The proof is split into two parts, the first one focuses when \(\lambda_1\) is a simple eigenvalue, i.e., is not associated with two linearly independent eigenvectors, and the second one concentrates when \(\lambda_1\) is not simple.