Asymptotic formula for the Morse index of radial solutions on expanding annuli: Allen-Cahn equation and scalar filed equation

Let \(A_R\subset \mathbb {R}^N\) $A_R\subset {\mathbb{R}}^N$ , \(N\ge 2\) $N\ge 2$ , be an annulus with inner radius R and outer radius \(R+1\) $R+1$ . We are concerned with the elliptic Neumann problem \(\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon ^2\Delta u+f(u)=0 & \text {for}\ x\in A_R,\\ \frac{\partial u}{\partial n}=0 & \text {for}\ x\in \partial A_R, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\epsilon }^2\mathrm{\Delta }u+f(u)=0\hfill & \text{for}x\in A_R,\hfill \\ \frac{\partial u}{\partial n}=0\hfill & \text{for}x\in \partial A_R,\hfill \end{array}\right)\end{array}$$ where \(\varepsilon >0\) $\epsilon >0$ is a small constant. In particular, the Allen-Cahn equation \(f(u)=u-u^3\) $f(u)=u-u^3$ and the scalar field equation \(f(u)=-u+u^p\) $f(u)=-u+u^p$ , \(p>1\) $p>1$ , are studied. We establish sharp asymptotic formulas of the Morse index of n-mode radial solutions as \(R\rightarrow \infty \) $R\to \infty$ . In the case of the scalar field equation the first n eigenvalues of the linearization around n-mode solutions of the one-dimensional problem \(\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon ^2u''-u+u^p=0 & \text {for}\ 0<x<1,\\ u'(0)=u'(1)=0 \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\epsilon }^2{u}^{\prime \prime }-u+u^p=0\hfill & \text{for}0<x<1,\hfill \\ u^{\prime }(0)=u^{\prime }(1)=0\hfill \end{array}\right)\end{array}$$ become important. We show that, as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , the first \(\ell \) $\ell$ eigenvalues converge to \(-(p+3)(p-1)/4\) $-(p+3)(p-1)/4$ and the other \(n-\ell \) $n-\ell$ eigenvalues converge to 0, where \(\ell \) $\ell$ is the number of the local maximum points of an n-mode solution u(x) on [0, 1].