G_{n}(z):=\frac{1}{2\pi}\int_{0}^{2\pi}G(z,\theta)e^{-in\theta}d\theta\,.$

The unique solution $v\in z^{1-m}H_{0}^{1}((0,1)\times\mathbb{T})$ to (7.24)
is therefore

(7.31) $v(z,\theta)=\sum_{n\in\mathbb{Z}}v_{n}(z)e^{in\theta}\,,$

where each Fourier component is the only solution $v_{n}\in
z^{1-m}H_{0}^{1}((0,1))$ to the ODE

(7.32) ${\mathcal{L}}_{n}v_{n}=G_{n}\qquad\textup{in }(0,1)\,,$

where

${\mathcal{L}}_{n}u:=z\partial_{z}^{2}u+(2m-2)\partial_{z}u-\alpha_{n}^{2}z\,u\,.$

Note that this is the ordinary differential operator (7.1) that we studied in
Subsection 7.1, and that the parameter

$\alpha_{n}:=\frac{\varepsilon|n|}{a_{-}}\in[0,\infty)\,,$

now depends on the Fourier frequency $n$ and on the scale $\varepsilon$.

Given an integer $j_{1}\geqslant 1$, for each Fourier mode we argue as in
Subsection 7.1, decomposing $G_{n}$ as

$G_{n}(z)=:\sum_{k=0}^{j_{1}-2}\frac{z^{k}}{k!}G_{n}^{(k)}(0)\widetilde{\chi}((1+\alpha_{n})z)+{\mathcal{R}}_{G_{n}}(z)\,.$

Thus the function $v_{n}\in z^{1-m}H_{0}^{1}((0,1))$ can be similarly written
as

(7.33)
$v_{n}=\sum_{k=0}^{j_{1}-2}\frac{1}{k!}\,v_{k,n}+v_{{\mathcal{R}},n}\,,$

where $v_{k,n},v_{{\mathcal{R}},n}\in z^{1-m}H_{0}^{1}((0,1))$ are the only
solutions to the ODEs

(7.34) $\displaystyle{\mathcal{L}}_{n}v_{k,n}$
$\displaystyle=z^{k}G_{n}^{(k)}(0)\widetilde{\chi}((1+\alpha_{n})z)\,,$ (7.35)
$\displaystyle{\mathcal{L}}_{n}v_{{\mathcal{R}},n}$
$\displaystyle={\mathcal{R}}_{G_{n}}$

in $(0,1)$. Note that the index $k$ will eventually range from 0 to $j_{1}-2$.

The case where $G(z)=\varepsilon zH(z)$ with $H,z\partial_{z}H\in
H^{j-2}((0,1)\times\mathbb{T})$ is handled the same way. We write $H$ as a
Fourier series

$H(z,\theta)=\sum_{n\in\mathbb{Z}}H_{n}(z)e^{in\theta}\,,\qquad
H_{n}(z):=\frac{1}{2\pi}\int_{0}^{2\pi}H(z,\theta)e^{-in\theta}d\theta\,,$

and note that the Fourier coefficients in (7.31) can be similarly written as
(7.33), where now $\widetilde{v}_{k,n},\widetilde{v}_{{\mathcal{R}},n}\in
z^{1-m}H_{0}^{1}((0,1))$ are the only solutions to

(7.36) $\displaystyle{\mathcal{L}}_{n}\widetilde{v}_{k,n}$
$\displaystyle=\varepsilon
z^{k+1}H_{n}^{(k)}(0)\widetilde{\chi}((1+\alpha_{n})z)\,,$ (7.37)
$\displaystyle{\mathcal{L}}_{n}\widetilde{v}_{{\mathcal{R}},n}$
$\displaystyle=\varepsilon z{\mathcal{R}}_{H_{n}}\,.$

Here, ${\mathcal{R}}_{H_{n}}$ is defined as

$H_{n}(z)=:\sum_{k=0}^{j_{1}-2}\frac{z^{k}}{k!}H_{n}^{(k)}(0)\widetilde{\chi}((1+\alpha_{n})z)+{\mathcal{R}}_{H_{n}}(z)\,,$

and

$v_{n}=\sum_{k=0}^{j_{1}-2}\frac{1}{k!}\widetilde{v}_{k,n}+\widetilde{v}_{{\mathcal{R}},n}\,.$

Step 2: Sum over Fourier modes. Lemma 7.5 now follows from Lemma 7.1 (with
$\alpha_{n}:=\varepsilon|n|/a_{-}$) using the Parseval identity.

First of all, observe that Lemma 7.1 (i.b) applied with $\ell:=0$ implies that

(7.38)
$\displaystyle\|v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}$
$\displaystyle\quad=2\pi\sum_{n\in\mathbb{Z}}\Big{(}\|v_{n}\|_{L^{2}((0,1))}^{2}+\|z\partial_{z}v_{n}\|_{L^{2}((0,1))}^{2}\Big{)}\lesssim
2\pi\sum_{n\in\mathbb{Z}}\|G_{n}\|_{L^{2}((0,1))}^{2}=\|G\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\,,$

and that, for all $j\geqslant 1$,

$\displaystyle\|\partial_{\theta}^{j}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}\partial_{\theta}^{j}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}$
$\displaystyle\quad=2\pi\sum_{n\in\mathbb{Z}}n^{2j}\Big{(}\|v_{n}\|_{L^{2}((0,1))}^{2}+\|z\partial_{z}v_{n}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\quad\lesssim\frac{2\pi}{\varepsilon^{2}}\sum_{n\in\mathbb{Z}}n^{2(j-1)}\|G_{n}\|_{L^{2}((0,1))}^{2}=\frac{1}{\varepsilon^{2}}\,\|\partial_{\theta}^{j-1}G\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\,.$

Likewise, Lemma 7.1 (iii.b) applied with $\ell:=0$ implies that, for all
$j\geqslant 2$,

$\|\partial_{\theta}^{j}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}\partial_{\theta}^{j}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\lesssim\frac{1}{\varepsilon^{2}}\,\|\partial_{\theta}^{j-2}H\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\,.$

Once we have proved (7.27) and (7.29), we focus on the more involved estimates
(7.28) and (7.30). Applying Lemma 7.1 (ii.b) with $\ell:=0$, we obtain that,
for all nonnegative integer $j_{2}$ with $j_{2}\leqslant j-1$

(7.39)
$\displaystyle\|\partial_{z}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}^{2}\partial_{z}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}$
$\displaystyle\quad=\sum_{n\in\mathbb{Z}}n^{2j_{2}}\Big{(}\|\partial_{z}v_{n}\|_{L^{2}((0,1))}^{2}+\|z\partial_{z}^{2}v_{n}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\quad\lesssim
2\pi\sum_{n\in\mathbb{Z}}n^{2j_{2}}\|G_{n}\|_{L^{2}((0,1))}^{2}=\|\partial_{\theta}^{j_{2}}G\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\,.$

Likewise, Lemma 7.1 (iv.b) applied $\ell:=0$ implies that, for all nonnegative
integer $j_{2}$ with $j_{2}\leqslant j-1$, it follows that

(7.40)
$\|\partial_{z}^{2}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}^{3}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\lesssim\|\partial_{\theta}^{j_{2}}H\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}\partial_{\theta}^{j_{2}}H\|_{L^{2}((0,1)\times\mathbb{T})}^{2}\,.$

We finally deal with the case where $j_{1}$ and $j_{2}$ are nonnegative
integers with $j_{1}\geqslant 2$ and $0\leqslant j_{2}\leqslant j-j_{1}$. By
direct computations, we get

(7.41)
$\displaystyle\|\partial_{z}^{j_{1}}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}+\|z\partial_{z}^{j_{1}+1}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}^{2}$
$\displaystyle\quad=2\pi\sum_{n\in\mathbb{Z}}n^{2j_{2}}\Big{(}\|v_{n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|zv_{n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\quad\lesssim\sum_{n\in\mathbb{Z}}n^{2j_{2}}\bigg{(}\sum_{k=0}^{j_{1}-2}\frac{1}{k!}\Big{(}\|v_{k,n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|zv_{k,n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\hskip
199.16928pt+\|v_{{\mathcal{R}},n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|zv_{{\mathcal{R}},n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\bigg{)}\,.$

On one hand, by Lemma 7.1 (ii.a) applied with $\ell:=j_{1}-1$, we infer that

$\displaystyle\sum_{n\in\mathbb{Z}}n^{2j_{2}}\bigg{(}\sum_{k=0}^{j_{1}-2}\frac{1}{k!}\Big{(}\|v_{k,n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|zv_{k,n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\quad\lesssim\sum_{k=0}^{j_{1}-2}\frac{1}{k!}(1+n^{2})^{j_{1}-1+j_{2}-k-\frac{1}{2}}|G_{n}^{(k)}(0)|^{2}\lesssim\|\partial_{z}^{k}G(0,\cdot)\|_{H^{j_{1}+j_{2}-k-\frac{3}{2}}(\mathbb{T})}^{2}\,.$

On the other hand, by Lemma 7.1 (ii.b) applied with $\ell:=j_{1}-1$, we get
that

$\displaystyle\sum_{n\in\mathbb{Z}}n^{2j_{2}}\big{(}\|v_{{\mathcal{R}},n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|zv_{{\mathcal{R}},n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\big{)}\lesssim\sum_{n\in\mathbb{Z}}n^{2j_{2}}\sum_{\nu=0}^{j_{1}-1}\bigg{(}(1+n^{2})^{j_{1}-1-\nu}\|{\mathcal{R}}_{G,n}^{(\nu)}\|_{L^{2}((0,1))}^{2}\bigg{)}$

Thus, it follows that

(7.42)
$\|\partial_{z}^{j_{1}}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}+\|z\partial_{z}^{j_{1}+1}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}\lesssim\|G\|_{H^{j-1}((0,1)\times\mathbb{T})}\,.$

The bound (7.27) then follows from (7.38), (7.39) and (7.42).

Let us now assume that $j_{1}\geqslant 3$. On one hand, by Lemma 7.1 (iv.a)
applied with $\ell:=j_{1}-2$, we get that

$\displaystyle\sum_{n\in\mathbb{Z}}n^{2j_{2}}\bigg{(}\sum_{k=0}^{j_{1}-2}\frac{1}{k!}\Big{(}\|\widetilde{v}_{k,n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|z\widetilde{v}_{k,n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\quad\lesssim\sum_{k=0}^{j_{1}-2}\frac{1}{k!}(1+n^{2})^{j_{1}-2+j_{2}-k-\frac{1}{2}}|H_{n}^{(k)}(0)|^{2}\lesssim\|\partial_{z}^{k}H(0,\cdot)\|_{H^{j_{1}+j_{2}-k-\frac{5}{2}}(\mathbb{T})}^{2}\,.$

On the other hand, by Lemma 7.1 (iv.b) applied with $\ell:=j_{1}-2$, we get

$\displaystyle\sum_{n\in\mathbb{Z}}n^{2j_{2}}\Big{(}\|\widetilde{v}_{{\mathcal{R}},n}^{(j_{1})}\|_{L^{2}((0,1))}^{2}+\|z\widetilde{v}_{{\mathcal{R}},n}^{(j_{1}+1)}\|_{L^{2}((0,1))}^{2}\Big{)}$
$\displaystyle\qquad\lesssim\sum_{n\in\mathbb{Z}}n^{2j_{2}}\sum_{\nu=0}^{j_{1}-2}\bigg{(}(1+n^{2})^{j_{1}-2-\nu}\Big{(}\|{\mathcal{R}}_{H,n}^{(\nu)}\|_{L^{2}((0,1))}^{2}+\|z{\mathcal{R}}_{H,n}^{(\nu+1)}\|_{L^{2}((0,1))}^{2}\Big{)}\bigg{)}\,.$

Thus, substituting into (7.41) with $v_{k,n}$ replaced by
$\widetilde{v}_{k,n}$, and $v_{{\mathcal{R}},n}$ replaced by
$\widetilde{v}_{{\mathcal{R}},n}$, we conclude that

(7.43)
$\|\partial_{z}^{j_{1}}\partial_{\theta}^{j_{2}}v\|_{L^{2}((0,1)\times\mathbb{T})}+\|z\partial_{z}^{j_{1}+1}\partial_{\theta}^{j_{2}}\|_{L^{2}((0,1)\times\mathbb{T})}\lesssim\|H\|_{H^{j-2}((0,1)\times\mathbb{T})}+\|z\partial_{z}H\|_{H^{j-2}((0,1)\times\mathbb{T})}\,.$

Combining (7.38), (7.40) and (7.43), we thus obtain (7.29). The lemma then
follows. ∎

##  8\. Spectral properties of $\mathfrak{L}_{a}$

In this section we study the spectral properties of the operator
$\mathfrak{L}_{a}$, depending on the parameters $a$ and $\ell$. Specifically,
we prove Propositions 2.10 and 2.11. These results show a phenomenon of
crossing of eigenvalues, which is essential for our bifurcation result.

The proof of Proposition 2.10 will follow from several lemmas. Let us recall
the positive symmetric bilinear form
${\mathcal{B}}:H_{0}^{1}(\Omega_{a})\times H_{0}^{1}(\Omega_{a})\to\mathbb{R}$
associated to the linear operator $\mathfrak{L}_{a}$, namely

$\displaystyle{\mathcal{B}}(v,w)$
$\displaystyle:=-\int_{\Omega}\big{(}L_{a}^{0,0}v+f_{a}^{\prime}(\Phi_{a,1},\widetilde{\psi}_{a})v\big{)}w\,(R+a-4)\,dR\,d\theta$
$\displaystyle=\int_{\Omega}\left[\partial_{R}v\,\partial_{R}w+\frac{\partial_{\theta}v\,\partial_{\theta}w}{(R+a-4)^{2}}-f_{a}^{\prime}(\Phi_{a,1},\widetilde{\psi}_{a})vw\right](R+a-4)\,dR\,d\theta\,.$

Also, we recall that there exists a sequence of eigenvalues of
$-\mathfrak{L}_{a}$ with finite multiplicity, which we denote by
$\lambda_{k}\equiv\lambda_{k}(a)$, such that

$\lambda_{1}<\lambda_{2}\leqslant\lambda_{3}\leqslant\cdots$

Of course, $\lambda_{k}(a)$ tends to infinity as $k\to\infty$, and depends
continuously on $a$.

As the potential is radial, the radial eigenfunctions of this operator, which
define an orthonormal basis of $L^{2}((1,7),(R+a-4)\,dR)$, can be equivalently
obtained from the symmetric bilinear form ${\mathcal{B}}^{\rm
rad}:H_{0}^{1}(1,7))\times H_{0}^{1}(1,7))\to\mathbb{R}$ given by

${\mathcal{B}}^{\rm
rad}(v,w):=\int_{1}^{7}\left[\partial_{R}v\,\partial_{R}w-f_{a}^{\prime}(\Phi_{a,1},\widetilde{\psi}_{a})vw\right](R+a-4)\,dR\,.$

The corresponding quadratic form is ${\mathcal{Q}}^{\rm
rad}(v):={\mathcal{B}}^{\rm rad}(v,v)$.

We denote the eigenvalues corresponding to the radial eigenfunctions by

$\lambda_{1}^{\rm rad}(a)<\lambda_{2}^{\rm rad}(a)<\cdots<\lambda_{k}^{\rm
rad}(a)<\cdots$

When there is no risk of confusion, we simply write $\lambda_{k}^{\rm
rad}\equiv\lambda_{k}^{\rm rad}(a)$. It is standard that $\lambda_{k}^{\rm
rad}(a)\to\infty$ as $k\to\infty$, and that $\lambda_{k}^{\rm
rad}(a)\neq\lambda_{j}^{\rm rad}(a)$ whenever $k\neq j$. To see this, note
that, the eigenvalue equation for a radial eigenfunction $\phi$ of
$\mathfrak{L}_{a}$ (which, by Proposition 6.9, is in
$\rho^{m-1}{\mathbf{X}}^{j}$) reduces to the ODE. The classical argument
showing that radial Dirichlet eigenvalues have multiplicity 1 then follows
from Frobenius’ theory for ODEs.

We start the proof of Proposition 2.10 with the following lemma:

######  Lemma 8.1.

For all $a\geqslant 4$, $\lambda_{1}^{\rm rad}(a)<\lambda_{2}^{\rm rad}(a)<0$.

###### Proof.

By composing with the diffeomorphism $\Phi_{a}$, it suffices to show that the
quadratic form

(8.1)
$\tilde{{\mathcal{Q}}}(v):=\int_{a_{-}}^{a^{+}}\left[(\partial_{r}v)^{2}-f_{a}^{\prime}(r,\psi_{a})v^{2}\right]r\,dr\,,$

is negative definite on a two dimensional subspace of
$H_{0}^{1}((a_{-},a_{+}))$. Recall that $\psi_{a}$ solves the equation

$\partial_{r}^{2}\psi_{a}+\frac{\partial_{r}\psi_{a}}{r}+f_{a}(r,\psi_{a}(r))=0\quad\textup{
in }(a_{-},a_{+})\,.$

Moreover, taking the derivative with respect to $r$, we get that
$\xi(r):=\partial_{r}\psi_{a}(r)$ solves

$\partial_{r}^{2}\xi+\frac{\partial_{r}\xi}{r}-\frac{\xi}{r^{2}}+f_{a}^{\prime}(r,\psi_{a}(r))\xi=0\quad\textup{
in }(a_{-},a_{+})\,.$

It is worth stressing that, in the above expression, we are using that
$\partial_{r}f_{a}(r,\psi_{a}(r))\equiv 0$ in $(a_{-},a_{+})$.

Next, we recall that $m_{a}$ is the unique maximum of the function $\psi_{a}$,
and define

$\xi_{1}(r):=\left\\{\begin{aligned} &\partial_{r}\psi_{a}(r)\,,\
&&r\in[a_{-},m_{a})\,,\\\
&0&&r\in[m_{a},a_{+})\,,\end{aligned}\right.\quad\textup{ and
}\quad\xi_{2}(r):=\left\\{\begin{aligned} &0\,,\ &&r\in[a_{-},m_{a}]\,,\\\
&\partial_{r}\psi_{a}(r)&&r\in(m_{a},a_{+}]\,.\end{aligned}\right.$

Clearly, $\xi_{1}$, $\xi_{2}\in H_{0}^{1}((a_{-},a_{+}))$ and

$\displaystyle\tilde{{\mathcal{Q}}}(\xi_{1})$
$\displaystyle:=\int_{a_{-}}^{m_{a}}\left[(\partial_{r}\xi_{1})^{2}\,-f_{a}^{\prime}(r,\psi_{a})\xi_{1}^{2}\right]r\,dr=\int_{a_{-}}^{m_{a}}\left(r(\partial_{r}\xi_{1})^{2}\,+r\xi_{1}\partial_{r}^{2}\xi_{1}+\xi_{1}\partial_{r}\xi_{1}-\frac{\xi_{1}^{2}}{r}\right)dr$
$\displaystyle=\int_{a_{-}}^{m_{a}}\left(\partial_{r}(r\xi_{1}\partial_{r}\xi_{1})-\frac{\xi_{1}^{2}}{r}\right)dr=-\int_{a_{-}}^{m_{a}}\frac{\xi_{1}^{2}}{r}\,dr<0\,.$

Analogously, it follows that

$\tilde{{\mathcal{Q}}}(\xi_{2})<0\,.$

Since $\xi_{1}$ and $\xi_{2}$ have disjoint support, we conclude that
$\tilde{{\mathcal{Q}}}(\xi)<0\,,$ for all $\xi\in{\rm span}\\{\xi_{1},\
\xi_{2}\\}\setminus\\{0\\}\,.$ This implies that $\lambda_{2}^{\rm rad}(a)<0$
and concludes the proof of the lemma. ∎

We are now interested in the behavior of the eigenvalues $\lambda_{k}^{\rm
rad}$ as $a\to+\infty$. On this purpose, let us recall (see Lemma 3.1) that
the limit function $\overline{\psi}$ is a solution to the limit problem

$\overline{\psi}^{\prime\prime}+\overline{f}(\overline{\psi})=0\quad\textup{
in }(1,7)\,.$

We denote the bilinear and quadratic forms associated with the linearized
operator, and defined on functions belonging to $H_{0}^{1}((1,7))$ by

$\overline{{\mathcal{B}}}(v,w):=\int_{1}^{7}\left(v^{\prime}(R)w^{\prime}(R)-\overline{f}^{\prime}(\overline{\psi}(R))v(R)w(R)\,\right)dR\,,\qquad\overline{{\mathcal{Q}}}(v):=\overline{{\mathcal{B}}}(v,v)\,.$

Just as in the case of the radial eigenvalues of $\mathfrak{L}_{a}$, it is
standard that there is an orthonormal basis of $L^{2}((1,7))$ consisting of
eigenfunctions, whose eigenvalues we label as

$\overline{\lambda}_{1}<\overline{\lambda}_{2}<\cdots<\overline{\lambda}_{k}<\cdots$

Furthemore, the eigenvalues tend to infinity and are simple:
$\overline{\lambda}_{j}\neq\overline{\lambda}_{k}$ for $j\neq k$.

We have the following important convergence result:

######  Lemma 8.2.

As $a\to\infty$, the following limits hold true:

$\lambda_{1}^{\rm rad}(a)\to\overline{\lambda}_{1}<0,\qquad\lambda_{2}^{\rm
rad}(a)\to\overline{\lambda}_{2}=0,\qquad\lambda_{3}^{\rm
rad}(a)\to\overline{\lambda}_{3}>0\,.$

###### Proof.

We first observe that, for any $v\in H^{1}_{0}((1,7))$,

$\displaystyle{\mathcal{Q}}^{\rm rad}(v)$
$\displaystyle=\int_{1}^{7}\left[(\partial_{R}v)^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))v^{2}\right](R+a-4)\,dR$
$\displaystyle=a\left[\int_{1}^{7}\left[(\partial_{R}v)^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))v^{2}\right]\,dR+o(1)\|v\|_{H^{1}(1,7)}^{2}\right].$

Moreover, by Lemma 3.1 (iv), we have that

$\displaystyle\left|\int_{1}^{7}\left[(\partial_{R}v)^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))v^{2}\right]\,dR-\int_{1}^{7}\left[(\partial_{R}v)^{2}-\overline{f}^{\prime}(\overline{\psi}(R))v^{2}\right]\,dR\,\right|$
$\displaystyle\quad=\bigg{|}\int_{1}^{7}\Big{(}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))+\overline{f}^{\prime}(\psi_{a}(R+a-4))\Big{)}v^{2}\,dR$
$\displaystyle\qquad\
+\int_{1}^{7}\Big{(}-\overline{f}^{\prime}(\psi_{a}(R+a-4))+\overline{f}^{\prime}(\overline{\psi}(R))\Big{)}v^{2}\,dR\,\bigg{|}$
$\displaystyle\quad\leqslant\frac{m-1}{a}\int_{1}^{7}\frac{v^{2}}{\rho}\,dR+o(1)\,\|v\|_{H^{1}(1,7)}^{2}=o(1)\|v\|_{H^{1}((1,7))}^{2}\,,\quad\textup{
as }a\to+\infty\,,$

and so that

${\mathcal{Q}}^{\rm
rad}(v)=a\,\big{[}\,\overline{{\mathcal{Q}}}(v)+o(1)\|v\|_{H^{1}((1,7))}^{2}\,\big{]}\quad\textup{
as }a\to+\infty\,.$

On the other hand, it is easy to see that

$\int_{1}^{7}v(R)^{2}(R+a-4)\,dR=a\,[1+o(1)]\int_{1}^{7}v(R)^{2}\,dR\quad\textup{
as }a\to+\infty\,.$

This implies the convergence of the corresponding Rayleigh quotients, in the
sense that

$\frac{{\mathcal{Q}}^{\rm
rad}(v)}{\|v\|^{2}_{L^{2}((1,7),\,(R+a-4)\,dR)}}\,\xrightarrow[]{a\to+\infty}\,\frac{\overline{{\mathcal{Q}}}(v)}{\
\|v\|^{2}_{L^{2}((1,7))}}\,,\quad\mbox{ for any fixed radial function }v\,.$

By the min-max characterization of the eigenvalues we infer that
$\lambda_{k}^{\rm rad}\to\overline{\lambda}_{k}$, as $a\to+\infty$, for any
fixed $k$. Thus, to conclude the proof we only need to verify that
$\overline{\lambda}_{2}=0$. On that purpose, observe that
$\overline{\psi}^{\prime}\in H_{0}^{1}((1,7))$ is a solution of the linearized
problem

$\xi^{\prime\prime}+\overline{f}^{\prime}(\overline{\psi})\xi=0\,.$

Hence, $\overline{\psi}^{\prime}$ is an eigenfunction with associated
eigenvalue $0$. Since $\overline{\psi}^{\prime}$ changes sign exactly once, we
conclude that $\overline{\lambda}_{2}=0$. ∎

As we can see in the previous result, the first radial eigenvalue is negative
and remains bounded away from $0$ as $a\to+\infty$. This is the reason for the
appearance of nonradial degeneracies, as we shall see. Our analysis has to be
precise enough to avoid interferences with the second radial eigenvalue, which
is converging to $0$ from below. In what follows we consider general
eigenvalues with $\ell$-symmetry of the operator $-\mathfrak{L}_{a}$, as
defined in (2.26) and below. In next lemma we fix the values of $a_{0}$ and
$\ell$ which will be used throughout this section.

######  Lemma 8.3.

Let $c:=\frac{1}{3}\min\\{-\overline{\lambda}_{1},\overline{\lambda}_{3}\\}$.
According to Lemma 8.2 we take $a_{0}\geqslant 4$ so that $|\lambda_{1}^{\rm
rad}-\overline{\lambda}_{1}|<c$ and $|\lambda_{3}^{\rm
rad}-\overline{\lambda}_{3}|<c$. Then, there exists $\ell\in\mathbb{N}$ such
that $\lambda_{3}^{\ell}(a_{0})>0$.

###### Proof.

If $\lambda_{3}^{\ell}(a_{0})=\lambda_{3}^{\rm rad}(a_{0})$ then it is
positive and we are done. If not, the corresponding eigenfunction $\phi_{3}$
can be written in Fourier as

$\phi_{3}(R,\theta)=\sum_{k\in\mathbb{N}}\varphi_{k}(R)\cos(k\ell\theta)\,.$

Hence, it suffices to show that for all functions of the form
$v(R,\theta)=\varphi(R)\cos(k\ell\theta)$ the quadratic form
${\mathcal{Q}}(v)$ is strictly positive for $\ell\in\mathbb{N}$ sufficiently
large. It turns out that, for $v$ as above,

$\displaystyle{\mathcal{Q}}(v)$
$\displaystyle=\int_{1}^{7}\int_{0}^{2\pi}\left[(\partial_{R}v)^{2}+\frac{(\partial_{\theta}v)^{2}}{(R+a-4)^{2}}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))v^{2}\right](R+a-4)\,dR\,d\theta$
$\displaystyle=\pi\int_{1}^{7}\left[(\partial_{R}\varphi)^{2}+\frac{\ell^{2}k^{2}}{(R+a-4)^{2}}\varphi^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))\varphi^{2}\right](R+a-4)\,dR.$

Thus, if we choose $\ell\in\mathbb{N}$ such that
$\frac{\ell^{2}}{(a_{0}-3)^{2}}+\overline{\lambda}_{1}-2c>0$, we conclude
that, for $v$ as above,

$\displaystyle{\mathcal{Q}}(v)$
$\displaystyle=\pi\int_{1}^{7}\left[(\partial_{R}\varphi)^{2}+\frac{\ell^{2}k^{2}}{(R+a-4)^{2}}\varphi^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))\varphi^{2}\right](R+a-4)\,dR$
$\displaystyle\geqslant\pi\int_{1}^{7}\left[(\partial_{R}\varphi)^{2}+(c-\lambda_{1}^{\rm
rad})\varphi^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))\varphi^{2}\right](R+a-4)\,dR$
$\displaystyle\geqslant\pi c\int_{1}^{7}\varphi^{2}\,(R+a-4)\,dR>0\,.$

The lemma then follows. ∎

In the next lemma we show that $\lambda_{3}^{\ell}(a)$ becomes negative if
$a>a_{0}$ is large enough.

######  Lemma 8.4.

Let $a_{0}$, $\ell$ and $c$ as in Lemma 8.3. Then, there exists $a_{2}>a_{0}$
such that $\lambda_{3}^{\ell}(a_{2})<0$.

###### Proof.

Let $a_{2}>a_{0}$ be such that
$\frac{\ell^{2}}{(a_{2}-3)^{2}}+\bar{\lambda}_{1}+2c<0$, and let $\phi_{1}$ be
the eigenfunction associated to $\lambda_{1}(a_{2})=\lambda_{1}^{\rm
rad}(a_{2})$. Then, we choose and fix the test function

$v(R,\theta):=\phi_{1}(R)\cos(\ell\theta)\,.$

By construction, $v$ is $L^{2}$-orthogonal to the two radial eigenfunctions
$\phi_{1}$, $\phi_{2}$ associated to the eigenvalues $\lambda_{1}^{\rm
rad}(a_{2})$, $\lambda_{2}^{\rm rad}(a_{2})$. Hence, it suffices to show that
${\mathcal{Q}}(v)<0$. Arguing as in the proof of Lemma 8.3, we then conclude
that

$\displaystyle{\mathcal{Q}}(v)$
$\displaystyle=\pi\int_{1}^{7}\left[(\partial_{R}\phi_{1})^{2}+\frac{\ell^{2}}{(R+a-4)^{2}}\phi_{1}^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))\phi_{1}^{2}\right](R+a-4)\,dR$
$\displaystyle\leqslant\pi\int_{1}^{7}\left[(\partial_{R}\phi_{1})^{2}+(-c-\lambda_{1}^{\rm
rad})\phi_{1}^{2}-f_{a}^{\prime}(R+a-4,\psi_{a}(R+a-4))\phi_{1}^{2}\right](R+a-4)\,dR$
$\displaystyle=-\pi c\int_{1}^{7}\phi_{1}^{2}\,(R+a-4)\,dR<0\,,$

and the lemma follows. ∎

We can now prove Proposition 2.10.

###### Proof of Proposition 2.10.

We fix $a_{0}$, $c$, $\ell$ and $a_{2}$ as in the previous lemmas and take
$\varepsilon_{0}>0$ such that $\varepsilon_{0}<\min\\{c,-\lambda^{\rm
rad}_{2}(a)\\}$ for all $a\in[a_{0},a_{2}]$. This can be ensured thanks to
Lemma 8.1. Then, we set

$A:=\big{\\{}a\in\mathbb{R},\ a\geqslant a_{0}:\
\lambda_{3}^{\ell}(a)<0\,\big{\\}}\,.$

Clearly, $a_{2}\in A$, which is then a non empty set, and so we can define

$\alpha:=\inf A\,.$

By definition, $\lambda_{3}^{\ell}(\alpha)=0$ and
$\lambda_{3}^{\ell}(a)\geqslant 0$ for all $a\in[a_{0},\alpha]$.

Now, given $\varepsilon\in(0,\varepsilon_{0})$, we can take $a_{1}>\alpha$
sufficiently close to $\alpha$ so that

  1. (a)

$\lambda_{3}^{\ell}(a_{1})<0$.

  2. (b)

$\lambda_{3}^{\ell}(a)>-\varepsilon$ for all $a\in[a_{0},a_{1}]$.

We have then proved assertions (i) and (ii) of Proposition 2.10. We now turn
our attention to (iii). Observe that, by our choice of $\varepsilon_{0}$ and
$c$ in Lemma 8.3, we have $\lambda_{3}^{\rm rad}(a)>0$ for any
$a\in[a_{0},a_{1}]$. Hence, if $\lambda_{3}^{\ell}(a)\leqslant 0$, it
corresponds to a nonradial eigenfunction $\phi_{3}$. Being $\phi_{3}$ the
first nonradial eigenfunction, it must be written as
$\phi_{3}(R,\theta)=\phi(R)\cos(\ell\theta)$, where $\phi(R)$ is the
eigenfunction corresponding to the first eigenvalue (which is equal to $0$)
associated to the bilineal form ${\mathcal{B}}^{\ell}:H_{0}^{1}((1,7))\times
H_{0}^{1}((1,7))\to\mathbb{R}$, given by

$\displaystyle{\mathcal{B}}^{\ell}(v,w)$
$\displaystyle:=\int_{1}^{7}\left[\partial_{R}v\,\partial_{R}w+\frac{\ell^{2}}{(R+a-4)^{2}}vw-
f_{a}^{\prime}(\Phi_{a,1},\widetilde{\psi}_{a})vw\right](R+a-4)\,dR\,.$

Again, this is a one-dimensional problem and its eigenfunction $\phi$ must be
unique. We thus conclude that $\lambda_{3}^{\ell}(a)$ is simple. Also, the
function $\phi$ must be positive, since it corresponds to the first
eigenvalue. On the other hand, by Proposition 6.9, we know that
$\phi_{3}=\rho^{m-1}w$ for some $w\in{\mathbf{X}}^{j}$. Moreover, $w$ is
continuous and cannot vanish on the boundary of $\Omega$. This implies the
desired estimate $\phi\geqslant C\rho^{m-1}$ for some $C>0$.

We finally show the validity of Proposition 2.10 (iv). We first prove that
$\lambda_{4}^{\ell}(a)>0$ for any $a\in[a_{0},\alpha]$. Recall that for any
$a\in[a_{0},\alpha]$, $\lambda_{3}^{\ell}(a)\geqslant 0$. The claims follows
immediately if $\lambda_{3}^{\ell}(a)>0$. Instead, if
$\lambda_{3}^{\ell}(a)=0,$ we conclude by its simplicity, which has been
proved above, and the strict inequality holds. It suffices now to take $a_{1}$
closer to $\alpha$, if necessary, so that (iv) is satisfied, and the proof is
concluded. ∎

We conclude this section with the proof of Proposition 2.11.

###### Proof of Proposition 2.11.

Let $(w_{n})_{n=1}^{\infty}\subset{\mathbf{X}}_{\ell}^{j}$ be as in (2.27).
First of all, observe that, for all $n\in\mathbb{N}$,

(8.2) $\displaystyle 0$ $\displaystyle={\mathcal{G}}_{a_{n}}(w_{n})$
$\displaystyle={\mathcal{G}}_{a_{n}}(0)+\int_{0}^{1}D{\mathcal{G}}_{a_{n}}(tw_{n})w_{n}\,dt=D{\mathcal{G}}_{a_{n}}(0)w_{n}+\int_{0}^{1}\big{(}D{\mathcal{G}}_{a_{n}}(tw_{n})-D{\mathcal{G}}_{a_{n}}(0)\big{)}w_{n}\,dt\,.$

Moreover, by Lemma 2.6, we have that

$\left\|\int_{0}^{1}\big{(}D{\mathcal{G}}_{a_{n}}(tw_{n})-D{\mathcal{G}}_{a_{n}}(0)\big{)}w_{n}\,dt\,\right\|_{{\mathbf{Y}}^{j}}=o(\|w_{n}\|_{{\mathbf{X}}^{j}})\,,\quad\textup{as
}n\to\infty\,.$

Hence, by Proposition 6.9, it follows that, for all $n\in\mathbb{N}$
sufficiently large,

(8.3)
$\|w_{n}\|_{{\mathbf{Y}}^{j}}\lesssim\|w_{n}\|_{{\mathbf{X}}^{j}}\lesssim\|\rho^{m-1}w_{n}\|_{H^{1}(\Omega)}\lesssim\|w_{n}\|_{{\mathbf{Y}}^{j}}\,.$

Now, we set $u_{n}:=w_{n}/\|w_{n}\|_{{\mathbf{X}}^{j}}$ for all
$n\in\mathbb{N}$. Since $(u_{n})_{n=1}^{\infty}$ is a bounded sequence, up to
a subsequence if necessary, we have

$u_{n}\rightharpoonup u_{0}\quad\textup{in
}{\mathbf{X}}^{j}\,,\quad\textup{and}\quad u_{n}\to u_{0}\quad\textup{in
}{\mathbf{Y}}^{j}\,,$

for some $u_{0}\in{\mathbf{X}}^{j}$. Note that the strong convergence in
${\mathbf{Y}}^{j}$ follows from the compact embedding proved in Lemma 4.4.
Furthermore, taking into account (8.3), we infer that $u_{0}\not\equiv 0$.

On the other hand, arguing as in (8.2), and using the continuous dependence of
$D{\mathcal{G}}_{a}(0)$ with respect to $a$, and the convergence $a_{n}\to
a^{*}$, we get that

$-D{\mathcal{G}}_{a^{*}}(0)u_{0}=0=\lambda_{3}^{\ell}(a^{*})\rho\,u_{0}\,.$

Hence, we conclude that $\rho^{m-1}u_{0}\equiv\phi_{3}$, with $\phi_{3}$ as in
Proposition 2.10, and so that $u_{0}\not\equiv 0$ on $\partial\Omega$. Even
more, we infer that $u_{0}(R,\theta)=\widetilde{\phi}(R)\cos(\ell\theta)$ for
some radial function $\widetilde{\phi}$ with
$\inf_{1<R<7}\,\widetilde{\phi}(R)>0$. This allows us to conclude that, for
all $n\in\mathbb{N}$ sufficiently large, the functions $b_{w_{n}}$ and
$B_{w_{n}}$ given in (2.19) with $w:=w_{n}$ are nonconstant. ∎

## Acknowledgements

This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme through
the grant agreement 862342 (A.E.). A.E. is also supported by the grant
PID2022-136795NB-I00 of the Spanish Science Agency and the ICMAT–Severo Ochoa
grant CEX2019-000904-S. A.J.F. is partially funded by Proyecto de
Consolidación Investigadora 2022, CNS2022-135640, MICINN. D.R. has been
supported by: the Grant PID2021-122122NB-I00 of the MICIN/AEI, the _IMAG-Maria
de Maeztu_ Excellence Grant CEX2020-001105-M funded by MICIN/AEI, and the
Research Group FQM-116 funded by J. Andalucía.

##  Appendix A Smooth compactly supported solutions given by
elliptic equations with autonomous nonlinearities are locally radial

Our objective in this short appendix is to show that one cannot obtain non-
radial smooth stationary Euler flows with compact support using the usual
formulation in terms of (autonomous) semilinear elliptic equations.

To see this, suppose that $v\in C^{1}(\mathbb{R}^{2})$ is a classical solution
of the stationary Euler equations (1.1) on $\mathbb{R}^{2}$, and that there
exists a region with $C^{1}$ boundary $\Omega\subset\mathbb{R}^{2}$ such that

(A.1) $v=\begin{cases}\nabla^{\perp}\bar{\psi}&\text{in }\Omega,\\\ 0&\text{in
}\mathbb{R}^{2}\backslash\Omega.\end{cases}$

Here $\bar{\psi}\in C^{2}(\overline{\Omega})$ is a solution of a semilinear
elliptic equation of the form

(A.2) $\Delta\psi+f(\psi)=0$

in the domain $\Omega$, for some continuous function $f\in C(\mathbb{R})$. We
assume that $\Omega$ is bounded, and that $\partial\Omega$ consists of
$J\geqslant 1$ connected components $\Gamma_{j}$.

The first observation is that one can equivalently assume that $v$ is globally
given by the perpendicular gradient of a solution of the semilinear equation
(A.2):

######  Proposition A.1.

The vector field (A.1) can be equivalently written as $v=\nabla^{\perp}\psi$,
where $\psi\in C^{2}(\mathbb{R}^{2})$ satisfies Equation (A.2) in all of
$\mathbb{R}^{2}$.

###### Proof.

Since $v\in C^{1}(\mathbb{R}^{2})$, it follows that
$\nabla\bar{\psi}|_{\partial\Omega}=0$ and
$\nabla^{2}\bar{\psi}|_{\partial\Omega}=0$, which in particular implies that
$\Delta\bar{\psi}|_{\partial\Omega}=0$, and that there exist constants $c_{j}$
such that $\bar{\psi}|_{\Gamma_{j}}=c_{j}$. Here, $\\{\Gamma_{j}\\}_{j=1}^{J}$
denote the connected components of $\partial\Omega$. By Equation (A.2) and the
continuity of $f$, one must then have $f(c_{j})=0$ for all $1\leqslant
j\leqslant J$. Let us then denote by $\Omega_{1},\ldots,\Omega_{J}$ the
different connected components of $\mathbb{R}^{2}\setminus\overline{\Omega}$,
relabeling the boundary components if necessary so that
$\partial\Omega_{j}=\Gamma_{j}$, and define the $C^{2}$ function

$\psi:=\begin{cases}\bar{\psi}&\text{in }\Omega\,,\\\ c_{j}&\text{in
}\Omega_{j}\text{ for each }1\leqslant j\leqslant J\,.\end{cases}$

The functions $\Delta\psi$ and $f(\psi)$ are continuous on $\mathbb{R}^{2}$,
identically zero in $\mathbb{R}^{2}\backslash\overline{\Omega}$, and coincide
with $\Delta\bar{\psi}$ and $f(\bar{\psi})$, respectively, in $\Omega$, where
they satisfy (A.2) by hypothesis. This immediately implies that $\psi$ is a
solution to (A.2) in the whole of $\mathbb{R}^{2}$ as claimed. ∎

######  Theorem A.2.

Let $\psi\in{C^{3}(\mathbb{R}^{2})}$ satisfy a semilinear equation of the form
(A.2) in $\mathbb{R}^{2}$ with $f\in C(\mathbb{R})$. If $\nabla\psi$ is
compactly supported, then $\psi$ is locally radial. In other words, define:

$D=\\{x\in\mathbb{R}^{2}:\ \nabla\psi(x)\neq 0\\}.$

Then,

$D=\bigcup_{i\in I}A_{i}\,,$

where $I$ is a countable set and where $\\{A_{i}\\}_{i\in I}$ are disjoint
annuli or disks. Moreover, $\psi$ is radially symmetric in each set $A_{i}$.

###### Proof.

Since $\nabla\psi$ has compact support, we have that $\psi$ is constant
outside a bounded set. By adding a constant if necessary, we can assume that
$\psi$ itself has compact support. Let us also point out that $f(0)=0$. We now
consider separately two cases.

Case 1: $\psi$ does not change sign. The case where $\psi\geqslant 0$
immediately follows from [1]. Obviously so does the case $\psi\leqslant 0$,
simply by considering the function $-\psi$.

Case 2: $\psi$ changes sign. We start by proving that $\nabla\psi=0$ on the
zero set $\psi^{-1}(0)$. For this, we argue by contradiction, and assume that
$\psi(p)=0$ and $\nabla\psi(p)\neq 0$ for some $p\in\mathbb{R}^{2}$. Then, we
follow the ideas of [18, 19]: for some $\delta>0$, let
$\sigma:(-\delta,\delta)\to\mathbb{R}^{2}$ be the solution to the ODE

$\left\\{\begin{aligned} \sigma^{\prime}(t)&=\nabla\psi(\sigma(t))\,,\quad
t\in(-\delta,\delta)\,,\\\ \sigma(0)&=p\,.\end{aligned}\right.$

As $\nabla\psi(p)\neq 0$, we have that $g^{\prime}(0)>0$, where
$g=\psi\circ\sigma$. By taking $\delta>0$ smaller if necessary and suitable
$\varepsilon_{i}>0$, we have that
$g:(-\delta,\delta)\to(-\varepsilon_{1},\varepsilon_{2})$ is a diffeomorphism.
In this way, we can write $f$ as

$f|_{(-\varepsilon_{1},\varepsilon_{2})}=-\Delta\psi\circ\sigma\circ g^{-1}.$

Since $\psi$ is $C^{3}$ in $\Omega$, then
$f|_{(-\varepsilon_{1},\varepsilon_{2})}$ is a $C^{1}$ function. But this
ensures that the problem $\Delta\psi+f(\psi)=0$ has a unique continuation
property on the zero level set, which implies that $\psi\equiv 0$, which is a
contradiction. Hence, we conclude that $\nabla\psi(x)=0$ at every point
satisfying $\psi(x)=0$.

Next, let us define

$\psi_{+}(x):=\max\\{\psi(x),0\\}=\left\\{\begin{aligned} &\psi(x)&&\text{if
}x\in\Omega_{+}\,,\\\ &0&&\text{if }x\notin\Omega_{+}\,,\end{aligned}\right.$

where $\Omega_{+}:=\\{x\in\Omega:\ \psi(x)>0\\}$. Clearly, $\psi=0$ on
$\partial\Omega_{+}$, so it follows that $\nabla\psi=0$ on
$\partial\Omega_{+}$. This implies that $\psi_{+}\in C^{1}(\mathbb{R}^{2})$,
and then it follows that it is a classical nonnegative solution to (A.1). As
in Case 1, we then conclude that $\psi_{+}$ is locally radial.

The same argument can be applied to $\psi_{-}(x):=\max\\{-\psi(x),0\\}$, so
the theorem follows. ∎

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