Some remarks about the Morse index for convex Hamiltonian systems

Abstract

We investigate the (linearized) Morse index of solutions to Hamiltonian systems, with a focus on convex Hamiltonian functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions and give an estimate of their Morse index.

1 Introduction

This paper deals with elliptic systems of Hamiltonian type

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = H_v(x,u,v)& \text { in } \Omega , \\ -\Delta v = H_u (x,u, v) \qquad & \text { in } \Omega ,\\ u=v=0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega$ is a bounded smooth domain in $\mathbb {R}^N$, $N\ge 2$, and H is continuous with respect to $x\in \Omega$ and $C^2$ with respect to $(u,v)\in \mathbb {R}^2$. We are particularly interested in systems where the Hamiltonian function H is convex w.r.t. the variables u, v, that is

$$\begin{aligned} H_{uu} , \, H_{vv} \ge 0 \ \text { and } \ H_{uu} H_{vv} \ge \left( H_{uv}\right) ^2 \end{aligned}$$

(1.2)

on $\Omega \times \mathbb {R}^2$. Notice that the convexity of H implies that the system is cooperative.

The system (1.1) is the Euler–Lagrange equation of the action functional

$$\begin{aligned} \mathcal {I}(u,v) =&\int _{\Omega } \nabla u \nabla v dx -\int _{\Omega } H(x,u,v) dx , \end{aligned}$$

(1.3)

meaning that weak solutions are critical points for ${\mathcal {I}}$. The principal part

$$\begin{aligned} {\mathcal {S}}_o (u,v) = \int _{\Omega } \nabla u \nabla v dx \end{aligned}$$

is strongly indefinite, as $\left( H^1_0\right) ^2$ splits into the direct sum of the two sets

$$\begin{aligned} H^{\pm }:= \left\{ (\phi , \pm \phi ) \,: \, \phi \in H^1_0\right\} \end{aligned}$$

where $\pm {\mathcal {S}}_o$ is coercive, respectively. Various approaches have been proposed to get rid of this difficulty and establish existence of solutions in suitable functional and variational frameworks, together with qualitative and symmetry properties. We refer to the surveys [5, 8] and the references therein for a comprehensive treatment of this topic. We also mention [6] concerning least energy sign changing solutions. To our purpose we emphasize that every choice of the functional setting produces continuous solutions and justifies the following assumption

$$\begin{aligned} \begin{aligned} H_{uu}(x,u(x), v(x)), \ H_{vv} (x,u(x), v(x)), \ H_{uv}(x,u(x), v(x)) \\ \text{ are } \text{ bounded } \text{ on } \Omega \text{. } \end{aligned} \end{aligned}$$

(1.4)

Our focus is on the Morse index of solutions. A first remark is that the quadratic form associated to the second derivative of $\mathcal I$ is

$$\begin{aligned} {{\mathcal {Q}}}_{{\mathcal {I}}}(\phi ,\psi )&= \langle \mathcal {I}''(u,v)(\phi ,\psi ), (\phi ,\psi )\rangle \nonumber \\&= 2\int _{\Omega } \nabla \phi \nabla \psi dx - \int _{\Omega } D^2 H(\phi ,\psi ) \!\cdot \! (\phi ,\psi ) dx \le 2\int _{\Omega } \nabla \phi \nabla \psi dx .\nonumber \\ \end{aligned}$$

(1.5)

by the convexity of H (here and henceforth we write $D^2 H$ meaning the Hessian matrix of $H(x,\cdot )$ with respect to the variables $(u,v)\in \mathbb {R}^2$). So, ${{\mathcal {Q}}}_{{\mathcal {I}}}$ is negative defined on the half-space $H^-$ and the “natural” Morse index of any weak solution, meaning the maximal dimension of a subspace where $-{\mathcal {I}}''$ is coercive, is not finite. Several notions have been proposed in the literature to overcome this problem and define a meaningful finite index. Among others, we mention the relative Morse index proposed by Abbondandolo [1], which measures the relative dimension of the negative eigenspace of ${\mathcal {I}}''$ with respect to $H^-$, and the reduced Morse index, which computes the standard Morse index of a reduced functional, corresponding to a scalar equation. The reduced functional and Morse index have been used, for instance, to estimate the relative Morse index from below and obtain a Liouville type result [10], and to construct multiple spike solutions [11]. A different approach consists in looking at the linearized Morse index, which is not directly related to the action functional ${\mathcal {I}}$, but provides a finite index which can be used, for instance, to prove that least energy solutions are foliated Schwartz symmetric, see [7].

In this paper we point out some general properties of the linearized Morse index for Hamiltonian systems, with an emphasis on the convex case (1.2). In Sect. 2, after recalling the definitions and basic properties, we show that the computation of the linearized Morse index reduces to the investigation of some scalar eigenvalue problems.

Proposition 1.1

Let (u, v) be a solution to (1.1) satisfying (1.4). Its linearized Morse index is the sum of the number (counting multiplicity) of negative eigenvalues of two scalar problems

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \phi + a \phi = \mu \phi \quad & \text { in } \Omega , \\ \phi =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.6)

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \psi + b\psi = \nu \psi \quad & \text { in } \Omega , \\ \psi =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.7)

where $\displaystyle a(x):=-\frac{1}{2}\big (H_{uu}(x,u(x),v(x)) + H_{vv}(x,u(x),v(x)) \big ) - H_{uv}(x,u(x),v(x))$,

and $\displaystyle b(x):=\frac{1}{2}\big (H_{uu}(x,u(x),v(x)) + H_{vv}(x,u(x),v(x)) \big ) - H_{uv}(x,u(x),v(x))$.

Moreover the negative eigenspace is spanned by functions of type $(\phi ,\phi )$ where $\phi$ solves (1.6), and $(\psi ,-\psi )$ where $\psi$ solves (1.7).

Under the convexity assumption (1.2), the linearized Morse index is equal to the number (counting multiplicity) of negative eigenvalues of (1.6), and the negative eigenspace is contained in $H^+$.

Next, in Sect. 3, we focus on radial solutions and give a formula for their Morse index which makes use of the notion of singular eigenvalues and extends to Hamiltonian systems the analogous formula for scalar equations, see [3]. For convex problems, we obtain the following

Proposition 1.2

Consider the system (1.1) with $\Omega$ radially symmetric and $H=H(|x|, u, v)$ satisfying (1.2). If (u, v) is a radial solution satisfying (1.4), then its linearized Morse index is given by

$$\begin{aligned} \mathop {\text {m}_{{\text {lin}}}}(u,v) = {\mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}} +\sum \limits _{k =1}^{\mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}} \sum \limits _{j=0}^{M_k} N_j, \end{aligned}$$

(1.8)

where $\mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}$ stands for the radial linearized Morse index, $N_j=\frac{(N+2j-2)(N+j-3)!}{(N-2)!j!}$,

$M_k= \min \left\{ n\in {\mathbb {N}}: n\ge \sqrt{\left( \frac{N-2}{2}\right) ^2-\widehat{\Lambda }^{{\text {rad}}}_k}-\frac{N}{2} \right\}$,

and $\widehat{\Lambda }^{{\text {rad}}}_k$ are the singular radial eigenvalues, characterized by

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \xi '\right) '- r^{N-1} a \xi = r^{N-3} \widehat{\Lambda }^{{\text {rad}}}\xi , \\ \xi \in \widehat{H}^1_{0,{\text {rad}}}.\end{array}\right. } \end{aligned}$$

In Sect. 4, we deal with a class of strongly coupled Hamiltonian systems with $H=H(u,v)$ and

$$\begin{aligned} H_u/u> 0&\text { and } H_v /v > 0 , \end{aligned}$$

(H1)

$$\begin{aligned} H_{uu}> \frac{H_u - H_{uv} v}{u}> 0&\text { and } H_{vv}> \frac{H_v - H_{uv} u}{v} > 0 , \end{aligned}$$

(H2)

for every $u, v \ne 0$. These assumptions are satisfied, for instance, by the Lane-Emden system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = |v|^{q-1} v& \text { in } \Omega , \\ -\Delta v = |u|^{p-1}u \qquad & \text { in } \Omega ,\\ u=v=0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

for $p, q >1$ and, more generally, by every fully coupled system for which

$$\begin{aligned}H(u,v)= F(v)+G(u)\end{aligned}$$

with $F''(v)> F'(v)/v >0$ and $G''(u)> G'(u)/u > 0$ for every $u, v \ne 0$.

We prove the following estimate.

Theorem 1.3

Consider the system (1.1) with $\Omega$ radially symmetric and $H=H(u, v)$ satisfying H1 and H2. If (u, v) is a classical radial solution and m is the number of nodal zones of u and v, then

$$\begin{aligned} \mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}\ge m \end{aligned}$$

(1.9)

and

$$\begin{aligned} \mathop {\text {m}_{{\text {lin}}}}(u,v) \ge \mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}+(m-1) N \ge m+(m -1) N. \end{aligned}$$

(1.10)

Theorem 1.3 is a consequence of the just stated general properties, but also of the strong coupling between u and v yielded by hypothesis H1. We think that this fact can be of some interest for itself, therefore we give right now a specific statement.

Proposition 1.4

Let $\Omega$ be radially symmetric, $H=H(|x|, u, v)$ continuous w.r.t. x and satisfying H1 for almost every $x\in \Omega$, and (u, v) a classical radial solution to (1.1). Then the following holds

• A. If one between u and v is strictly positive (or negative), then both u and v are strictly positive (or negative), and have only one critical point, which coincides with the origin if $\Omega$ is a ball.

• B. If one between u and v has exactly $m\ge 2$ nodal zones and $m-1$ internal zeros, then both u and v have exactly m nodal zones and $m-1$ internal zeros, they have the same sign in their first nodal zone (and therefore in the following ones), and have exactly one critical point inside each nodal zone. Moreover the nodal zones are intertwined, meaning that the $i^{th}$ nodal zones of u and v have a non-empty intersection, which contains both the critical points of u and v.

2 The linearized Morse index and the symmetric eigenvalues

The notion of linearized Morse index is focused on the linearization of problem (1.1) near a solution (u, v), i.e.

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi = H_{uv} \phi + H_{vv}\psi & \text { in } \Omega , \\ -\Delta \psi = H_{uu}\phi + H_{uv}\psi \qquad & \text { in } \Omega ,\\ \phi =\psi =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(2.1)

and on the related bilinear and quadratic form

$$\begin{aligned} {{\mathcal {B}}}_{\text {lin}}\left( (\phi ,\psi ), (\xi ,\eta )\right)&= \int _{\Omega } \left( \nabla \phi \nabla \xi + \nabla \psi \nabla \eta \right) dx \nonumber \\&\quad - \int _{\Omega } \left( H_{uu} \phi \eta +H_{vv} \psi \xi +H_{uv}\left( \phi \xi +\psi \eta \right) \right) dx, \end{aligned}$$

(2.2)

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )&= \int _{\Omega } \left( |\nabla \phi |^2 + |\nabla \psi |^2\right) dx\nonumber \\&\quad - \int _{\Omega } \left( (H_{uu}+H_{vv}) \phi \psi + H_{uv} ( \phi ^2+\psi ^2 \right) dx, \end{aligned}$$

(2.3)

defined respectively on $\left( H^1_0\right) ^2\times \left( H^1_0\right) ^2$ and $\left( H^1_0\right) ^2$.

The linearized Morse index is the maximal dimension of a subspace of $\left( H^1_0\right) ^2$ where the quadratic form ${{\mathcal {Q}}}_{\text {lin}}$ is negative defined. We will denote it by $\mathop {\text {m}_{{\text {lin}}}}(u,v)$, henceforth.

Let us remark that ${{\mathcal {Q}}}_{\text {lin}}$ differs from the related quadratic form ${{\mathcal {Q}}}_{{\mathcal {I}}}$ introduced in (1.5). Interestingly,

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\phi ,-\phi ) = 2\int _{\Omega } |\nabla \phi |^2 dx + \int _{\Omega } \left( H_{uu}+H_{vv} - 2H_{uv} \right) \phi ^2 dx = - {{\mathcal {Q}}}_{{\mathcal {I}}}(\phi ,-\phi ), \nonumber \\ \end{aligned}$$

(2.4)

while in the orthogonal complement $H^{+}$ the two quadratic forms coincide:

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\phi ,\phi ) = 2\int _{\Omega } |\nabla \phi |^2 dx - \int _{\Omega } \left( H_{uu}+H_{vv}+ 2H_{uv} \right) \phi ^2 dx = {{\mathcal {Q}}}_{{\mathcal {I}}}(\phi ,\phi ). \nonumber \\ \end{aligned}$$

(2.5)

Hence the notion of linearized Morse index can provide a finite index which is meaningful also from the point of view of the action functional. In order to take advantage of the theory of bounded self-adjoint operators, however, one can not rely on the bilinear form ${{\mathcal {B}}}_{\text {lin}}$, which is not symmetric, but instead on its symmetrization

$$\begin{aligned} \begin{aligned} {{\mathcal {B}}}_{\text {sym}}\left( (\phi ,\psi ), (\xi ,\eta )\right) = \int _{\Omega } \left( \nabla \phi \nabla \xi + \nabla \psi \nabla \eta \right) dx \\ - \frac{1}{2}\int _{\Omega } \left( \Delta H (\phi \eta + \psi \xi ) + 2H_{uv} ( \phi \xi +\psi \eta ) \right) dx,\end{aligned} \end{aligned}$$

(2.6)

which provides the same quadratic form as

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi ) ={{\mathcal {B}}}_{\text {sym}}\left( (\phi ,\psi ), (\phi ,\psi )\right) . \end{aligned}$$

The standard spectral decomposition theory applies provided that there exists $k\in \mathbb {R}$ such that the continuous, symmetric bilinear form ${{\mathcal {B}}}_{\text {sym}}+ k \langle , \rangle _{(L^2)^2}$ is coercive. It certainly holds true if the solution (u, v) fulfills (1.4). In that way there exists a non-decreasing sequence of eigenvalues $\Lambda _n^{{\text {sym}}}\rightarrow +\infty$. The first eigenvalue is the minimum of the Rayleigh quotient

$$\begin{aligned} {\Lambda }^{{\text {sym}}}_1:= \inf \left\{ \dfrac{{{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )}{\Vert \phi \Vert _2^2+\Vert \psi \Vert _2^2}\, : \, \phi ,\psi \in H^1_0 \right\} \end{aligned}$$

(2.7)

which is attained by a nontrivial function $(\phi _1,\psi _1)$ (an eigenfunction) which satisfies

$$\begin{aligned} {{\mathcal {B}}}_{\text {sym}}\left( (\phi _1,\psi _1), (\xi ,\eta )\right) =\Lambda _1^{{\text {sym}}} \int _{\Omega } \left( \phi _1\xi +\psi _1\eta \right) dx \end{aligned}$$

for every $\xi , \eta \in H^1_0$, i.e. solves in weak sense

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi - H_{uv} \phi - \frac{1}{2}\Delta H \psi = {\Lambda }\phi \qquad & \text { in } \Omega , \\ -\Delta \psi -\frac{1}{2}\Delta H \phi - H_{uv} \psi = {\Lambda }\psi & \text { in } \Omega , \\ \phi =\psi =0 & \text { on } \partial \Omega \end{array}\right. } \end{aligned}$$

(2.8)

for ${\Lambda }={\Lambda }^{{\text {sym}}}_1$. Conversely, any weak solution of (2.8) with ${\Lambda }={\Lambda }^{{\text {sym}}}_1$ realizes the minimum in (2.7). Next, the following eigenvalues are defined by

$$\begin{aligned} {\Lambda }^{{\text {sym}}}_{k}:= \inf \left\{ \dfrac{{{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )}{\Vert \phi \Vert _2^2+\Vert \psi \Vert _2^2}\, : \, \phi ,\psi \in H^1_0 \, \int _{\Omega } \left( \phi \phi _n +\psi \psi _n \right) dx = 0 \text { if } n=1,\dots k-1 \right\} \end{aligned}$$

with $k\ge 2$, and are attained by eigenfunctions $(\phi _k,\psi _k)$ which solve (2.8) for ${\Lambda }={\Lambda }^{{\text {sym}}}_k$. Eventually, the family of eigenfunctions forms an orthogonal basis for $\left( H^1_0\right) ^2$ and induces a splitting

$$\begin{aligned} \left( H^1_0\right) ^2 = W^-\oplus W^+\oplus W^0.\end{aligned}$$

Here $W^{+}$, $W^-$ and $W^0$ are the spaces spanned by the eigenfunctions $(\phi _k,\psi _k)$ with $\Lambda _k^{\text {sym}} <0$ (resp., $>0$, $=0$), so that ${{\mathcal {Q}}}_{\text {lin}}$ is negative (resp., positive) defined on $W^{-}$ ( resp, $W^+$). Moreover $W^-$ has finite dimension because $\Lambda _k^{{\text {sym}}}\rightarrow +\infty$. Summing up, the linearized Morse index of a solution (u, v) is the dimension of $W^-$, i.e. the number of negative eigenvalues of problem (2.8), each counted with multiplicity. We refer to [7] for rigorous proofs and more details.

2.1 Reduction to a scalar eigenvalue problem

Projecting the eigenvalue problem (2.8) into $H^+\oplus H^-$ gives rise to a decoupled system. If (u, v) is a solution to (1.1), we introduce the functions

$$\begin{aligned} a(x)=&-\frac{1}{2} \left( H_{uu}(x,u(x),v(x)) + H_{vv}(x,u(x),v(x))\right) - H_{uv}(x,u(x),v(x)) , \end{aligned}$$

(2.9)

$$\begin{aligned} b(x)=&\frac{1}{2}\big (H_{uu}(x,u(x),v(x)) + H_{vv}(x,u(x),v(x)) \big ) - H_{uv}(x,u(x),v(x)) , \nonumber \\ \end{aligned}$$

(2.10)

and the scalar eigenvalue problems

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \xi + a \xi = \mu \xi \quad & \text { in } \Omega , \\ \xi =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(2.11)

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \eta + b\eta = \nu \eta \quad & \text { in } \Omega , \\ \eta =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(2.12)

It is very easy to prove the following interesting property.

Lemma 2.1

Let (u, v) a weak solution to (1.1) satisfying (1.4). A number $\Lambda$ is a symmetric eigenvalue if and only if it is an eigenvalue for at least one between (2.11) or (2.12), and the related eigenfunctions of (2.8) are linear combinations of functions of type $(\xi ,\xi )$ and/or $(\eta ,-\eta )$, respectively. In particular the multiplicity of $\Lambda$ as an eigenvalue of (2.8) is the sum of its multiplicity according to (2.11) and (2.12), counting 0 if $\Lambda$ is not an eigenvalue for (2.11), or respectively (2.12).

Proof

For $(\phi ,\psi )\in \left( H^1_0\right) ^2$ we write $\xi = \frac{1}{2} (\phi +\psi )$ and $\eta =\frac{1}{2}(\phi -\eta )$, so that $P_{H^+}(\phi ,\psi )=(\xi ,\xi )$, $P_{H^-}(\phi ,\psi )=(\eta ,-\eta )$, and $(\phi ,\psi )= (\xi +\eta , \xi -\eta )$. By a trivial computation, the system (2.8) translates into the decoupled system

$$\begin{aligned}{\left\{ \begin{array}{ll} -\Delta \xi + a \xi = \Lambda \xi \qquad & \text { in } \Omega , \\ -\Delta \eta +b \eta = \Lambda \eta & \text { in } \Omega , \\ \xi =\eta =0 & \text { on } \partial \Omega \end{array}\right. } \end{aligned}$$

and the claim follows. $\square$

Proof of Proposition 1.1

The first statement, concerning any Hamiltonian function, follows readily by Lemma 2.1 and the characterization of the linearized Morse index by means of the symmetric eigenvalues.

It remains to check that the convexity assumption (1.2) implies that $b\ge 0$ pointwise, so that the eigenvalue problem (2.12) only has nonnegative eigenvalues.

If $H_{uu}=0$, then $H_{uv}=0$ and $b=H_{vv} \ge 0$. Otherwise $H_{uu}> 0$, and

$$\begin{aligned} 2 H_{uu} b = \left( H_{uu}\right) ^2 + H_{uu} H_{vv} - 2H_{uu} H_{uv} \underset{(1.2)}{\ge }\left( H_{uu} - H_{uv}\right) ^2 \ge 0 . \end{aligned}$$

$\square$

In particular in the convex case, Proposition 1.1 shows that the negative eigenspace of the quadratic form $\mathcal Q_{{\text {lin}}}$, i.e. $W^-$, is contained in $H^+$. Moreover, as far as negative eigenvalues are concerned, the symmetric eigenfunctions inherit all the well-known properties of the eigenfunctions for compact scalar problems: the first eigenvalue is simple and has a positive eigenfunction, and eigenfunctions related to the $n^{th}$ eigenvalues have at most n nodal zones (component-wise).

2.2 Linearized and relative Morse index

Here we briefly recall the notion of relative Morse index introduced by Abbondandolo [1] for Hamiltonian systems, and compare it with the linearized Morse index.

If $(u,v) \in \left( H^1_0\right) ^2$ is a weak solution of (1.1), we write ${\mathcal {S}}$ for the self-adjoint realization of the symmetric bilinear form defined by ${\mathcal {I}}''(u,v)$ on $H^1_0$, i.e.

$$\begin{aligned} \langle {\mathcal {S}} (\phi ,\psi ), (\xi ,\eta )\rangle = \langle (\phi , \psi ) , {\mathcal {S}} (\xi , \eta )\rangle = \langle {\mathcal {I}}''(u,v)(\phi ,\psi ), (\xi ,\eta )\rangle \\ = \int _{\Omega }\left( \nabla \phi \nabla \eta +\nabla \psi \nabla \xi \right) dx - \int _{\Omega } D^2H(u,v) (\phi ,\psi )\cdot (\xi ,\eta ) dx . \end{aligned}$$

${\mathcal {S}}$ is the difference between the strongly indefinite, but invertible, operator

$$\begin{aligned} \langle {\mathcal {S}}_o(\phi ,\psi ) , (w,z)\rangle =&\int _{\Omega } \left( \nabla \phi \nabla z + \nabla \psi \nabla w \right) dx , \end{aligned}$$

and the operator

$$\begin{aligned} \langle {\mathcal {S}}_{H}(\phi ,\psi ) , (w,z)\rangle =&\int _{\Omega } D^2H(u,v) (\phi ,\psi )\!\cdot \!(w,z) dx , \end{aligned}$$

which is compact under suitable assumptions on the Hamiltonian H (see, for instance, [1, Proposition 3.2.1]). So ${\mathcal {S}}$ is a Fredholm operator and determines an unique $\mathcal S$-invariant orthogonal splitting

$$\begin{aligned} \left( H^1_0\right) ^2 = V^- \oplus V^+ \oplus {\textrm{Ker}}({\mathcal {S}}) \end{aligned}$$

in such a way that ${{\mathcal {Q}}}_{{\mathcal {I}}}$ is negative (resp., positive) defined on $V^-$ (resp., $V^+$). As mentioned in the Introduction, the dimension of $V^-$, i.e. the “natural” Morse index, is infinite for every solution (u, v) under assumption (1.2).

The relative Morse index is the relative dimension of $V^-$ with respect to $H^-$, that is

$$\begin{aligned} {\textrm{m}}_{H^-}(u,v):={\dim }(V^-, H^-):={\dim }(V^-\cap H^+) - {\dim }((V^-)^{\perp }\cap H^-). \nonumber \\ \end{aligned}$$

(2.13)

The convexity assumption (1.2) implies that $(V^-)^{\perp }\cap H^-=\{(0,0)\}$, then in this case (2.13) simplifies into

$$\begin{aligned} {\textrm{m}}_{H^-}(u,v)={\dim }(V^-\cap H^+). \end{aligned}$$

(2.14)

In any case

$$\begin{aligned} {\mathrm m}_{H^-} (u,v) \le {\mathrm m}_{{\text {lin}}} (u,v). \end{aligned}$$

(2.15)

Indeed by definition ${\mathrm m}_{H^-} (u,v) \le {\dim }(V^-\cap H^+)$, and obviously $Q_{{\mathcal {I}}}$ is negative on $V^-\cap H^+$. By (2.5), also ${{\mathcal {Q}}}_{\text {lin}}$ is negative, hence ${\dim }(V^-\cap H^+)\le \mathop {\text {m}_{{\text {lin}}}}(u,v)$.

We explicitly remark that, although in the convex case ${{\mathcal {Q}}}_{{\mathcal {I}}}$ is negative on the subspaces $W^-\subset H^+$ and $H^-$, one can not infer that ${{\mathcal {Q}}}_{{\mathcal {I}}}$ is negative on $W^-\oplus H^-$. Indeed ${{\mathcal {Q}}}_{\text {lin}}(\xi +\eta , \xi -\eta ) = {{\mathcal {Q}}}_{\text {lin}}(\xi ,\xi )+ {{\mathcal {Q}}}_{\text {lin}}(\eta , -\eta )$, but

$$\begin{aligned} {{\mathcal {Q}}}_{{\mathcal {I}}}(\xi +\eta , \xi -\eta )&= {{\mathcal {Q}}}_{{\mathcal {I}}}(\xi ,\xi )+ {{\mathcal {Q}}}_{{\mathcal {I}}}(\eta , -\eta )+2\langle {{\mathcal {I}}}''(\xi ,\xi ), (\eta ,-\eta )\rangle \\&={{\mathcal {Q}}}_{{\mathcal {I}}}(\xi ,\xi )+ {{\mathcal {Q}}}_{{\mathcal {I}}}(\eta , -\eta )+ 2 \int _\Omega \left( H_{vv}-H_{uu}\right) \xi \eta dx. \end{aligned}$$

3 Radial solutions: a decomposition formula for the Morse index

In the present section we take that $\Omega$ is radially symmetric (a ball or a spherical shell) and $H=H(|x|, u, v)$. If (u, v) is a radial solution, one can look at the restriction of the quadratic form ${{\mathcal {Q}}}_{\text {lin}}$ to the space of radial functions and define the radial linearized Morse index $\mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}$ as the maximal dimension of a subspace of $\left( H^1_{0,{\text {rad}}}\right) ^2$ where ${{\mathcal {Q}}}_{\text {lin}}$ is negative, and the radial symmetric eigenvalues ${\Lambda }_{{\text {rad}}}^{{\text {sym}}}$ by minimizing the Rayleigh quotient first on $\left( H^1_{0,{\text {rad}}}\right) ^2$ and then on its subspaces. The eigenvalues are characterized by means of a radial differential problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \phi '\right) '- r^{N-1} \left( H_{uv} \phi + \frac{1}{2}\Delta H \psi \right) = r^{N-1} \Lambda \phi , \\ -\left( r^{N-1} \psi '\right) '- r^{N-1} \left( \frac{1}{2}\Delta H \phi + H_{uv} \psi \right) = r^{N-1} \Lambda \psi , \\ \phi , \psi \in H^1_{0,{\text {rad}}}, \end{array}\right. } \end{aligned}$$

(3.1)

that can be projected onto $H^{\pm }_{{\text {rad}}}$ giving rise to the decoupled system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \xi '\right) '+ r^{N-1} a \xi = r^{N-1}\Lambda \xi , \\ -\left( r^{N-1} \eta '\right) '+ r^{N-1} b \eta = r^{N-1}\Lambda \eta , \\ \xi , \eta \in H^1_{0,{\text {rad}}} \end{array}\right. } \end{aligned}$$

with a and b as in (2.9), (2.10). In particular, the analogous of Proposition 1.1 holds.

Proposition 3.1

Let the set $\Omega$ be radially symmetric, $H=H(|x|, u, v)$, and take (u, v) a continuous radial solution of (1.1). Under assumption (1.2), the radial linearized Morse index is equal to the number of negative eigenvalues of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \xi '\right) '+ r^{N-1} a \xi = r^{N-1}\mu \xi , \\ \xi \in H^1_{0,{\text {rad}}}, \end{array}\right. } \end{aligned}$$

(3.2)

each counted with multiplicity. Moreover the negative radial symmetric eigenvalues coincide with the negative eigenvalues of (3.2), and the related eigenfunction are $(\xi ,\xi )$, where $\xi \in H^1_{0,{\text {rad}}}$ solves (3.2) with $\mu =\Lambda ^{{\text {rad}}}_{{\text {sym}}}$.

We mention in passing that (3.2) is a Sturm-Liouville problem, hence Picone’s comparison Principle applies and gives that the negative radial eigenvalues are simple, and if the $n^{th}$ radial eigenvalue is negative, then its eigenfunctions have exactly n nodal zones.

3.1 Singular symmetric eigenvalues

The notion of singular eigenvalues, an effective tool introduced to deal with scalar equations, also applies to systems. Let us briefly recall the definitions and main properties, and refer to [3] and the references therein for a detailed account. Let

$$\begin{aligned} \widehat{L}= \left\{ \phi \in L^2 \,: \, \int _{\Omega } \frac{1}{|x|^2} \phi ^2dx <\infty \right\} , \end{aligned}$$

endowed with the scalar product

$$\begin{aligned} \langle \phi ,\eta \rangle _{\widehat{L}}:= \int _{\Omega } \frac{1}{|x|^2} \phi \eta dx \end{aligned}$$

and the induced norm

$$\begin{aligned} \Vert \phi \Vert _{\widehat{L}}:= \left( \langle \phi ,\phi \rangle _{\widehat{L}}\right) ^{\frac{1}{2}}. \end{aligned}$$

Next, let $\widehat{H}^1_0:= H^1_0 \cap \widehat{L}$ and $\widehat{H}^1_{0,{\text {rad}}}:= H^1_{0,{\text {rad}}} \cap \widehat{L}$.

A number $\widehat{\Lambda }$ is said a singular symmetric eigenvalue if the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi - H_{uv} \phi - \frac{1}{2}\Delta H \psi = \frac{1}{|x|^2}{\Lambda }\phi \qquad & \text { in } \Omega , \\ -\Delta \psi -\frac{1}{2}\Delta H \phi - H_{uv} \psi = \frac{1}{|x|^2}{\Lambda }\psi & \text { in } \Omega , \\ \phi , \psi \in \widehat{H}^1_0 & \end{array}\right. } \end{aligned}$$

(3.3)

has a nontrivial weak solution, i.e. $(\phi ,\psi ) \in \left( \widehat{H}^1_0\right) ^2{\setminus }\{(0,0)\}$ with

$$\begin{aligned} {{\mathcal {B}}}_{\text {sym}}\left( (\phi ,\psi ), (\xi ,\eta )\right) = \widehat{\Lambda }\left( \langle \phi ,\eta \rangle _{\widehat{L}} + \langle \psi ,\xi \rangle _{\widehat{L}} \right) \end{aligned}$$

for every $\xi , \eta \in \widehat{H}^1_0$. In that case, any function $(\phi ,\psi )$ solving (3.3) is called a singular eigenfunction related to $\widehat{\Lambda }$.

The same arguments of the scalar case (see, for instance, [3, Propositions 3.1, 3.2 and Lemma 3.3]) show that if

$$\begin{aligned} \widehat{\Lambda }_1:=\inf \left\{ \dfrac{{{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )}{\Vert \phi \Vert _{\widehat{L}}^2 + \Vert \psi \Vert _{\widehat{L}}^2 }\,: \, \phi ,\psi \in \widehat{H}^1_0 \right\} < \left( \frac{N-2}{2}\right) ^2, \end{aligned}$$

then it is attained by a nontrivial function $(\phi ,\psi )$ which solves (3.3) with ${\Lambda }=\widehat{\Lambda }_1$, so that $\widehat{\Lambda }_1$ and $(\phi ,\psi )$ are respectively an eigenvalue and a related eigenfunction. Next we can define iteratively

$$\begin{aligned} \widehat{\Lambda }_{n}:= \inf \left\{ \dfrac{{{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )}{\Vert \phi \Vert _{\widehat{L}}^2 + \Vert \psi \Vert _{\widehat{L}}^2} \,: \, \phi ,\psi \in \widehat{H}^1_0, \, \langle \phi ,\phi _k\rangle _{\widehat{L}}+ \langle \psi ,\psi _k\rangle _{\widehat{L}} =0 \text { for } k=1,\dots n-1 \right\} . \end{aligned}$$

As far as $\widehat{\Lambda }_{n} < \left( \frac{N-2}{2}\right) ^2$, it is attained by a nontrivial function $(\phi ,\psi )$ which solves (3.3) and then $\widehat{\Lambda }_{n}$ is a singular eigenvalue according to the previous definition.

Conversely, if $\left( \Lambda , \phi ,\psi \right) \in \left( -\infty , \left( \frac{N-2}{2}\right) ^2\right) \times \left( \widehat{H}^1_0\right) ^2{\setminus }\{(0,0)\}$ solves (3.3), then there exists n such that $\Lambda = \widehat{\Lambda }_{n}$ according to (3.1).

When the set $\Omega$ is radially symmetric, $H=H(|x|, u, v)$, and (u, v) is a radial solution, one can define the singular radial symmetric eigenvalues and eigenfunctions as solutions to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \phi '\right) '- r^{N-1} \left( H_{uv} \phi + \frac{1}{2}\Delta H \psi \right) = r^{N-3} \widehat{\Lambda }^{{\text {rad}}} \phi , \\ -\left( r^{N-1} \psi '\right) '- r^{N-1} \left( \frac{1}{2}\Delta H \phi + H_{uv} \psi \right) = r^{N-3} \widehat{\Lambda }^{{\text {rad}}} \psi , \\ \phi , \psi \in \widehat{H}^1_{0,{\text {rad}}}, \end{array}\right. } \end{aligned}$$

(3.4)

or as minima of the respective Rayleigh quotients, provided that the minima are below the value $\left( \frac{N-2}{2}\right) ^2$. In particular

$$\begin{aligned} \widehat{\Lambda }_{n}^{{\text {rad}}}:= \min \left\{ \max \limits _{(\phi ,\psi ) \in V_n\setminus \{0\} } \dfrac{{{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )}{\Vert \phi \Vert _{\widehat{L}}^2 + \Vert \psi \Vert _{\widehat{L}}^2}: \, V_n \text{ n-dim. } \text{ subspace } \text{ of } \left( \widehat{H}^1_{0,{\text {rad}}} \right) ^2 \right\} . \nonumber \\ \end{aligned}$$

(3.5)

An important property of singular eigenvalues is the following.

Proposition 3.2

The linearized Morse index and the radial linearized Morse index are, respectively, the number of negative singular symmetric eigenvalues and of negative singular radial symmetric eigenvalue, each counted with multiplicity.

The proof is exactly the same of [9], and uses the continuity of the eigenvalues with respect to the set $\Omega$.

3.2 Decomposition formula

The advantage in dealing with singular eigenvalues is that they decompose in a radial and an angular part. To be more specific, let us introduce some notations. We write $Y_j$ for the Spherical Harmonics, i.e. the eigenfunctions of the Laplace-Beltrami operator on the sphere $S^{N-1}$. Of course the operator $\left( -\Delta _{S^{N-1}}\right) ^{-1}$ is positive compact and self-adjoint in $L^2(S^{N-1})$ and so it admits a sequence of eigenvalues $0={\lambda }_1<{\lambda }_2\le {\lambda }_j$ and eigenfunctions $Y_j(\theta )$ which form an Hilbert basis for $L^2(S^{N-1})$. Namely they satisfy

$$\begin{aligned} -\Delta _{S^{N-1}}Y_j(\theta )={\lambda }_j Y_j(\theta )\ \text { for }\theta \in S^{N-1}. \end{aligned}$$

(3.6)

The eigenvalues ${\lambda }_j$ are given by the well known values

$$\begin{aligned} {\lambda }_j:=j(N+j-2)\ \ \text { for }j=0,1,\dots \end{aligned}$$

(3.7)

each of which has multiplicity

$$\begin{aligned} N_j:={\left\{ \begin{array}{ll} 1 & \text { when }j=0,\\ \frac{(N+2j-2)(N+j-3)!}{(N-2)!j!} & \text { when }j\ge 1. \end{array}\right. } \end{aligned}$$

(3.8)

The same arguments of [3, Proposition 4.1] prove the following

Proposition 3.3

Take $\Omega$ a radially symmetric domain, $H=H(|x|, u, v)$, and (u, v) a radial solution to (1.1) satisfying (1.4). The singular symmetric eigenvalues $\widehat{\Lambda }_n<\left( \frac{N-2}{2}\right) ^2$ can be decomposed in radial and angular part as

$$\begin{aligned} \widehat{\Lambda }_n=\widehat{\Lambda }_k^{{\text {rad}}}+{\lambda }_j \quad \text { for some} k\ge 1\text { and }j\ge 0. \end{aligned}$$

(3.9)

Conversely, if a singular radial symmetric eigenvalue $\widehat{\Lambda }_k^{{\text {rad}}}$ according to (3.4) is such that $\widehat{\Lambda }_k^{{\text {rad}}}< \left( \frac{N-2}{2}\right) ^2 -{\lambda }_j$ for some $j\ge 0$, then $\widehat{\Lambda }_n$ given by (3.9) is a singular symmetric eigenvalue.

Moreover the set of solutions to (3.3) with $\widehat{\Lambda }= \widehat{\Lambda }_n$ is spanned by functions of type

$$\begin{aligned} (Y_j(\theta )\phi _k(r),Y_j(\theta )\psi _k(r)), \end{aligned}$$

(3.10)

where $Y_j$ solves (3.6) and $(\phi _k,\psi _k)$ solves (3.4).

Eventually, Propositions 3.2 and 3.3 yield an useful formula to compute the Morse index of a radial solution (u, v). If $\widehat{\Lambda }^{{\text {rad}}}_k$ is a radial singular eigenvalue, we write $\textsf {m}_k$ for its multiplicity, i.e. for the dimension of the solution set of (3.4), so that

$$\begin{aligned} \mathop {\text {m}_{{\text {lin}}}^{{\text {rad}}}}(u,v)= \sum \limits _{k \,: \, \widehat{\Lambda }^{{\text {rad}}}_k <0} \textsf {m}_k. \end{aligned}$$

Making also use of the explicit formulas (3.7), (3.8), one ends up with

Corollary 3.4

Take $\Omega$ a radially symmetric domain and $H=H(|x|, u, v)$; if (u, v) is a radial solution to (1.1) satisfying (1.4), then

$$\begin{aligned} \mathop {\text {m}_{{\text {lin}}}}(u,v) = \sum \limits _{k \, : \, \widehat{\Lambda }^{{\text {rad}}}_k <0} \sum \limits _{j=0}^{M_k} \textsf {m}_k N_j , \end{aligned}$$

(3.11)

where $M_k= \min \left\{ n\in {\mathbb {N}}: n\ge \sqrt{\left( \frac{N-2}{2}\right) ^2-\widehat{\Lambda }^{{\text {rad}}}_k}-\frac{N}{2} \right\}$ and $N_j$ as in (3.8).

Next, also problems (3.3) and (3.4) transform into decoupled systems after projection onto $H^+$ and $H^-$. Let a and b the functions introduced in (2.9); it is easy to check the following interesting properties.

Lemma 3.5

A number $\Lambda <\left( \frac{N-2}{2}\right) ^2$ is a singular symmetric eigenvalue if and only if it is an eigenvalue for at least one between

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \xi + a \xi = \dfrac{1}{|x|^2}\mu \xi \qquad & \text { in } \Omega , \\ \xi =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(3.12)

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\Delta \eta +b \eta = \dfrac{1}{|x|^2}\nu \eta \qquad & \text { in } \Omega , \\ \eta =0 & \text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(3.13)

and the related eigenfunctions of (3.3) are linear combinations of functions of type $(\xi ,\xi )$ and/or $(\eta ,-\eta )$, respectively.

Under the convexity assumption (1.2), the linearized Morse index of any solution (u, v) is equal to the number of negative eigenvalues of the singular scalar problem (3.12), each counted with multiplicity.

If (u, v) is a radial solution, one can investigate the radial Morse index by means of the radial versions of (3.12), (3.12), that is

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\left( r^{N-1} \xi '\right) '- r^{N-1} a \xi = r^{N-3} \mu \xi , & \\ \xi \in \widehat{H}^1_{0,{\text {rad}}} & \end{array}\right. } \end{aligned}$$

(3.14)

$$\begin{aligned} & {\left\{ \begin{array}{ll} -\left( r^{N-1} \eta '\right) '- r^{N-1} b \eta = r^{N-3} \nu \eta , & \\ \eta \in \widehat{H}^1_{0,{\text {rad}}}.& \end{array}\right. } \end{aligned}$$

(3.15)

Even though the latter Sturm-Liouville problems are singular at $r=0$, the main comparison properties still hold (see [3, Sections 3,4]). Next lemma immediately follows.

Lemma 3.6

Consider the system (1.1) with $\Omega$ radially symmetric and $H=H(|x|, u, v)$. If (u, v) is a radial solution satisfying (1.4), then for every radial singular eigenvalue $\widehat{\Lambda }_k^{{\text {rad}}}<\left( \frac{N-2}{2}\right) ^2$ one of the following items applies:

• (i) either $\widehat{\Lambda }_k^{{\text {rad}}}$ is simple and the respective eigenfunction is $(\phi _k, \phi _k)$, where the scalar function $\phi _k$ solves (3.14) for $\mu =\widehat{\Lambda }_k^{{\text {rad}}}$,

• (ii) or $\widehat{\Lambda }_k^{{\text {rad}}}$ is simple and the respective eigenfunction is $(\psi _k, -\psi _k)$, where the scalar function $\psi _k$ solves (3.15) for $\nu =\widehat{\Lambda }_k^{{\text {rad}}}$,

• (iii) or $\widehat{\Lambda }_k$ has multiplicity two and the respective eigenspace is spanned by one eigenfunction of type (i) and one of type (ii).

Moreover under the convexity assumption (1.2) only item (i) may happen, so that every radial singular eigenvalue is simple and the respective eigenfunction has the form $(\phi _k,\phi _k)$, where $\phi _k$ has exactly k nodal zones. I

Inserting these properties into the representation formula (3.11) proves Proposition 1.2.

4 Strongly coupled problems

In the following we take $u, v \in H^1_{0,{\text {rad}}}$ a radial solution to (1.1), and we enforce the convexity assumption (1.2) by asking that

$$\begin{aligned} H_u/u> 0&\text { and } H_v /v > 0 , \end{aligned}$$

(H1)

$$\begin{aligned} H_{uu}> \frac{H_u - H_{uv} v}{u}> 0&\text { and } H_{vv}> \frac{H_v - H_{uv} u}{v} > 0 , \end{aligned}$$

(H2)

for every $u, v \ne 0$. Though Theorem 1.3 is proved in full only for H not depending on x, the preliminary properties, and in particular Proposition 1.4, hold also for $H=H(|x|, u, v)$ continuous w.r.t. x and satisfying H1 and H2 for almost every $x\in \Omega$. We introduce some notations before entering the details.

Notations 4.1

In the following $\Omega$ can be either a ball $\Omega =\{x\in \mathbb {R}^N: |x|<R\}$, either a spherical shell $\Omega =\{x\in \mathbb {R}^N: \delta<|x|<R\}$. We will use the notations

• $s_1<\dots s_{m-1} \text{ and } t_1<\dots t_{n-1}$ for the internal zero points of u and v, respectively,

• $s_0=t_0=0$ or $s_0=t_0=\delta$ according if $\Omega$ is a ball or a spherical shell, and $s_m=t_n=R$,

• $\sigma _0<\sigma _1<\dots \sigma _{m-1} \text{ and } \tau _0< \tau _1<\dots \tau _{n-1}$ for the maximum/minimum points in each nodal zone of u and v, respectively.

We also point out that in radial coordinates (1.1) reads as

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} u'\right) '= r^{N-1} H_v(r,u,v), \\ -\left( r^{N-1} v'\right) '= r^{N-1} H_u(r,u,v). \end{array}\right. } \end{aligned}$$

(4.1)

Let us describe the profile of any radial solution and prove Proposition 1.4.

Proof of Proposition 1.4 Part A

To fix idea, we take that $v>0$.

If $\Omega$ is a ball, since u is smooth and radial then $u'(0)=0$. Integrating the first equation in (4.1) in (0, r) gives

$$\begin{aligned} -r^{N-1} u'(r) = \int _{\sigma _0}^r \rho ^{N-1} H_v(\rho , u(\rho ), v(\rho )) d\rho > 0, \end{aligned}$$

(4.2)

by assumption H1, so that u is strictly decreasing, and then, in turn, positive, and $\sigma _0=0$ is the unique critical point. Repeating the same computation with reversed role for v and u shows that also v as an unique critical point at the origin and completes this part of the proof.

If $\Omega$ is a spherical shell, let $\sigma = \inf \left\{ r\in (\delta , r) \,: \, u'(r)=0\right\}$. By continuity $u'(\sigma )=0$, and the previous argument show that u is strictly decreasing, and then, in turn, positive on $(\sigma , R)$. On the other side, when $r\in (\delta , \sigma )$, the same computations give

$$\begin{aligned} r^{N-1} u'(r) = \int _r^{\sigma } \rho ^{N-1} H_v(\rho , u(\rho ), v(\rho )) d\rho > 0, \end{aligned}$$

(4.3)

so that u is strictly increasing in $(\delta , \sigma )$ and therefore positive, again. Summing up, u(r) is strictly positive on $(\delta , R)$ and has an unique critical point at $\sigma _0=\sigma$. Repeating the same computation with reversed role for v and u shows that also v as an unique critical point and completes the proof of part A. $\square$

Proof of Proposition 1.4 Part B

The case $m=2$. To fix idea, we take that there exists $t\in (0,1)$ such that $v(t)=0$, $v>0$ at the left of t and $v<0$ at the right of t.

If $\Omega$ is a ball, the relation (4.2) still holds for $r\in (0,t)$, so that u is strictly decreasing in (0, t), and 0 is a local maximum point for u. Besides $u'(r)<0$ for every $r\in (0,1)$ may not occur, otherwise u is positive on [0, 1) and part A forbids v to have two nodal zones. Hence there exists $\sigma \in (t,R)$ such that $u'<0$ on $(0,\sigma )$ and $u'(\sigma )=0$. Taking $r\in (\sigma , R)$ and integrating again the first equation in (4.1) now gives

$$\begin{aligned} -r^{N-1} u'(r) = \int _{\sigma }^r \rho ^{N-1} H_v(\rho , u(\rho ), v(\rho )) d\rho < 0, \end{aligned}$$

so that u is strictly increasing on $(\sigma , R)$. Therefore the function u has exactly one critical point $\sigma \in (t,R)$, where the minimum is achieved. Moreover u can not be negative in [0, 1) (otherwise the case A could apply), so that u is positive in the first nodal zone and changes sign exactly once. Eventually, switching the role of u and v and repeating the same reasoning allows to conclude.

If $\Omega$ is a spherical shell, we write as before $\sigma = \inf \left\{ r\in (\delta , r) \,: \, u'(r)=0\right\}$. If $\sigma \ge t$ integrating in the interval between $(\sigma , r)$, one sees that u is strictly increasing on $(\sigma , R)$, and therefore negative. Because $u(\delta )=0$ and u has not any critical point in $(\delta , \sigma )$, it follows that u is strictly negative on $(\delta , R)$ and case A implies that the same holds for v, which is not true. Hence $\sigma < t$, and then u is strictly decreasing on $(\sigma , t)$ and increasing on $(\delta ,\sigma )$. Again, case A yields that u must have a further critical point in (t, R), and the conclusion follows like the ball.

The case $m\ge 3$. We proceed by induction on the number of nodal zones, assuming that the claim is true for solutions with less that m nodal zones and deducing that it is fulfilled also by solutions with m nodal zones.

Now there exist $t_0<t_1<\dots t_{m-1} <t_m=R$ such that $v>0$ on $(t_0,t_1)$ and on any interval of type $(t_i,t_{i+1})$ with i even, and $v<0$ on on any interval of type $(t_i,t_{i+1})$ with i odd, with $v(t_i)=0$ for $i=1,\dots , m$, and also $v(t_0)=0$ if $\Omega$ is a spherical shell. We also write $\tau _i$ for a maximum (if i is even) or minimum (if i is odd) point of v chosen in the interval $[t_i, t_{i+1}]$.

First we take that $\Omega$ is a ball. Reasoning as in the cases $m=2$ one sees that there exists $\sigma _1> t_1$ such that $u'(r)<0$ for $r\in (0,\sigma _1)$ and $u'(\sigma _1)=0$. Let us show that

$$\begin{aligned} \sigma _1 < t_2. \end{aligned}$$

(4.4)

If $\sigma _1 \ge t_{m-1}$, integrating the first equation in (4.1) in $(\sigma _1,r)$ for any $r>\sigma _1$ gives

$$\begin{aligned} -r^{N-1} u'(r) = \int _{\sigma _1}^r \rho ^{N-1} H_v(\rho ,u(\rho ),v(\rho )) d\rho < 0 \text{ if } \text{ m } \text{ is } \text{ even, } \text{ or } >0 \text{ if } m \text{ is } \text{ odd. } \end{aligned}$$

In the first case, $\sigma _1$ is the only one critical point of u in (0, R) and it is a minimum point, so that u has at most two nodal zones. In the second case, u is decreasing on (0, R) and therefore has only one nodal zone. Anyway, we can use the induction basis after switching the role of u and v and obtain the contradiction that v has no more than two nodal zones. Hence $\sigma _1 <t_{m-1}$.

If $m=3$ (4.4) is already proved, otherwise a similar reasoning shows that whenever $\sigma _1\ge t_2$, then u has less than m nodal zones, and the induction bases yields the contradiction that also v has less then m nodal zones. Afterward, repeating the arguments of the case $m=2$ one sees that u is strictly decreasing on $(0,\sigma _1)$ and increasing on $(\sigma _1, \sigma _2)$, $\sigma _2\in (t_2, R)$ being a critical point. If $m=3$, then u strictly decreasing on $(\sigma _2, R)$ and the proof is completed. Otherwise one sees that $\sigma _2 <t_3$ and the proof can be concluded after a finite number of steps.

If $\Omega$ is a spherical shell, combining the arguments used to prove the case $m=2$ and (4.4) one sees that $\sigma = \inf \left\{ r\in (\delta , r) \,: \, u'(r)=0\right\}$ satisfies $\sigma <t_1$, and then the proof follows like the ball. $\square$

Thanks to Proposition 1.4, the two components of every radial solution have the same number of nodal zones. Henceforth we shall say “a solution (u, v) with m nodal zones”, meaning that both u and v have m nodal zones.

4.1 Proof of Theorem 1.3

We begin by estimating the radial Morse index and proving (1.9).

Proposition 4.2

Let $\Omega$ be radially symmetric, $H=H(|x|, u, v)$ continuous w.r.t. x and satisfying H1, H2 for almost every $x\in \Omega$. If (u, v) is a classical radial solution to (1.1) with m nodal zones, then its radial Morse index is at least m.

Proof

Let

$$\begin{aligned} \begin{array}{cc} u_i(r) ={\left\{ \begin{array}{ll} u(r) & \text { in } [s_{i-1},s_i], \\ 0 & \text { elswhere, } \end{array}\right. }\; & \; v_i(r) ={\left\{ \begin{array}{ll} v(r) & \text { in } [t_{i-1},t_i], \\ 0 & \text { elswhere, } \end{array}\right. } \end{array} \end{aligned}$$

for $i=1,\dots m$. Here we have used Notations 4.1. It is clear that $u_i, v_i\in H^1_{0,{\text {rad}}}$ and $(u_i,v_i)$, $(u_j,u_j)$ are orthogonal for $i\ne j$, meaning that

$$\begin{aligned} \int _{s_0}^{s_m} r^{N-1} \left( u_iu_j + v_iv_j\right) dr =0. \end{aligned}$$

It suffices to check that

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(u_i,v_i) < 0 \quad \text{ for } i=1,\dots m. \end{aligned}$$

(4.5)

Using $(u_i,v_i)$ as a test function in the weak formulation of (4.1) gives

$$\begin{aligned} \int _{s_0}^{s_m} r^{N-1} \left( |u'_i|^2 + |v'_i|^2\right) dr = \int _{s_0}^{s_m} r^{N-1} \left[ H_v u_i + H_u v_i \right] dr, \end{aligned}$$

hence

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(u_i,v_i) =&\int _{s_0}^{s_m}\!\!r^{N-1} \left( H_v - H_{uv} u_i - H_{vv} v_i \right) u_i dr\\&+ \int _{s_0}^{s_m} \!\!r^{N-1} \left( H_u - H_{uu} u_i - H_{uv} v_i \right) v_i dr . \end{aligned}$$

We only estimate the first integral of the right side:

$$\begin{aligned} \int _{s_0}^{s_m}\!\!r^{N-1} \left( H_v - H_{uv} u_i - H_{vv} v_i \right) u_i \, dr&= \int _{I} \!\!r^{N-1} \left( H_v - H_{uv} u - H_{vv} v \right) u \, dr \\&\quad + \int _{J} \!\!r^{N-1} \left( H_v - H_{uv} u\right) u \, dr \end{aligned}$$

where $I=(s_{i-1},s_i)\cap (t_{i-1},t_i)$ and $J= (s_{i-1},s_i){\setminus }(t_{i-1},t_i)$. Note that on the set I the functions u and v have the same sign by Proposition 1.4, so that

$$\begin{aligned} \int _{I} \!\!r^{N-1} \left( H_v - H_{uv} u - H_{vv} v \right) u \, dr = \int _{I} \!\!r^{N-1} \left( \frac{H_v - H_{uv} u}{v} - H_{vv} \right) u v \, dr <0 \end{aligned}$$

by assumption H2. On the other hand, $J \subset (t_{i-2}, t_{i-1}) \cup (t_i, t_{i+1})$. Indeed by Proposition 1.4 it is known that $(s_{i},s_{i+1})\cap (t_{i}, t_{i+1})$ is not empty, therefore $s_{i}\le t_{i+1}$, and similarly $s_{i-1}\ge t_{i-2}$. Hence $uv<0$ on J, and

$$\begin{aligned} \int _{J} \!\!r^{N-1} \left( H_v - H_{uv} u \right) u \, dr = \int _{J} \!\!r^{N-1} \frac{H_v - H_{uv} u}{v} u v \, dr <0 \end{aligned}$$

by assumption H2. $\square$

Next we give an upper bound for the singular radial eigenvalues.

Let $\xi =u'$ and $\eta =v'$; an easy computation shows that, if $H=H(u,v)$ does not depend on x, then

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( r^{N-1} \xi '\right) '- r^{N-1} \left( H_{uv} \xi + H_{vv}\eta \right) = -(N-1) r^{N-3} \xi & 0<r< 1, \\ -\left( r^{N-1} \eta '\right) '- r^{N-1} \left( H_{uu} \xi + H_{uv}\eta \right) = -(N-1) r^{N-3} \eta & 0<r < 1, \end{array}\right. } \end{aligned}$$

(4.6)

in weak sense, that is

$$\begin{aligned} \begin{aligned} \int _0^1 r^{N-1} \left( \xi '\phi ' + \eta '\psi ' \right) dr - \int _0^1 r^{N-1} \left( H_{uv} \left( \xi \phi +\eta \psi \right) + H_{uu} \xi \psi +H_{vv} \eta \phi \right) dr \\ = -(N-1) \int _0^1 r^{N-3} \left( \xi \phi +\eta \psi \right) dr \end{aligned}\nonumber \\ \end{aligned}$$

(4.7)

for every $\phi ,\psi \in \widehat{H}^1_{0,{\text {rad}}}$.

Proposition 4.3

Let assumption H2 hold, and take (u, v) a classical radial solution with $m\ge 2$ nodal zones. Then

$$\begin{aligned} \widehat{\Lambda }_1^{{\text {rad}}}< \dots<\widehat{\Lambda }_{m-1}^{{\text {rad}}} <- (N-1). \end{aligned}$$

(4.8)

Proof

Using Notations 4.1, we define the auxiliary functions

$$\begin{aligned} \begin{array}{cc} \xi _k(r) ={\left\{ \begin{array}{ll} \xi (r) & \text { in } [\sigma _{k-1},\sigma _k], \\ 0 & \text { elswhere, } \end{array}\right. }\; & \; \eta _k(r) ={\left\{ \begin{array}{ll} \eta (r) & \text { in } [\tau _{k-1},\tau _k], \\ 0 & \text { elswhere, } \end{array}\right. } \end{array} \end{aligned}$$

for $k=1\dots m-1$. It is not difficult to check that $\xi _k, \eta _k\in \widehat{H}^1_{0,{\text {rad}}}$ (see [4, Lemma 3.2]), in addition $(\xi _k,\eta _k)$ and $(\xi _j,\eta _j)$ are orthogonal in $(\widehat{L})^2$ if $k\ne j$. Moreover using $(\xi _k, \eta _k)$ as a test function in (4.7) gives

$$\begin{aligned} \int _{s_0}^{s_1} r^{N-1} \left( |\xi '_k|^2 + |\eta '_k|^2\right) dr= & \int _{s_0}^{s_1} r^{N-1} \left( H_{uu} \xi \eta _k +H_{vv} \eta \xi _k + H_{uv} \left( \xi _k^2+\eta _k^2\right) \right) dr \nonumber \\ & -(N-1) \int _{s_0}^{s_1} r^{N-3} \left( \xi _k^2 + \eta _k^2\right) dr \end{aligned}$$

(4.9)

Let us check that

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\xi _k, \eta _k)\le - (N-1) \int _{s_0}^{s_1} r^{N-3} (\xi _k^2 + \eta _k^2) dr \end{aligned}$$

(4.10)

for every $k=1, \dots , m-1$. Thanks to (4.9) we have

$$\begin{aligned} {{\mathcal {Q}}}_{\text {lin}}(\xi _k, \eta _k)&= \int _{s_0}^{s_1} r^{N-1} \left[ H_{uu} (\xi -\xi _k) \eta _k + H_{vv} \xi _k (\eta - \eta _k) \right] dr\\&\quad -(N-1) \int _{s_0}^{s_1} r^{N-3} \left( \xi _k^2 + \eta _k^2\right) dr \end{aligned}$$

We compute, as an example,

$$\begin{aligned} \int _{s_0}^{s_1} r^{N-1} H_{vv} \xi _k (\eta - \eta _k) dr = \int _I r^{N-1} H_{vv} u' v' dr \end{aligned}$$

for $I= (\sigma _{k-1},\sigma _k) \setminus (\tau _{k-1},\tau _k)$. First we consider the case $\sigma _{k-1} < \tau _{k-1}$ and look at the sub-interval $I_k=(\sigma _{k-1}, \tau _{k-1} )$. Proposition 1.4 implies that $u'v'<0$ in $I_k$: indeed $I_k$ is contained in the $k^{th}$ nodal zone of u and also of v, where u and v have the same sign, say they are positive. In particular, both $\sigma _{k-1}$ and $\tau _{k-1}$ are maximum points, therefore $u'(r)<0$ for $r\in (\sigma _{k-1}, \sigma _k)$ and $v'(r)>0$ for $r\in (\tau _{k-2}, \tau _{k-1})$. Using also the hypothesis H2 we conclude that

$$\begin{aligned} \int _{I_k} r^{N-1} H_{vv} u' v' dr < 0 \end{aligned}$$

if $\sigma _{k-1} < \tau _{k-1}$, while there is no contribution from this term if $\sigma _{k-1} \ge \tau _{k-1}$. Similarly the other terms can be computed, and (4.10) follows. Remark that equality holds only if $\sigma _{k-1}=\tau _{k-1}$ and $\sigma _k= \tau _k$.

In the first instance, (4.10) yields that $\widehat{\Lambda }_1^{{\text {rad}}} \le -(N-1)$. But if $\widehat{\Lambda }_1^{{\text {rad}}} = -(N-1)$, then the functions $(\xi _k,\eta _k)$ should solve (3.14), which is not possible since they are equal to zero in an interval.

If $m\ge 3$, we look at the subspace of $\left( \widehat{ H}^1_{0,{\text {rad}}}\right) ^2$ generated by $(\xi _k,\eta _k)$ with $k=1,\dots m-1$, that we denote by V. Since both the components of $(\xi _k,\eta _k)$ and $(\xi _j,\eta _j)$ have contiguous support when $k\ne j$, it is clear that V has dimension $m-1$ and ${{\mathcal {Q}}}_{\text {lin}}(\phi ,\psi )\le - (N-1) \int _0^1 r^{N-3} (\phi ^2+\psi ^2) dr$ for every $(\phi ,\psi )\in V$. Hence $\widehat{\Lambda }_{m-1}^{{\text {rad}}} \le -(N-1)$ thanks to (3.5). Again, $\widehat{\Lambda }_{m-1}^{{\text {rad}}} = -(N-1)$ may not occur, otherwise one function of type $(\xi _k,\eta _k)$ should solve (3.14). $\square$

Putting the estimate (4.8) inside the Morse index formula (1.8), and recalling also Proposition 4.10, yields (1.10) and conclude the proof of Theorem 1.3.