Entire radial and nonradial solutions for systems with critical growth

Abstract

In this paper we establish existence of radial and nonradial solutions to the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_1 = F_1(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ -\Delta u_2 = F_2(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0 &{}\quad \text {in }{\mathbb R}^N,\\ u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

where $F_1,F_2$ are nonlinearities with critical behavior.

1 Introduction

The aim of this paper is to prove existence of radial and nonradial solutions to some nonlinear systems

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_1 = F_1(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ -\Delta u_2 = F_2(u_1,u_2) &{}\quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0 &{}\quad \text {in }{\mathbb R}^N,\\ u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

where $F_1,F_2$ are nonlinearities with critical behavior in the Sobolev sense, $N\geqslant 3$ and $D^{1,2}({\mathbb R}^N) = \bigl \{u\in L^{2^*}({\mathbb R}^N)\text { such that }|\nabla u| \in L^2({\mathbb R}^N)\bigr \}$ with $2^*=\frac{2N}{N-2}$. A common feature of the systems that we will study is their invariance by translations and dilations. Papers on existence or qualitative properties of solutions to systems with critical growth in ${\mathbb R}^N$ are very few, due to the lack of compactness given by the Talenti bubbles and the difficulties arising for the lack of good variational methods. The first example of system which we consider is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_1 =\alpha u_1^{ 2^*-1} +(1-\alpha ) u_1^\frac{2}{N-2}u_2^\frac{N}{N-2} &{}\quad \text {in }{\mathbb R}^N,\\ -\Delta u_2 =\alpha u_2^{ 2^*-1}+(1-\alpha ) u_2^\frac{2}{N-2}u_1^\frac{N}{N-2} &{}\quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0, \quad u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(1.1)

where $N\ge 3$ and $\alpha$ is a real parameter. This system, also known as Gross–Pitaevskii, arises in many physical contexts such as nonlinear optics and the Hartree–Fock theory, see [11] for its derivation, and it is very studied mainly in the cubic case, which corresponds to the critical case in ${\mathbb R}^4$ or on bounded domains where the cubic exponent is subcritical in ${\mathbb R}^3$. It is coupled when $1-\alpha \ne 0$ and cooperative when $1-\alpha >0$. Physically, this condition means the attractive interaction of the states $u_1$ and $u_2$, while $1-\alpha <0$ means the repulsive interaction between them. Note that System (1.1) has a gradient structure with the energy functional

$$\begin{aligned} \mathcal {E}(u_1,u_2) = \frac{1}{2} \int _{{\mathbb R}^N} |\nabla u_1|^2+|\nabla u_2|^2 - \frac{N-2}{2N} \int _{{\mathbb R}^N} \alpha \bigl (u_1^{2^*}+u_2^{2^*}\bigr ) + (1-\alpha ) \Bigl (u_1^\frac{N}{N-2} u_2^\frac{N}{N-2}\Bigr ) \end{aligned}$$

even if it is not so easy to apply variational methods to find solutions. System (1.1) was already considered in [10] where the existence of infinitely many nontrivial solutions is obtained using a perturbation argument.

Another particular case of (2.3) is the following generalization of the system considered by O. Druet, E. Hebey [6], namely

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1=\left[ \left( \alpha u_1^{2} + (1-\alpha ) u_2^{2}\right) ^2\right] ^{\frac{1}{N-2}}u_1 &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta u_2= \left[ \left( (1-\alpha ) u_1^{2} + \alpha u_2^{2}\right) ^2\right] ^{\frac{1}{N-2}}u_2 &{} \quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0, \quad u_1,u_2\in D^{1,2}({\mathbb R}^N). \end{array}\right. } \end{aligned}$$

(1.2)

In [6] the case of $\alpha =\frac{1}{2}$ was studied and the stability of solutions on manifolds was considered. Further, the radial symmetry and uniqueness of the solutions in ${\mathbb R}^N$ is proved.

We also mention the paper [3] where the radial symmetry of solutions is proved for a particular critical nonlinearity.

The starting point of our study is the paper [8] where we studied the existence of radial solutions for the $k\times k$ system of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u_i = \sum \limits _{j=1}^ka_{ij}u_j^{2^* - 1} &{}\quad \text {in }{\mathbb R}^N,\\ u_i>0 &{}\quad \text {in }{\mathbb R}^N,\\ u_i\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(1.3)

for $i=1,\ldots ,k$, where $N\ge 3$ and the matrix $A := (a_{ij})_{i,j=1,\ldots ,k}$ is symmetric and satisfies

$$\begin{aligned} \sum \limits _{j=1}^k a_{ij} = 1\text { for any } i = 1,\ldots ,k. \end{aligned}$$

(1.4)

Note that the case $k=2$ and $A = \bigl ({\begin{matrix}0&{}1\\ 1&{}0 \end{matrix}}\bigr )$ is known in the literature as nonlinearity belonging to the critical hyperbola.

Under the assumption (1.4) it is straightforward that system (1.3) always admits the trivial solutions

$$\begin{aligned} u_1 = \cdots =u_k = U_{\delta ,y}(x) := \frac{\left[ N(N-2)\delta ^2\right] ^{\frac{N-2}{4}}}{ (\delta ^2+|x-y|^2)^{\frac{N-2}{2}}} \end{aligned}$$

for any $\delta >0$ and $y\in {\mathbb R}^N$. To simplify the notation, let

$$\begin{aligned} U(x) := U_{1,0}(x) = \frac{\left[ N(N-2)\right] ^{\frac{N-2}{4}}}{ (1+|x|^2)^{\frac{N-2}{2}}}. \end{aligned}$$

(1.5)

A careful study of the linearized system of (1.3) at this trivial solution allows us to prove the existence of nontrivial radial solutions when the eigenvalues of the matrix A reach some specific values using bifurcation theory.

Note that System (1.3) does not have a variational structure and indeed our methods do not require it.

Even if the existence of radial solutions to some of the previous examples (1.1)–(1.3) is a new result, the main interest is the existence of nonradial ones. Nonradial solutions may be found mainly for noncooperative systems where the lack of the maximum principle can give a symmetry breaking of the solutions. Indeed, in [6] and [3], the radial symmetry of the solutions is proved in a particular cooperative case.

In this paper we want to purse several goals. First, we want to introduce a new setting which allows us to consider Systems (1.1)–(1.3) jointly. Indeed all these problems admit the trivial solutions $u_1 =u_2 = U_{\delta ,y}(x)$ which is the starting point to apply the bifurcation theory like in [8]. A general treatment of these problems is possible since we significantly improve the final part of the paper [8] showing that the Lagrange multiplier introduced to “kill” the direction of dilation invariance coming from the critical Sobolev exponent is indeed a natural constraint if we allow some invariance (Kelvin invariance) on the solutions. This lets us switch from a local bifurcation result in [8] to a global one.

This invariance is a good tool to overcome the degeneracy of critical problems in ${\mathbb R}^N$ which are invariant under dilation and can also be applied to the result in [5], where a Pohozaev identity gives the result only locally.

Another technical problem arises since our nonlinearities in general are not ${\mathcal C}^1$ at zero. This problem was already noticed by [10] and indeed their existence results are given in dimension 3 where they are able to define and to invert the linearized operator associated to their system. To overcome this problem we use a different functional setting that allows us to work only with positive values of $u_1$ and $u_2$. Observe that the functional setting of our operator is a delicate part of the proof.

Secondly we continue the study in [8] and we address to the existence of nonradial solutions to (1.3) using in a tricky way some even and odd symmetries. Obviously our solutions cannot be invariant with respect to odd symmetries since we are looking for positive ones. But we can introduce a suitable setting (see Eq. 2.9) in which we can make use of this invariance. This is a new aspect that has never been investigated before and that can shed light on how solutions of systems of this type are.

This use of the symmetries is the key point that allows us to distinguish between radial and nonradial solutions.

A crucial step of our method is the characterization of the kernel of the linearized operator associated to our systems. Actually, in [8], we find radial solutions using the classical Crandall–Rabinowitz Theorem which requires a one dimensional kernel. This is achieved by restricting the problem to radially symmetric functions and “killing” the direction of scale invariance.

Considering also nonradial functions the dimension of the kernel increases dramatically and it becomes very hard to control it. Moreover it is not clear whether the solution obtained considering this new kernel is nonradial. As said before, the use of suitable even and odd symmetries is significant and allows us to prove that in many cases the kernel contains only nonradial functions and it is odd dimensional. To exploit them, we need some invariance on the operator associated to our problem. This invariance naturally appears in the case of a $2\times 2$ system while it not clear whether it applies in the general case of more equations as (1.3). For this reason we focus hereafter on the case  $2\times 2$ and we believe that a further study is needed to understand the general case. To compute the dimension of the kernel in these symmetric spaces we need a classification of symmetric spherical harmonics in ${\mathbb S}^N$ and indeed this is part of Sects. 4 and 5.

Finally we also give an asymptotic expansion of the solutions near the bifurcation point so as to better understand them. In this way we can distinguish different nonradial solutions by their symmetries and expansions.

2 Statement of the main results

Let us introduce our abstract setting. We consider

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1 = F_1(\alpha , u_1, u_2) &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta u_2 = F_2(\alpha , u_1, u_2) &{} \quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0,\quad u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(2.1)

where the $F_i$ satisfy the following assumptions: for all $\alpha \in {\mathbb R}$ and for $i = 1,2$,

• (F1) the derivatives $\partial _\alpha F_i$, $\partial _{u} F_i$ and $\partial _{\alpha u} F_i$ of the map $F_i : {\mathbb R}\times (0, +\infty )^2 \rightarrow {\mathbb R}: (\alpha , u) \mapsto F_i(\alpha , u)$ exist and are continuous;

• (F2) for all $\alpha \in {\mathbb R}$, there exists a neighborhood $\mathcal A$ of $\alpha$ and a constant C such that, for all $\alpha \in \mathcal A$ and $(u_1, u_2) \in (0, +\infty )^2$, $|\partial _u F_i(\alpha , u_1, u_2)| \le C (u_1^{2^* - 2} + u_2^{2^*-2})$ and $|\partial _{\alpha u} F_i(\alpha , u_1, u_2)| \le C (u_1^{2^* - 2} + u_2^{2^*-2})$;

• (F3) $F_i(\alpha , 1,1) = 1$;

• (F4) $F_i(\alpha , \lambda u_1, \lambda u_2) = \lambda ^{2^*-1} \, F_i(\alpha , u_1, u_2)$ for all $\lambda > 0$ and $(u_1, u_2) \in (0, +\infty )^2$;

• (F5) $F_1(\alpha , u_1, u_2) = F_2(\alpha , u_2, u_1)$ for all $(u_1, u_2) \in (0, +\infty )^2$;

• (F6) for all $\alpha$, $\partial _\alpha \beta (\alpha ) > 0$ where $\beta (\alpha ) := \partial _{u_1} F_1(\alpha , 1,1) - \partial _{u_2} F_1(\alpha , 1,1)$.

By (F3) it is straightforward that System (2.1) admits, for any $\alpha \in {\mathbb R}$, the trivial solution $(u_1, u_2) = (U,U)$ and (F4) says that our system is scale invariant. Further, in view of Eq. (2.1), it is also translation invariant.

This generalization encompasses the following Schrodinger system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1=\alpha u_1^{2^*-1} + (1-\alpha ) u_1^pu_2^{2^*-1-p} &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta u_2=(1- \alpha ) u_1^{2^*-1-p}u_2^p+\alpha u_2^{2^*-1} &{} \quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0, \quad u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(2.2)

with $0\leqslant p<2^*-1$ and $\alpha$ is a real parameter. When $p=0$ System (2.2) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1=\alpha u_1^{2^*-1} + (1-\alpha ) u_2^{2^*-1} &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta u_2=(1- \alpha ) u_1^{2^*-1}+\alpha u_2^{2^*-1} &{} \quad \text {in }{\mathbb R}^N,\\ u_1\geqslant 0,\ u_2\geqslant 0, \quad u_1,u_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(2.3)

while for $p=\frac{2}{N-2}$ we get System (1.1). Moreover System (2.1) includes System (1.2).

Our first result is the generalization of the local radial bifurcation result obtained in [8] for (2.3) to a global one for System (2.1). An important role in our results will be played by the Jacobi polynomials $P_j^{(\beta , \gamma )}$ that we introduce now. They are defined as

$$\begin{aligned} P_m^{(\beta , \gamma )}(\xi ) = \sum _{s=0}^m \left( {\begin{array}{c}m+\beta \\ s\end{array}}\right) \left( {\begin{array}{c}m+\gamma \\ m-s\end{array}}\right) \left( \frac{\xi -1}{2}\right) ^{m-s} \left( \frac{\xi +1}{2}\right) ^s \end{aligned}$$

(2.4)

for $m\in {\mathbb N}$, $\beta ,\gamma \in {\mathbb R}^+$ and $\xi \in {\mathbb R}$.

Theorem 2.1

Assume (F1)–(F6). The point $(\alpha ^*, U,U)$ is a radial bifurcation point from the curve of trivial solutions $(\alpha ,U,U)$ to System (2.1) if $\alpha ^*$ satisfies

$$\begin{aligned} \beta (\alpha ^*) = \frac{(2n+N)(2n+N-2)}{N(N-2)} \end{aligned}$$

(2.5)

for some $n \in {\mathbb N}$, where $\beta$ is defined in (F6). More precisely there exists a continuously differentiable curve defined for $\varepsilon$ small enough

$$\begin{aligned} (-\varepsilon _0,\varepsilon _0) \rightarrow {\mathbb R}\times \bigl (D^{1,2}_{\text {rad}}({\mathbb R}^N)\bigr )^2 : \varepsilon \mapsto \bigl (\alpha (\varepsilon ), u_1(\varepsilon ), u_2(\varepsilon )\bigr ) \end{aligned}$$

passing through $(\alpha ^*,U,U)$, i.e., $\bigl (\alpha (0),u_1(0), u_2(0)\bigr ) = (\alpha ^*,U,U)$, such that, for all $\varepsilon \in (-\varepsilon _0,\varepsilon _0)$, $(u_1(\varepsilon ), u_2(\varepsilon ))$ is a radial solution to (2.1) with $\alpha = \alpha (\varepsilon )$. Moreover,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1(\varepsilon ) = U + \varepsilon W_n(|x|) + \varepsilon \phi _{1,\varepsilon }(|x|),\\ u_2(\varepsilon ) = U - \varepsilon W_n(|x|) + \varepsilon \phi _{2,\varepsilon }(|x|), \end{array}\right. } \end{aligned}$$

(2.6)

with $W_n$ being the function

$$\begin{aligned} \begin{aligned} W_n(|x|)&:= \frac{1}{\left( 1+|x|^2\right) ^{\frac{N-2}{2}}} \, P_n^{\left( \frac{N-2}{2}, \frac{N-2}{2} \right) } \left( \frac{1-|x|^2}{1+|x|^2}\right) \end{aligned} \end{aligned}$$

(2.7)

where $\phi _{1,\varepsilon },\phi _{2,\varepsilon }$ are functions uniformly bounded in $D^{1,2}({\mathbb R}^N)$ with respect to $\varepsilon \in (-\varepsilon _0, \varepsilon _0)$, and such that $\phi _{i,0}=0$ for $i=1,2$. Finally the bifurcation is global and the Rabinowitz alternative holds.

The values $\alpha ^*$ in (2.5) are all of those for which the linearized system at the trivial solution (U, U) is non-invertible showing that condition (2.5) is also necessary.

Corollary 2.2

For any $n\in {\mathbb N}$, let

$$\begin{aligned} \alpha ^*_{n}={\left\{ \begin{array}{ll} \frac{(2n+N-2)(2n+N)}{2N(N+2-p(N-2))} + \frac{N+2}{2 (N+2-p(N-2))} - \frac{p(N-2)}{N+2-p(N-2)} &{}\quad \text {in }~(2.2),\\[2mm] \frac{(2n+N-2)(2n+N)}{2N^2} + \frac{N-2}{2N} &{}\quad \text {in }~(1.1),\\[2mm] \frac{(2n+N-2)(2n+N)+N(6-N)}{8N} &{}\quad \text {in }~(1.2). \end{array}\right. } \end{aligned}$$

(2.8)

Then $(\alpha ^*, U,U)$ is a radial bifurcation point of Systems (2.2), (1.1) and (1.2) from its curve of trivial solutions $(\alpha ,U,U)$ if $\alpha ^* = \alpha ^*_n$ for some $n\in {\mathbb N}$. Moreover, the expansion around the bifurcation point given by Theorem 2.1 holds and the curve is global.

Remark 2.3

An interesting fact is that in (2.2) the exponent p does not enter in a relevant way in the proof of the previous results and indeed the solutions we find have, near a bifurcation point, the same expansion for every value of p. In this way we have a path of solutions connecting (2.3) with (1.1) showing that these solutions are not due the variational structure of (2.3).

The next step is to find nonradial solutions. In [9] was proved that in the cooperative case (i.e., when $1-\alpha >0$), System (2.2) admits only radial solutions. Note that, for all $n \ge 1$, $1 - \alpha ^*_n < 1 - \alpha ^*_1 = 0$ where $\alpha ^*_n$ is defined by (2.8). Then $\alpha ^*_n$ are good “candidates” to find nonradial solutions. Moreover, at each value $\alpha ^*_n$ the linearized system possesses many nonradial solutions and the kernel becomes richer and richer as $n\rightarrow \infty$ (see Proposition 3.1). However, one technical problem in looking for nonradial solutions is that the kernel of the linearized problem at a degeneracy point always contains the radial function $W_n$ defined by (2.7). So our aim becomes to choose a suitable subspace of the kernel in which $W_n$ does not lies. This will be done by using in a tricky way some odd-symmetries. It is possible indeed to apply such symmetries to a linear combination of the components $u_1,u_2$ even if the solutions we are interested in are positive.

Here is our basic idea: if one writes

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1 = U+\frac{z_1 +z_2}{2} \\ u_2 = U+\frac{z_1 -z_2}{2} \end{array}\right. } \end{aligned}$$

(2.9)

then the system satisfied by $z_1,z_2$ admits solutions obtained by imposing the following symmetries on $(z_1, z_2)$:

$$\begin{aligned} \forall (x', x_N) \in {\mathbb R}^N,\qquad z_1(x', x_N) = z_1(|x'|, -x_N) \text { and } z_2(x', x_N) = -z_2(|x'|, -x_N) , \end{aligned}$$

(2.10)

(more general symmetries will be imposed later; see Sect. 4.2 for more details). The crucial remark is that the new system in $(z_1,z_2)$ obtained by (2.9) is invariant for the symmetries in (2.10) (see (3.1)–(3.5)). This use of odd symmetries is unclear if we considered directly System (2.1).

In order to state our first nonradial bifurcation result, we use in ${\mathbb R}^N$ the spherical coordinates $(r,\varphi ,\theta _1,\dots ,\theta _{N-2})\in [0,+\infty )\times [0,2\pi )\times [0,\pi )^{N-2}$. We have

Theorem 2.4

Assume (F1)–(F6) and let $\alpha ^*_n$ be the unique solution to (2.5) for some $n \in {\mathbb N}$. The point $(\alpha ^*_n, U,U)$ is a nonradial bifurcation point for the curve of trivial solutions $(\alpha ,U,U)$ to System (2.1) when $n \bmod 4 \in \{1,2\}$. More precisely, there exist a continuum $\mathcal {C}$ of nonradial solutions $(u_1,u_2)$ to System (2.1), bifurcating from $(\alpha ^*_n,U,U)$; the bifurcation is global and the Rabinowitz alternative holds. Finally for any sequence of solutions $(\alpha _k, u_{1,k}, u_{2,k}) \rightarrow (\alpha ^*_n, U, U)$, we have that (up to a subsequence)

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{1,k} = U + \varepsilon _k Z_n\left( x\right) + o(\varepsilon _k) ,\\ u_{2,k} = U - \varepsilon _k Z_n\left( x\right) + o(\varepsilon _k) , \end{array}\right. } \end{aligned}$$

(2.11)

as $k\rightarrow \infty$ where $\varepsilon _k = \Vert z_{2,k}\Vert _X \rightarrow 0$ (see (2.9) and (3.13)) and $Z_n\not \equiv 0$ is the function

$$\begin{aligned} Z_n(x) = \sum _{h=1,\ h \text {\ odd}}^n a_h \frac{r^h}{(1+r^2)^{h+\frac{N-2}{2}}} P_{n-h}^{\left( h+\frac{N-2}{2}, h+\frac{N-2}{2}\right) } \left( \frac{1-r^2}{1+r^2}\right) P_h^{\left( \frac{N-3}{2},\frac{N-3}{2}\right) } (\cos \theta _{N-2}) \end{aligned}$$

(2.12)

for some coefficients $a_h\in {\mathbb R}$.

Observe that the functions $P_h^{\left( \frac{N-3}{2},\frac{N-3}{2}\right) } (\cos \theta _{N-2})$ are the spherical harmonics that are $O(N-1)$-invariant.

Corollary 2.5

Let $n\in {\mathbb N}$ and $\alpha ^*_n$ as defined in Corollary 2.2. Then the same claims of Theorem 2.4 hold for Systems (2.2) and (1.2).

It is possible to prove a similar result using more symmetries. Here we ask the following ones: $\forall x = (x', x_{N-m+1}, \ldots , x_N)\in {\mathbb R}^{N-m}\times {\mathbb R}^m$,

$$\begin{aligned}&z_1(x) = z_1(|x'|,\pm x_{N-m+1}, \ldots ,\pm x_N), \text { and }\\&z_2(x', x_{N-m+1}, x_N) = - z_2(|x'|, -x_{N-m+1}, \ldots ,x_N),\\&\ldots \\&z_2(x', x_{N-m+1}, x_N) = - z_2(|x'|, x_{N-m+1}, \ldots ,- x_N) \bigr \}. \end{aligned}$$

Imposing these symmetries on the functions $z_1,z_2$ defined in (2.9), we get the following result:

Theorem 2.6

Let $1\leqslant m\leqslant N$ and let $\alpha ^*_n$ be the unique solution to (2.5) for some $n\geqslant m$. Suppose that

$$\begin{aligned} \left( {\begin{array}{c}m+\left\lfloor \frac{n-m}{2}\right\rfloor \\ m\end{array}}\right) \text { is an odd integer.} \end{aligned}$$

(2.13)

Then for any m there exists a continuum $\mathcal {C}_m$ of nonradial solutions that satisfies System (2.1), bifurcating from $(\alpha ^*_n,U,U)$ and the bifurcation is global and the Rabinowitz alternative holds. Moreover the continua $\mathcal {C}_m$ are distinct and we have that, up to a subsequence, $(u_1,u_2)$ has the same expansion as in (2.11) where

$$\begin{aligned} Z_n(x)=\sum _{h=1}^n a_h \frac{r^h}{(1+r^2)^{h+\frac{N-2}{2}}} P_{n-h}^{\left( h+\frac{N-2}{2}, h+\frac{N-2}{2}\right) } \left( \frac{1-r^2}{1+r^2}\right) Y_h(\theta ) \end{aligned}$$

(2.14)

and the spherical harmonics $Y_h(\theta )$ are $O(N-m)$ invariant and odd in the last m variables.

Corollary 2.7

Let $n\in {\mathbb N}$ and $\alpha ^*_n$ as defined in Corollary 2.2. Then the same claims of Theorem 2.6 hold for System (2.2) and (1.2).

For the reader’s convenience, we state the previous theorem when $m=2$.

Corollary 2.8

Let $m=2$ in Theorem 2.6. Then if

$$\begin{aligned} n \bmod 8 \in \{2,3,4,5\} \end{aligned}$$

(2.15)

the claim of Theorem 2.6 holds and $Z_n$ in this case is given by

$$\begin{aligned} Z_n(x)=\sum _{h=1}^n a_h \frac{r^h}{(1+r^2)^{h+\frac{N-2}{2}}} P_{n-h}^{\left( h+\frac{N-2}{2}, h+\frac{N-2}{2}\right) } \left( \frac{1-r^2}{1+r^2}\right) Y_h(\theta ) \end{aligned}$$

(2.16)

for some coefficients $a_h\in {\mathbb R}$, where $Y_h(\theta )$ are spherical harmonics which are $O(N-2)$ invariant and are odd with respect to $x_N$ and to $x_{N-1}$.

We conclude by giving one more existence result which produces a nonradial solutions for every value of n. These solutions are found imposing an odd symmetry with respect to an angle in spherical coordinates and also a periodicity assumption. They are different from the previous ones since they have a different expansion.

Theorem 2.9

Assume (F1)–(F6) and $\alpha ^*_n$ be the unique solution to (2.5) for some $n\in {\mathbb N}$. Then for any $n\in {\mathbb N}$, $n \ge 2$, there exists a continuum ${\mathcal D}_n$ of nonradial solutions to System (2.1), bifurcating from $(\alpha ^*_n, U,U)$. When $\varepsilon$ is small enough this continuum is a continuously differentiable curve

$$\begin{aligned} (-\varepsilon _0,\varepsilon _0) \rightarrow {\mathbb R}\times \bigl (D^{1,2}_{\text {rad}}({\mathbb R}^N)\bigr )^2 : \varepsilon \mapsto \bigl (\alpha (\varepsilon ), u_1(\varepsilon ), u_2(\varepsilon )\bigr ) \end{aligned}$$

passing through $(\alpha ^*_n,U,U)$, i.e., $\bigl (\alpha (0),u_1(0), u_2(0)\bigr ) = (\alpha ^*_n,U,U)$, such that, for all $\varepsilon \in (-\varepsilon _0,\varepsilon _0)$, $(u_1(\varepsilon ), u_2(\varepsilon ))$ is a nonradial solution to (2.1) with $\alpha = \alpha (\varepsilon )$. Moreover,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1(\varepsilon ) = U + \varepsilon Z_n(x) + \varepsilon \phi _{1,\varepsilon }(x),\\ u_2(\varepsilon ) = U - \varepsilon Z_n(x) + \varepsilon \phi _{2,\varepsilon }(x), \end{array}\right. } \end{aligned}$$

with

$$\begin{aligned} Z_n(r,\varphi ,\Theta ) = a \frac{r^n}{\left( 1+r^2\right) ^{n+\frac{N-2}{2}}} \sin (n\varphi )(\sin \theta _1)^n\ldots (\sin \theta _{N-2})^n, \qquad a\in {\mathbb R}, \end{aligned}$$

(2.17)

(here we use the spherical coordinates $(r, \varphi , \Theta ) = (r,\varphi ,\theta _1,\dots ,\theta _{N-2})$ in ${\mathbb R}^N$). Moreover the bifurcation is global and the Rabinowitz alternative holds.

Remark 2.10

Note that the function $Y_n(\varphi ,\Theta ) = \sin (n\varphi )(\sin \theta _1)^n\ldots (\sin \theta _{N-2})^n$ is the unique spherical harmonic of order n which is odd and periodic of period $\frac{2\pi }{n}$ with respect to the angle $\varphi$. Moreover, in Cartesian coordinates we have that $Y_n(x) = \mathfrak {I}m(x_1+ix_2)^n$.

Corollary 2.11

Let $n\in {\mathbb N}$ and $\alpha ^*_n$ as defined in Corollary 2.2. Then the same claims of Theorem 2.9 hold for System (2.2) and (1.2).

Remark 2.12

It is difficult to give a formula with the exact number of solutions which takes in account all the previous theorems. Here we describe a particular case: choose $n=4$ in (2.5) and $N\ge 4$ then we have the existence of at least five solutions bifurcating by (U, U) as follows:

(i):

one radial solution (Theorem 2.1),

(ii):

one nonradial solution with $z_1$ even in all the coordinates and $z_2$ odd with respect to $x_{N-1}$ and $x_N$ and even in other coordinates (Corollary 2.8),

(iii):

one nonradial solution with $z_2$ odd with respect to $x_{N-3},\ldots ,x_N$ and even in other coordinates (Theorem 2.6 with $m=3$),

(iv):

one nonradial solution in ${\mathbb R}^N$ with $N\geqslant 4$ with $z_2$ odd with respect to $x_{N-4},\ldots , x_N$ and even in other coordinates (Theorem 2.6 with $m=4$),

(v):

one nonradial solution where $z_1$ and $z_2$ are periodic of period $\frac{2\pi }{4}$ with respect to the angle $\varphi$ and $z_2$ is odd in $\varphi$ (Theorem 2.9).

In the following table, which does not pretend to be exhaustive, we show the number of solutions bifurcating from (U, U) arising from Theorems 2.1–2.9.

	N $=$ 3	N $=$ 4	N $=$ 5
n $=$ 2	4	4	4
n $=$ 3	4	4	4
n $=$ 4	4	5	5
n $=$ 5	4	5	6
n $=$ 6	3	4	5
n $=$ 7	2	3	3

Remark 2.13

Note that the our results for System (1.1) hold for any dimension $N\geqslant 3$, extending some recent results of [10]. Finally, as observed in [10], when the dimension $N\ge 4$, System (1.1) becomes linear or sublinear in some of its components and this fact produces problem in defining and estimating the linearization. In some sense, we can say that the bifurcation theory suits well this problem. We remark moreover that the solutions founded in [10] are always different from ours since their expansion is of the following type $u_1=U+\varepsilon \phi _1$ and $u_2=\sum _k U_{\delta _k,y_k}+\varepsilon \phi _2$.

The paper is organized as follows: in Sect. 3 we recall some preliminaries and introduce the functional setting to find the nonradial solution. In Sect. 4 we define the symmetric spaces and prove Theorems 2.1, 2.4 and 2.6. In Sect. 5 we prove Theorem 2.9.

3 Preliminary results and the functional setting

To study System (2.1), we perform the following change of variables

$$\begin{aligned} {\left\{ \begin{array}{ll} z_1 = u_1 + u_2 - 2U, \\ z_2 = u_1 - u_2, \end{array}\right. } \end{aligned}$$

(3.1)

that turns (2.1) into the system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta z_1 = f_1(|x|,z_1,z_2) &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta z_2 = f_2(|x|,z_1,z_2) &{} \quad \text {in }{\mathbb R}^N,\\ z_1,z_2\in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(3.2)

where

$$\begin{aligned} f_1(|x|,z_1,z_2)&:= F_1\Bigl (\alpha , U+\frac{z_1+z_2}{2},\ U+\frac{z_1-z_2}{2} \Bigr ) \nonumber \\&\quad + F_2\Bigl (\alpha , U+\frac{z_1+z_2}{2},\ U+\frac{z_1-z_2}{2} \Bigr ) - 2 U^{2^*-1} , \end{aligned}$$

(3.3)

$$\begin{aligned} f_2(|x|,z_1,z_2)&:= F_1\Bigl (\alpha , U+\frac{z_1+z_2}{2},\ U+\frac{z_1-z_2}{2} \Bigr ) \nonumber \\&\quad - F_2\Bigl (\alpha , U+\frac{z_1+z_2}{2},\ U+\frac{z_1-z_2}{2} \Bigr ). \end{aligned}$$

(3.4)

One important feature in looking for nonradial solutions is that, using (F5), this change of variables gives the following invariance:

$$\begin{aligned} \begin{aligned} f_1(|x|,z_1,-z_2)&= f_1(|x|,z_1,z_2),\\ f_2(|x|,z_1,-z_2)&= -f_2(|x|,z_1,z_2). \end{aligned} \end{aligned}$$

(3.5)

Solutions to (2.1) are zeros of the operator

$$\begin{aligned} T(\alpha ,z_1,z_2) := \begin{pmatrix} z_1 - (-\Delta )^{-1} \bigl (f_1(|x|,z_1,z_2)\bigr )\\ z_2 - (-\Delta )^{-1} \bigl (f_2(|x|,z_1,z_2)\bigr ) \end{pmatrix}. \end{aligned}$$

Clearly, $T(\alpha , 0,0) = (0,0)$ for all $\alpha \in {\mathbb R}$ (thanks to (F3) and (F4)). A necessary condition for the bifurcation is that the linearized operator $\partial _zT(\alpha , 0,0)$ is not invertible. This corresponds to study the system:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w_1 = \frac{\partial f_1}{\partial z_1}(|x|,0,0) \, w_1 + \frac{\partial f_1}{\partial z_2}(|x|,0,0) \, w_2 &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta w_2=\frac{\partial f_2}{\partial z_1}(|x|,0,0) \, w_1 + \frac{\partial f_2}{\partial z_2}(|x|,0,0) \, w_2 &{} \quad \text {in }{\mathbb R}^N,\\ w_1,w_2 \in D^{1,2}({\mathbb R}^N). \end{array}\right. } \end{aligned}$$

(3.6)

A simple computation shows

$$\begin{aligned} \frac{\partial f_1}{\partial z_1}(\alpha ,0,0)&=\frac{1}{2}\left[ \frac{\partial F_1}{\partial u_1}(\alpha ,U,U) + \frac{\partial F_1}{\partial u_2}(\alpha ,U,U) + \frac{\partial F_2}{\partial u_1}(\alpha ,U,U) + \frac{\partial F_2}{\partial u_2}(\alpha ,U,U)\right] ,\\ \frac{\partial f_1}{\partial z_2}(\alpha ,0,0)&=\frac{1}{2}\left[ \frac{\partial F_1}{\partial u_1}(\alpha ,U,U) - \frac{\partial F_1}{\partial u_2}(\alpha ,U,U) + \frac{\partial F_2}{\partial u_1}(\alpha ,U,U) - \frac{\partial F_2}{\partial u_2}(\alpha ,U,U)\right] , \end{aligned}$$

and a very similar expression holds for $\frac{\partial f_2}{\partial z_i}(\alpha ,0,0)$ for $i=1,2$. First observe that from (F5) we get

$$ \frac{\partial F_1}{\partial u_1}(\alpha ,U,U)=\frac{\partial F_2}{\partial u_2}(\alpha ,U,U)\quad \text { and } \quad \frac{\partial F_1}{\partial u_2}(\alpha ,U,U)=\frac{\partial F_2}{\partial u_1}(\alpha ,U,U). $$

Then, differentiating (F4) with respect to $\lambda$ we get

$$ (\partial _{u_1}F_1 + \partial _{u_2}F_1)(\alpha , U,U) = (2^*-1) U^{2^*-2} F_1(\alpha , 1, 1) = \frac{N+2}{N-2} U^{\frac{4}{N-2}}. $$

Moreover, using again (F4):

$$\begin{aligned} \partial _{u_j}F_i(\alpha , \lambda u_1, \lambda u_2) = \lambda ^{2^*-2} \partial _{u_j}F_i(\alpha , u_1, u_2) \quad \text {for } i = 1,2 \text { and } j = 1,2, \end{aligned}$$

and in particular

$$ \partial _{u_j}F_i(\alpha , U, U) = U^{2^*-2} \, \partial _{u_j}F_i(\alpha , 1,1). $$

Putting together all these remarks, it is straightforward that system (3.6) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w_1 = \frac{N+2}{N-2} \, U^{\frac{4}{N-2}} w_1 &{} \quad \text {in }{\mathbb R}^N,\\ -\Delta w_2=\beta (\alpha ) \, U^{\frac{4}{N-2}} w_2 &{} \quad \text {in }{\mathbb R}^N,\\ w_1,w_2 \in D^{1,2}({\mathbb R}^N), \end{array}\right. } \end{aligned}$$

(3.7)

with $\beta (\alpha )$ defined in (F6).

System (3.7) is degenerate for any $\alpha$, since the problem is invariant by translations and dilations. Indeed, it is well known that the first equation admits the solutions $W(x):=\frac{1-|x|^2}{\left( 1+|x|^2\right) ^{N/2}}$ and $W_i(x)=\frac{\partial U}{\partial x_i}$ for $i=1,\ldots , N$. The second equation instead has solutions if and only if $\beta (\alpha )$ is an eigenvalue of the linearized equation of the classical critical problem at the standard bubble U. Using the classification of the eigenvalues and eigenfunctions in [8, Theorem 1.1], one gets that the second equation admits nontrivial solutions if and only if $\beta (\alpha ) = \lambda _n \frac{N+2}{N-2}$ with $\lambda _n := \frac{(2n+N-2)(2n+N)}{N(N+2)}$ for some $n\in {\mathbb N}$. So we have the following classification result for (3.7).

Proposition 3.1

Let $\beta _n$ be given by

$$\begin{aligned} \beta _n := \frac{(2n+N)(2n+N-2)}{N(N-2)} . \end{aligned}$$

(3.8)

• (i) When $\beta (\alpha ) \ne \beta _n$ for all $n \in {\mathbb N}$, all solutions to (3.7) are given by

  $$\begin{aligned} (w_1,w_2) = \left( \sum _{i=1}^N a_i\frac{\partial U}{\partial x_i} + b W, \ 0\right) \end{aligned}$$

  (3.9)

  for some real constants $a_1,\ldots ,a_N,b$, where W is the radial function defined by

  $$\begin{aligned} W(x) :=\frac{1}{d} \left( x\cdot \nabla U + \frac{N-2}{2} U\right) = \frac{1 - |x|^2}{(1 + |x|^2)^{N/2}} \end{aligned}$$

  (3.10)

  with $d := \tfrac{1}{2} N^{(N-2)/4} (N-2)^{(N+2)/4}$.

• (ii) When $\beta (\alpha ) = \beta _n$ for some $n \in {\mathbb N}$, all solutions to (3.7) are given by

  $$\begin{aligned} (w_1,w_2) = \left( \sum _{i=1}^N a_i\frac{\partial U}{\partial x_i} + b W, \ \sum _{k=0}^n A_k W_{n,k}(r) Y_k(\theta ) \right) \end{aligned}$$

  (3.11)

  for some real constants $a_1,\ldots ,a_N,b, A_{0},\ldots ,A_n$, where $W_{n,k}$ are

  $$\begin{aligned} W_{n,k}(r) := \frac{r^k}{(1+r^2)^{k+\frac{N-2}{2}}} P_{n-k}^{\left( k+\frac{N-2}{2}, k+\frac{N-2}{2}\right) } \left( \frac{1-r^2}{1+r^2}\right) \end{aligned}$$

  (3.12)

  for $k=0,\ldots ,n$. Here, as usual, $Y_k(\theta )$ denotes a spherical harmonic related to the eigenvalue $k(k+N-2)$ and $P_j^{(a, b)}$ are the Jacobi polynomials.

In [8] we restricted to the radial functions and since the kernel of the second equation in (3.7) at the values $\beta _n$ is one dimensional, Crandall–Rabinowitz’ Theorem allowed us to prove the bifurcation result. In the nonradial setting, the kernel of the second equation in (3.7) is very rich. We prove a bifurcation result using the Leray Schauder degree, when this kernel has an odd dimension.

Of course, in this case, we need some compactness of the operator T. Since we seek positive solutions to System (2.1) and the maximum principle does not apply, the standard space $D^{1,2}({\mathbb R}^N)$ does not seem to be the best one. For this reason we use a suitable weighted functional space. Set

$$\begin{aligned} D := \left\{ u\in L^{\infty }({\mathbb R}^N) \Bigm | \sup \limits _{x\in {\mathbb R}^N}\frac{|u(x)|}{U(x)}<+\infty \right\} \end{aligned}$$

endowed with the norm $\Vert u\Vert _D := \sup _{x\in {\mathbb R}^N}\frac{|u(x)|}{U(x)}$ and define

$$\begin{aligned} X := D^{1,2}({\mathbb R}^N)\cap D. \end{aligned}$$

(3.13)

Then X is a Banach space when equipped with the norm $\Vert u\Vert _X := \max \{\Vert u\Vert _{1,2}, \Vert u\Vert _D\}$ where $\Vert u\Vert _{1,2} = (\int _{{\mathbb R}^N} |\nabla u|^2)^{1/2}$ is the classical norm in $D^{1,2}({\mathbb R}^N)$.

Definition 3.2

Let us denote by ${\mathcal X}$ the space

$$\begin{aligned} {\mathcal X}:= \left\{ (z_1, z_2) \in X^2 \bigm | \exists \delta > 0,\ |z_2| \le (2-\delta )U + z_1 \right\} \end{aligned}$$

and define the operator

$$\begin{aligned} T : {\mathbb R}\times {\mathcal X}\rightarrow X\times X \end{aligned}$$

as

$$\begin{aligned} T(\alpha ,z_1,z_2) := \begin{pmatrix} z_1 - (-\Delta )^{-1} \bigl (f_1(|x|,z_1,z_2) \bigr ) \\ z_2 - (-\Delta )^{-1} \bigl (f_2(|x|,z_1,z_2) \bigr ) \end{pmatrix} . \end{aligned}$$

(3.14)

Note that if $(z_1,z_2) \in {\mathcal X}$, both quantities $U+\frac{z_1+z_2}{2}$ and $U+\frac{z_1-z_2}{2}$ are positive so that $F_i(\alpha , U+\frac{z_1+ z_2}{2}, U+\frac{z_1- z_2}{2})$ are well defined on ${\mathbb R}^N$ and $C^1$. Moreover, ${\mathcal X}$ is an open subset of $X^2$.

The zeros of the operator T correspond to the solutions to System (2.1). As said before, Problem (2.1) is degenerate for any $\alpha$. To overcome this degeneracy we will use some symmetry and invariance properties. The solutions we will find will inherit the symmetry and the invariance. To overcome the degeneracy of the first equation in (3.7), which is due to the scale invariance of the problem, we use the Kelvin transform k(z) of z, namely

$$\begin{aligned} k(z)(x) := \frac{1}{|x|^{N-2}} \, z\left( \frac{x}{|x|^2}\right) \end{aligned}$$

(3.15)

and we denote by $X_k^{\pm }\subseteq X$ the subset of functions in X which are invariant (up to the sign) by a Kelvin transform, i.e.

$$\begin{aligned} X_k^{+} := \{z\in X \mathrel | k(z) = z \} \quad \text {and}\quad X_k^{-} := \{z\in X \mathrel | k(z) = -z \} . \end{aligned}$$

(3.16)

Observe that $U\in X_k^+$, $W \in X_k^-$ and, using the fact that the Jacobi polynomials $P_j^{(a, b)}$ are even if j is even and odd if j is odd, an easy computation shows that $W_{n,k} \in X_k^+$ if $n-k$ is even while $W_{n,k} \in X_k^-$ if $n-k$ is odd.

First we prove some properties of the operator T.

Lemma 3.3

The operator T given by (3.14) is well defined and continuous from ${\mathbb R}\times {\mathcal X}$ to $X^2$. Moreover, $\partial _\alpha T$, $\partial _z T$ and $\partial _{\alpha z} T$ exist and are continuous. Finally, T maps ${\mathbb R}\times \bigl ({\mathcal X}\cap (X_k^+ \times X_k^{\pm }) \bigr )$ to $X_k^+\times X_k^{\pm }$.

Proof

First notice that, (F4) implies $\lim _{\lambda \rightarrow 0} F_i(\alpha , \lambda u_1, \lambda u_2) = 0$. Thus, using (F2), one gets

$$\begin{aligned} |F_i(\alpha , u_1, u_2)|&= \biggl |\int _0^1 \partial _\lambda \bigl ( F_i(\alpha , \lambda u_1, \lambda u_2) \bigr ) \,\mathrm {d}\lambda \biggr | \nonumber \\&\le \int _0^1 \bigl |\partial _{u_1} F_i(\alpha , \lambda u_1, \lambda u_2)[u_1] + \partial _{u_2} F_i(\alpha , \lambda u_1, \lambda u_2)[u_2]\bigr | \,\mathrm {d}\lambda \nonumber \\&\le C(u_1^{2^*-2} + u_2^{2^*-2}) (u_1 + u_2) \nonumber \\&\le C(u_1^{2^*-1} + u_2^{2^*-1}). \end{aligned}$$

(3.17)

(Different occurrences of C may denote different constants.) Given that $z_1,z_2$ and U belong to X, (3.17) implies that $\bigl |F_i\bigl (\alpha , U + \frac{z_1+z_2}{2}, U + \frac{z_1-z_2}{2}\bigr )\bigr |\leqslant CU^{2^*-1}$ and thus, using (3.3) and (3.4),

$$\begin{aligned} |f_i(|x|, z_1, z_2)| \le CU^{2^*-1}\quad \text { for }i=1,2. \end{aligned}$$

Then $f_i(|x|, z_1, z_2)$ belong to $L^{\frac{2N}{N+2}}({\mathbb R}^N)$ and there exists a unique $g_i\in D^{1,2}({\mathbb R}^N)$ for $i=1,2$ such that $g_i$ is a weak solution to

$$\begin{aligned} -\Delta g_i = f_i(|x|,z_1,z_2)\quad \hbox {in }{\mathbb R}^N. \end{aligned}$$

(3.18)

The solution $g_i$ enjoys the following representation:

$$\begin{aligned} g_i(x) = \frac{1}{\omega _N(N-2)}\int _{{\mathbb R}^N} \frac{1}{|x-y|^{N-2}} f_i(|y|, z_1,z_2) \,\mathrm {d}y \end{aligned}$$

where $\omega _N$ is the area of the unit sphere in ${\mathbb R}^N$. This implies

$$\begin{aligned} \left| g_{i}(x)\right| \le C\int _{{\mathbb R}^N}\frac{1}{|x-y|^{N-2}}U^{2^*-1}(y) \,\mathrm {d}y = CU(x) \end{aligned}$$

and $g_i\in X$ showing that T is well defined from ${\mathcal X}$ to $X\times X$.

Next we have to show that the operator T maps Kelvin invariant (up to a sign) functions into functions that are Kelvin invariant (with the same sign). It is enough to show that $\bigl ( (-\Delta )^{-1} (f_1(|x|,z_1,z_2)), (-\Delta )^{-1} (f_2(|x|,z_1,z_2)) \bigr )$ maps ${\mathcal X}\cap (X_k^+ \times X_k^{\pm })$ into $X_k^+\times X_k^{\pm }$. Assume $(z_1,z_2)\in {\mathcal X}\cap (X_k^+ \times X_k^{\pm })$ and let, as before, $g_i = (-\Delta )^{-1}(f_i(|x|,z_1,z_2))$. Then $g_i\in X$ is a weak solution to (3.18) and letting $\widetilde{g}_i := k(g_i)$, the Kelvin transform of $g_i$ we have that $\widetilde{g}_i$ weakly solves

$$\begin{aligned} -\Delta \widetilde{g}_i = - \frac{1}{|x|^{N+2}} \Delta g_i\Bigl (\frac{x}{|x|^2}\Bigr ) = \frac{1}{|x|^{N+2}} \, f_i\left( \frac{x}{|x|^2}, z_1\Bigl (\frac{x}{|x|^2}\Bigr ), z_2\Bigl (\frac{x}{|x|^2}\Bigr ) \right) \end{aligned}$$

An easy consequence of (F4) is that

$$\begin{aligned}&\frac{1}{|x|^{N+2}} \, F_i\left( \alpha , \Bigl (U + \frac{z_1+z_2}{2}\Bigr ) \Bigl (\frac{x}{|x|^2}\Bigr ) , \Bigl (U + \frac{z_1-z_2}{2}\Bigr )\Bigl (\frac{x}{|x|^2}\Bigr ) \right) \\&\quad =F_i\Bigl (\alpha ,k(U)+\frac{k(z_1)+k(z_2)}{2}, k(U)+\frac{k(z_1)-k(z_2)}{2}\Bigr ). \end{aligned}$$

This, together with the fact that U and $z_1$ are Kelvin invariant while $z_2$ is Kelvin invariant up to a sign (depending which space $X_k^{\pm }$ we are dealing with) shows that

$$\begin{aligned} \frac{1}{|x|^{N+2}} \, f_i\left( \frac{x}{|x|^2}, z_1\Bigl (\frac{x}{|x|^2}\Bigr ), z_2\Bigl (\frac{x}{|x|^2}\Bigr ) \right) = f_i\bigl (|x|,z_1(x),\pm z_2(x)\bigr ) \end{aligned}$$

where ± depends on the space $X_k^{\pm }$ we consider. Then, using (3.5), it follows that

$$\begin{aligned} \frac{1}{|x|^{N+2}} \, f_1\left( \frac{1}{|x|}, z_1\Bigl (\frac{x}{|x|^2}\Bigr ), z_2\Bigl (\frac{x}{|x|^2}\Bigr ) \right) = f_1\bigl (|x|,z_1(x),z_2(x)\bigr ) \end{aligned}$$

while

$$\begin{aligned} \frac{1}{|x|^{N+2}} \, f_2\left( \frac{x}{|x|^2}, z_1\Bigl (\frac{x}{|x|^2}\Bigr ), z_2\Bigl (\frac{x}{|x|^2}\Bigr )\right) = \pm f_2\bigl (|x|,z_1(x),z_2(x)\bigr ). \end{aligned}$$

This implies that $\widetilde{g}_1$ weakly solves $-\Delta \widetilde{g}_1=f_1(|x|,z_1(x),z_2(x))$ and $\widetilde{g}_2$ solves $-\Delta \widetilde{g}_2=\pm f_2(|x|,z_1(x),z_2(x))$. The uniqueness of solutions in $D^{1,2}({\mathbb R}^N)$ then implies $\widetilde{g}_1 = g_1$ and $\widetilde{g}_2 = \pm g_2$ which shows that $g_1\in X_k^+$ and $g_2\in X_k^{\pm }$. This concludes the first part of the proof.

Let us now prove the continuity of T on ${\mathbb R}\times {\mathcal X}$. Let $\alpha _n \rightarrow \alpha$ in ${\mathbb R}$ and $(z_{1,n}, z_{2,n}) \rightarrow (z_1, z_2)$ in ${\mathcal X}$ as $n \rightarrow \infty$, and set

$$\begin{aligned} g_{i,n} := (-\Delta )^{-1} f_{i,n} \quad \text {where } f_{i,n}(x) := f_i(|x|,z_{1,n},z_{2,n}) \text { with } \alpha = \alpha _n. \end{aligned}$$

Since $z_{i,n} \rightarrow z_i$ in $D^{1,2}({\mathbb R}^N)$, the convergence also holds in $L^{2^*}({\mathbb R}^N)$. Using (3.17) and Lebesgue’s dominated convergence theorem and its converse, one deduces that $f_{i,n} \rightarrow f_i$ in $L^{\frac{2N}{N+2}}$. Therefore $g_{i,n} \rightarrow g_i$ in $D^{1,2}$ and $T(\alpha _n, z_n) \rightarrow T(\alpha , z)$ in $D^{1,2}$. Now let us show the convergence in D. We have that

$$\begin{aligned} \begin{aligned} \frac{|g_{i,n}(x)-g_i(x)|}{U(x)}&\le \frac{1}{\omega _N(N-2)U(x)}\int _{{\mathbb R}^N} \frac{1}{|x-y|^{N-2 }}\frac{|f_{i,n}(y)-f_i(y)|}{U(y)^{2^*-1}} \, U(y)^{2^*-1} \,\mathrm {d}y\\&\le C\sup \limits _{y\in {\mathbb R}^N} \frac{|f_{i,n}(y)-f_i(y)|}{U(y)^{2^*-1}}. \end{aligned} \end{aligned}$$

(3.19)

Moreover, using (F4), one gets

$$\begin{aligned}&\frac{|f_{i,n}(y) - f_i(y)|}{U(y)^{2^*-1}} \le \sum _{i=1}^2 \Bigl | F_i\Bigl (\alpha _n, 1 + \frac{z_{1,n} + z_{2,n}}{2 U},\ 1 + \frac{z_{1,n} - z_{2,n}}{2 U} \Bigr ) \\&\quad - F_i\Bigl (\alpha , 1 + \frac{z_1 + z_2}{2 U},\ 1 + \frac{z_1 - z_2}{2 U} \Bigr ) \Bigr | \end{aligned}$$

Thanks to the convergence in D, ${z_{j,n}}/{U} \rightarrow {z_{j}}/{U}$ uniformly for $j=1,2$. Thus $1 + \frac{z_{1,n} \pm z_{2,n}}{2 U} \rightarrow 1 + \frac{z_{1} \pm z_{2}}{2 U}$ uniformly on ${\mathbb R}^N$. The continuity of the maps $F_i$ then imply that both terms of the sum converge uniformly to 0.

The existence and continuity of the derivatives is proved in a similar way. $\square$

Next we show a compactness result for the operator $(z_1,z_2) \mapsto \left( (-\Delta )^{-1}f_1,(-\Delta )^{-1}f_2 \right)$. Here we need some decay estimates on solutions of a semilinear elliptic equation.

Lemma 3.4

([13]) If $0< p < N$ and h is a non negative, radial function belonging to $L^1({\mathbb R}^N)$, then

$$\begin{aligned} \int _{{\mathbb R}^N} \frac{h(y)}{|x-y|^{p}} \,\mathrm {d}y = O\biggl ( \frac{1}{|x|^{p}} \biggr ) \quad \text {as } |x| \rightarrow +\infty . \end{aligned}$$

Now we can prove our compactness result:

Lemma 3.5

For all $\alpha$, the operator

$$\begin{aligned} M(z_1,z_2) := \left( (-\Delta )^{-1} f_1(|x|,z_1,z_2) , (-\Delta )^{-1}f_2(|x|,z_1,z_2) \right) \end{aligned}$$

(3.20)

is compact from ${\mathcal X}$ to $X^2$.

Proof

1. From Lemma 3.3, we have that $M : {\mathcal X}\rightarrow X^2$ is continuous. Now let $(z_n) = (z_{1,n},z_{2,n})$ be a bounded sequence in ${\mathcal X}$ and let us prove that, up to a subsequence, $g_n := M(z_n)$ converges strongly to some $g\in X\times X$. On one hand, since $(z_n)$ is bounded in $D^{1,2}\times D^{1,2}$, going if necessary to a subsequence, one can assume that $(z_n)$ converges weakly to some $z=(z_1,z_2)$ in $D^{1,2}\times D^{1,2}$ and $z_n \rightarrow z$ almost everywhere. On the other hand, $( \Vert z_n\Vert _{D\times D} )$ is also bounded which means that $|z_{i,n}| \le C U$ where C is independent of i and n and so, using (3.17), $|f_i(|x|, z_n)| \le C U^{2^*-1}$. Lebesgue’s dominated convergence theorem then implies that $f_i(|x|, z_n)$ converges strongly to $f_i(|x|, z)$ in $L^{\frac{2N}{N+2}}$ for $i=1,2$. From the continuity of $(-\Delta )^{-1} : L^{\frac{2N}{N+2}} \rightarrow D^{1,2}$, one concludes that $g_n \rightarrow g$ in $D^{1,2}\times D^{1,2}$. The inequality $|z_{i,n}| \le C U$ also implies

$$\begin{aligned} |g_{i,n}(x)| \le C \int _{{\mathbb R}^N} \frac{1}{|x-y|^{N-2}} \, |f_i(z_n(y))| \,\mathrm {d}y \le C \int _{{\mathbb R}^N}\frac{U^{2^*-1}(y)}{|x-y|^{N-2}} \,\mathrm {d}y = C \, U(x), \end{aligned}$$

and passing to the limit yields $g_i \in D$.

2. It remains to show that $\Vert g_n - g\Vert _{D\times D} \rightarrow 0$. First, Hölder’s inequality allows to get the estimate:

$$\begin{aligned}&|g_{i,n}(x)-g_i(x)|\\&\quad \le C\int _{{\mathbb R}^N} \frac{1}{|x-y|^{N-2}} \, \bigl |f_i(|y|, z_n(y))-f_i(|y|, z(y))\bigr | \,\mathrm {d}y \\&\quad = C\int _{{\mathbb R}^N} \frac{U^{2^*-1-\varepsilon }(y)}{|x-y|^{N-2}} \, \frac{\bigl |f_i(|y|, z_n(y))-f_i(|y|, z(y))\bigr |}{ U^{2^*-1-\varepsilon }(y)} \,\mathrm {d}y\\&\quad \le C \left( \int _{{\mathbb R}^N} \left| \frac{U^{\frac{N+2}{N-2}-\varepsilon }(y)}{|x-y|^{N-2}}\right| ^{\frac{q}{q-1}} \right) ^{\frac{q-1}{q}} \left( \int _{{\mathbb R}^N} \left| \frac{\bigl |f_i(|y|, z_n(y))-f_i(|y|, z(y))\bigr |}{ U^{2^*-1}(y)} \, U^{\varepsilon }(y) \right| ^q\right) ^{\frac{1}{q}} \end{aligned}$$

where $\varepsilon > 0$ will be chosen small and $q > 1$ large such that $\varepsilon q= 2^*$. Note that (3.17) implies $|f_i(|y|, z(y))| \le C (U + |z_1| + |z_2|)^{2^* - 1} + C U^{2^*-1}$ and so the ratio in the right integral is bounded on ${\mathbb R}^N$. Thus the integrand of the right integral is bounded by $C^q\, U^{\varepsilon q}(y) \le C U^{2^*}(y) \in L^1({\mathbb R}^N)$ where C is independent of n. Lebesgue’s dominated convergence theorem then implies that this integral converges to 0 as $n \rightarrow \infty$.

The proof will be complete if we show:

$$\begin{aligned} \begin{aligned} \int _{{\mathbb R}^N} \left| \frac{U^{\frac{N+2}{N-2}-\varepsilon }(y)}{|x-y|^{N-2 }} \right| ^{\frac{q}{q-1}} \,\mathrm {d}y \le \frac{C}{(1+|x|)^{(N-2)\frac{q}{q-1}}} = C U^{\frac{q}{q-1}}(x). \end{aligned} \end{aligned}$$

(3.21)

This inequality follows from Lemma 3.4 because $h := U^{\left( \frac{N+2}{N-2}-\varepsilon \right) \frac{q}{q-1}} \in L^1({\mathbb R}^N)$ i.e., $(N-2) \bigl ( \frac{N+2}{N-2}-\varepsilon \bigr ) \frac{q}{q-1} > N$, and $(N-2) \frac{q}{q-1} < N$ are possible if $\varepsilon$ is small enough and q is large enough. $\square$

4 The role of symmetries

The operator T is a compact perturbation of the identity and, as proved in Lemma 3.3, maps ${\mathbb R}\times \bigl ({\mathcal X}\cap (X_k^+ \times X_k^{\pm }) \bigr )$ into $X_k^+\times X_k^{\pm }$.

We want to find solutions to our problem as zeroes of T and we will use the bifurcation theory. As explained in the introduction, we want to find both radial and nonradial solutions. In particular, to obtain the nonradial ones, we use some symmetry properties of the operator T that can be obtained by (3.5).

We state the definition in a general way and we will then apply to some specific cases so to obtain different solutions. Let us introduce some notations. Let $\mathcal {S}$ be a subgroup of O(N), where O(N) is the orthogonal group of ${\mathbb R}^N$, and let

$$\begin{aligned} X_{\mathcal {S}} := \bigl \{v\in X_k^+ \bigm | \forall s\in \mathcal {S},\ \forall x \in {\mathbb R}^N,\ v(s^{-1}(x)) = v(x) \bigr \} \end{aligned}$$

(4.1)

be the set of functions invariant by the action of $\mathcal {S}$. Let $\sigma : \mathcal {S}\rightarrow \{-1,1\}$ be a group morphism and define a second action of $\mathcal {S}$ on X by

$$\begin{aligned} (s \diamond v)(x) := \sigma (s) \, v\bigl (s^{-1}(x)\bigr ). \end{aligned}$$

(4.2)

The invariant subspace of $X_k^+ \times X_k^{\pm }$ of interest is

$$\begin{aligned} \begin{aligned} {\mathcal Z}:&= \bigl \{z = (z_1, z_2) \in X_k^+\times X_k^{\pm } \bigm | \forall s\in \mathcal {S},\&z_1(s^{-1}(x)) = z_1(x) \text { and }\\&\sigma (s)\, z_2(s^{-1}(x)) = z_2(x) \bigr \}. \end{aligned} \end{aligned}$$

(4.3)

Then we can prove the following result:

Lemma 4.1

The operator T defined in (3.14) maps ${\mathbb R}\times ({\mathcal X}\cap {\mathcal Z})$ into ${\mathcal Z}$.

Proof

We will show that $T = (T_1, T_2)$ is equivariant under the action of $\mathcal {S}$, namely

$$\begin{aligned} T_1\bigl (\alpha , z_1(s^{-1}(x)), \sigma (s)z_2 (s^{-1}(x))\bigr )&=T_1\bigl (\alpha , z_1(x), z_2 (x)\bigr ) , \\ \text {and}\quad T_2\bigl (\alpha , z_1(s^{-1}(x)), \sigma (s)z_2 (s^{-1}(x))\bigr )&=\sigma (s) \, T_2\bigl (\alpha , z_1(x), z_2 (x)\bigr ). \end{aligned}$$

Let $z = (z_1, z_2) \in {\mathcal X}$. First, notice that, thanks to (3.5), the functions $f_1$ and $f_2$ defined in (3.3)–(3.4) satisfy

$$\begin{aligned} f_1\bigl (|x|, z_1(s^{-1}(x)), \sigma (s)z_2 (s^{-1}(x))\bigr )&= f_1(|x|, z(x)) , \\ \text {and}\quad f_2\bigl (|x|, z_1(s^{-1}(x)), \sigma (s)z_2 (s^{-1}(x))\bigr )&= \sigma (s) f_2(|x|, z(x)). \end{aligned}$$

Second, because the Laplacian is equivariant under the action of the group O(N), it readily follows that $(-\Delta )^{-1} \bigl (\sigma (s) f(s(x))\bigr ) = \sigma (s) \bigl ((-\Delta )^{-1} f(s(x))\bigr )$ for any $\sigma$, $s\in \mathcal {S}$ and $f \in L^{2N/(N+2)}$.

Putting these observations together concludes the proof. $\square$

Lemma 4.2

Assume $\beta (\alpha ) \ne \beta _n$ for all $n \in {\mathbb N}$, with $\beta _n$ be as defined in (3.8), and that the subspace of solutions in $X_\mathcal {S}$ to the first equation of (3.7) has only the trivial solution. Still denote T the operator defined in (3.14) restricted to ${\mathcal X}\cap {\mathcal Z}$. Then the linear map $\partial _z T(\alpha , 0,0) : {\mathcal Z}\rightarrow {\mathcal Z}$ is invertible, where $\partial _z T(\alpha , 0,0)$ is the Fréchet derivative of T with respect to z at $(\alpha ,0,0)$.

Proof

For any $(w_1, w_2) \in X^2$, one has, see (3.7),

$$\begin{aligned} \partial _z T(\alpha , 0,0) \begin{pmatrix} w_1\\ w_2 \end{pmatrix} = \begin{pmatrix} w_1-(-\Delta )^{-1} \Bigl ( \frac{N+2}{N-2} U^{\frac{4}{N-2}} \, w_1 \Bigr ) \\ w_2 - (-\Delta )^{-1} \Bigl (\beta (\alpha ) \, U^{\frac{4}{N-2}} \, w_2 \Bigr ) \end{pmatrix} \end{aligned}$$

(4.4)

with $\beta (\alpha )$ as defined in (F6). Since $\partial _z T(\alpha , 0,0)$ is a compact perturbation of the identity (see Lemma 3.5 in [8]), it suffices to prove that $\ker \bigl (\partial _z T( \alpha ,0,0)\bigr ) = \{(0,0)\}$ in ${\mathcal Z}$ whenever $\beta (\alpha ) \ne \beta _n$. Let $(w_1, w_2) \in {\mathcal Z}\subseteq X_k^+ \times X_k^{\pm }$. Notice that $\partial _z T( \alpha ,0,0) \left( {\begin{matrix} w_1 \\ w_2\end{matrix}}\right) = \left( {\begin{matrix} 0\\ 0 \end{matrix}}\right)$ if and only if $(w_1, w_2)$ is a solution to (3.7). By assumption we have that $w_1\equiv 0$ and Proposition 3.1 says that the only solutions to the second equation are given by (3.9) as we assumed $\beta (\alpha ) \ne \beta _n$. This gives the claim. $\square$

Remark 4.3

From Lemma 4.2 we have that, when $\beta (\alpha ) \ne \beta _n$ for all n,

$$\begin{aligned} \deg \bigl (T(\alpha ,\cdot ), \widetilde{B}, 0\bigr ) = \deg \bigl (\partial _z T(\alpha ,0,0), \widetilde{B}, 0 \bigr ) = (-1)^{m(\alpha )} \end{aligned}$$

(4.5)

where $\widetilde{B}$ is a suitable ball in ${\mathcal Z}$ centered at the origin and $m(\alpha )$ the sum of the algebraic multiplicities of all eigenvalues $\lambda$ belonging to (0, 1) of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta w_1 = \lambda \, \tfrac{N+2}{N-2} U^{\frac{4}{N-2}} w_1 &{} \quad \text {in }{\mathbb R}^N, \\ \displaystyle -\Delta w_2 = \lambda \, \beta (\alpha ) \, U^{\frac{4}{N-2}} w_2 &{} \quad \text {in }{\mathbb R}^N,\\ (w_1,w_2)\in {\mathcal Z}. \end{array}\right. } \end{aligned}$$

(4.6)

Proposition 4.4

Assume the same hypotheses as in Lemma 4.2. Let $n \in {\mathbb N}$ and $\alpha ^*_n$ be such that $\beta (\alpha ^*_n) = \beta _n$ (recall that $\beta _n$ is defined in (3.8)). For $\varepsilon > 0$ small enough, the following holds

$$\begin{aligned} m(\alpha ^*_n +\varepsilon ) = m(\alpha ^*_n -\varepsilon ) + \gamma (n) \end{aligned}$$

(4.7)

where $\gamma (n)$ is the algebraic multiplicity of the solutions to $-\Delta w = \beta _n U^{\frac{4}{N-2}} \, w$ such that $(0,w)\in {\mathcal Z}$.

Proof

As the first equation of (4.6) does not depend on $\alpha$, its contribution is the same to the values $m(\alpha ^*_n\pm \varepsilon )$. Concerning the second one, since $\beta (\alpha )$ is a continuous increasing function we have get that $\beta (\alpha ^*_n +\varepsilon )\searrow \beta (\alpha ^*_n)$ and then the contribution of the second equation to $m(\alpha ^*_n +\varepsilon )$ is given by the algebraic multiplicity of the eigenvalues $\lambda =\left\{ \frac{1}{\beta (\alpha ^*_n +\varepsilon )},\ldots , \frac{\beta _n}{\beta (\alpha ^*_n+\varepsilon )}\right\}$. In the same way, for $\varepsilon$ small enough we have that $m(\alpha ^*_n -\varepsilon )$ is given by the algebraic multiplicity of the eigenvalues $\lambda =\left\{ \frac{1}{\beta (\alpha ^*_n +\varepsilon )},\ldots , \frac{\beta _{n-1}}{\beta (\alpha ^*_n +\varepsilon )}\right\}$. This gives the claim. $\square$

Proposition 4.5

Assume the same hypotheses as in Lemma 4.2 and let us suppose that $\gamma (n)$ is an odd integer. Then the point $(\alpha ^*_n, U,U)$ is a bifurcation point from the curve of trivial solutions $(\alpha ,U,U)$ to System (2.1). Moreover the bifurcation is global, the Rabinowitz alternative holds, and for any sequence $(\alpha _k, u_{1,k}, u_{2,k})$ of solutions converging to $(\alpha ^*_n, U,U)$, we have that

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{1,k} = U+\frac{z_{1,k} +z_{2,k}}{2} \\ u_{2,k} = U+\frac{z_{1,k} -z_{2,k}}{2} \end{array}\right. } \end{aligned}$$

and, up to a subsequence,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{1,k} = U + \varepsilon _k Z_n + o(\varepsilon _k) ,\\ u_{2,k} = U - \varepsilon _k Z_n + o(\varepsilon _k) , \end{array}\right. } \end{aligned}$$

(4.8)

as $k\rightarrow \infty$ where $Z_n$ is a solution to the second equation in (3.7) such that $(0, Z_n) \in {\mathcal Z}$, $\Vert Z_n\Vert _X = 1$ and $\varepsilon _k = \Vert z_{2,k}\Vert _X \rightarrow 0$.

Proof

From (4.5) and (4.7), it is standard to see that the curve of trivial solutions for the operator $T : {\mathbb R}\times ({\mathcal X}\cap {\mathcal Z}) \rightarrow {\mathcal Z}$ bifurcates at the values $\alpha ^*_n$ with $\beta (\alpha ^*_n) = \beta _n$ for any n such that $\gamma (n)$ is odd, see [12, Theorem II.3.2] and the bifurcation is global. The Rabinowitz alternative finally follows from [12, Theorem II.3.3].

Next let us show the expansion (2.11). Let $(z_{1,k},z_{2,k})$ be solutions obtained by the bifurcation result to (2.1) as $\alpha _k\rightarrow \alpha ^*_n$ (recall that $(z_{1,k},z_{2,k})\rightarrow (0,0)$ in the space X). First we show that

$$\begin{aligned} \frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert } \le C \end{aligned}$$

(4.9)

where C is a constant independent of k and $\Vert \cdot \Vert = \Vert \cdot \Vert _X$. First, notice that $z_{2,k} \not \equiv 0$ because, if it was, $z_{1,k} \in X_\mathcal {S}$ would satisfy $-\Delta z_1 = f_1(|x|, z_1, 0)$ but the assumption that the first equation of (3.7) has only the trivial solution in $X_\mathcal {S}$ implies that this equation only has trivial solutions for $\alpha \approx \alpha ^*_n$. This contradicts the fact that $(z_{1,k}, z_{2,k})$ lies on the branch of nontrivial solutions.

To show (4.9), let us argue by contradiction: let us suppose that, up to subsequence, $\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert } \rightarrow +\infty$. Set $w_{1,k}=\frac{z_{1,k}}{\Vert z_{1,k}\Vert }$, $w_{2,k}=\frac{z_{2,k}}{\Vert z_{2,k}\Vert }$. The system satisfied by $w_{1,k}$ and $w_{2,k}$ is

$$\begin{aligned} -\Delta w_{1,k}&{=} \frac{1}{\Vert z_{1,k}\Vert } \left[ F_1\left( \alpha _k, U + \Vert z_{1,k}\Vert \frac{w_{1,k}+\frac{\Vert z_{2,k}\Vert }{\Vert z_{1,k}\Vert }w_{2,k}}{2},\ U + \Vert z_{1,k}\Vert \frac{w_{1,k} - \frac{\Vert z_{2,k}\Vert }{\Vert z_{1,k}\Vert }w_{2,k}}{2}\right) \right. \nonumber \\&\quad + F_2\left( \alpha _k, U+\Vert z_{1,k}\Vert \frac{w_{1,k}+\frac{\Vert z_{2,k}\Vert }{\Vert z_{1,k}\Vert }w_{2,k}}{2},\ U + \Vert z_{1,k}\Vert \frac{w_{1,k}-\frac{||z_{2,k}||}{||z_{1,k}||}w_{2,k}}{2}\right) \nonumber \\&\quad - \left. 2 U^{2^*-1}\Bigr ) \right] \end{aligned}$$

(4.10a)

$$\begin{aligned} -\Delta w_{2,k}&{=} \frac{1}{\Vert z_{2,k}\Vert }\left[ F_1\left( \alpha _k, U + \Vert z_{2,k}\Vert \frac{\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }w_{1,k} + w_{2,k}}{2},\ U + \Vert z_{2,k}\Vert \frac{\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }w_{1,k} - w_{2,k}}{2}\right) \right. \nonumber \\&\left. \quad - F_2\left( \alpha _k, U + \Vert z_{2,k}\Vert \frac{\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }w_{1,k} + w_{2,k}}{2},\ U + \Vert z_{2,k}\Vert \frac{\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }w_{1,k} - w_{2,k}}{2}\right) \right] \end{aligned}$$

(4.10b)

$$\begin{aligned}&\Vert w_{1,k}\Vert = \Vert w_{2,k}\Vert = 1 \end{aligned}$$

(4.10c)

Going if necessary to a subsequence, we can assume $w_{1,k} \rightharpoonup w_1$ and $w_{2,k} \rightharpoonup w_2$ in $D^{1,2}$ for some $(w_1, w_2) \in {\mathcal Z}$. Arguing as in the first part of the proof of Lemma 3.5, we deduce that $w_{1,k} \rightarrow w_1$ and $w_{2,k} \rightarrow w_2$ in $L^{2^*}({\mathbb R}^N)$ and in $D^{1,2}$. Using that $F_i(\alpha _k,U,U)=U^{2^*-1}$ for $i=1,2$, we can pass to the limit on Eq. (4.10a) and show that $w_1 \in X_\mathcal {S}$ satisfies

$$\begin{aligned} -\Delta w_1 =\left[ \frac{\partial F_1}{\partial u_1}\bigl (\alpha ^*_n, U,U\bigr ) + \frac{\partial F_1}{\partial u_2} \bigl (\alpha ^*_n, U,U\bigr ) + \frac{\partial F_2}{\partial u_1} \bigl (\alpha ^*_n, U,U\bigr ) + \frac{\partial F_2}{\partial u_2} \bigl (\alpha ^*_n, U,U\bigr ) \right] \frac{w_1}{2} \end{aligned}$$

Moreover, arguing as in the second part of the proof of Lemma 3.5 on (4.10a), we can show that $\Vert w_{1,k} - w_1\Vert _D \rightarrow 0$. Thus $w_{1,k} \rightarrow w_1$ in X and $\Vert w_1\Vert = 1$. As in Sect. 3, using the properties of F we have that $w_1\in X_\mathcal {S}$ satisfies

$$\begin{aligned} -\Delta w_1 = \frac{N+2}{N-2} \, U^{\frac{4}{N-2}} \, w_1 \quad \text {in }{\mathbb R}^N, \end{aligned}$$

This is a contradiction since in $X_\mathcal {S}$ the previous equation admits only the trivial solution. So (4.9) holds.

Hence, up to a subsequence, we have that $\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert } \rightarrow \delta \ge 0$. Passing to the limit in (4.10b), we get that

$$\begin{aligned} -\Delta w_2 = \frac{\partial F_1}{\partial u_1}\bigl (\alpha ^*_n, U,U\bigr ) \frac{\delta w_1+w_2}{2} + \frac{\partial F_1}{\partial u_2}\bigl (\alpha ^*_n, U,U\bigr ) \frac{\delta w_1-w_2}{2}\\ -\frac{\partial F_2}{\partial u_1}\bigl (\alpha ^*_n, U,U\bigr ) \frac{\delta w_1+w_2}{2} - \frac{\partial F_2}{\partial u_2}\bigl (\alpha ^*_n, U,U\bigr ) \frac{\delta w_1-w_2}{2}. \end{aligned}$$

and, arguing again as in the second part of the proof of Lemma 3.5, $w_{2,k}\rightarrow w_2$ in X with $\Vert w_2\Vert = 1$. As before, using the properties of F, we have that $w_2$ solves

$$\begin{aligned} -\Delta w_2 = \beta (\alpha ) \, U^{\frac{4}{N-2}} w_2 \quad \text {in }{\mathbb R}^N, \end{aligned}$$

and hence $w_2 = Z_n$ where $Z_n$ is a solution to the second equation in (3.7) such that $(0, Z_n) \in {\mathcal Z}$ and $\Vert Z_n\Vert = 1$. Then $z_{2,k}= \Vert z_{2,k}\Vert (Z_n+o(1))$. Next we show that

$$\begin{aligned} z_{1,k} = o(1) \Vert z_{2,k}\Vert . \end{aligned}$$

(4.11)

This is clear if $\lim \limits _{k\rightarrow +\infty }\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }=0$ since in this case

$$\begin{aligned} \Vert z_{1,k}\Vert = \frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert } \Vert z_{2,k}\Vert = o(1) \Vert z_{2,k}\Vert . \end{aligned}$$

(4.12)

On the other hand, it is not possible that $\frac{\Vert z_{1,k}\Vert }{\Vert z_{2,k}\Vert }\ge D>0$ because in this case we can pass to the limit in (4.10a) and as before we get a contradiction. This shows (4.11). Coming back to the definition of $(u_{1,k},u_{2,k})$ we have that (4.8) holds with $\varepsilon _k = \Vert z_{2,k}\Vert$. $\square$

Now we specify some subgroups $\mathcal {S}$ that satisfy the assumptions of Lemma 4.2. Observe that when $\beta (\alpha )\ne \beta _n$ the second equation in (3.7) does not possess solutions. The first equation instead admits in $X_k^+$ the solutions $\sum _{i=1}^N a_i\frac{x_i}{(1+|x|^2)^{N/2} }$. Then, the assumptions of Lemma 4.2 are satisfied if the functions $\frac{x_i}{(1+|x|^2)^{N/2}}$ do not belong to $X_S$. The first example is the radial case which allows to prove Theorem 2.1. The other examples, which are provided for every $N\ge 3$, prove the existence of different nonradial solutions.

4.1 The radial case

Following the previous notation we let $\mathcal {S}=O(N)$ and $\sigma : \mathcal {S}\rightarrow \{-1,1\}$ be the group morphism such that $\sigma (s) := 1$ for all $s \in O(N)$. Thus

$$\begin{aligned} X_{\mathcal {S}}&= \bigl \{ v \in X \bigm | \forall x\in {\mathbb R}^N,\ v(x) = v(|x|) \bigr \},\\ {\mathcal Z}\equiv {\mathcal Z}_{{\text {rad}}}^{\pm }&= \bigl \{z\in X_k^+ \times X_k^{\pm } \bigm | \forall x\in {\mathbb R}^N,\ z(x) = z(|x|) \bigr \}. \end{aligned}$$

Proof of Theorem 2.1

To prove the bifurcation result we define the operator T in (3.14) in the space ${\mathcal Z}_{{\text {rad}}}^+\subseteq X_k^+\times X_k^+$ when n is even and in the space ${\mathcal Z}_{{\text {rad}}}^-\subseteq X_k^+\times X_k^-$ when n is odd. Recalling the discussion at the beginning of Sect. 3, we have that the linearized operator $\partial _zT(\alpha ,0,0)$ is invertible if and only if system (3.7) does not admit solutions in ${\mathcal Z}^+_{{\text {rad}}}$ when n is even (${\mathcal Z}^-_{{\text {rad}}}$ in case of n odd). From Proposition 3.1 we know that the first equation in (3.7) does not depend on $\alpha$ and admits the unique radial solution W(|x|) which does not belong to $X_k^+$. The second equation in (3.7) instead admits solutions if and only if $\beta (\alpha )=\beta _n$ and the corresponding radial solution is $W_n(|x|) := W_{n,0}(r)$. Hence the assumption of Lemma 4.2 are satisfied. Moreover from (3.12) and the definition of the Jacobi polynomials we have that $W_n\in X_k^+$ if n is even and $W_n\in X_k^-$ if n is odd showing that $\gamma (n)=1$ for any n. Further, using the monotonicity of $\beta (\alpha )$, the global bifurcation result and the Rabinowitz alternative follows from Theorem II.3.2 and Theorem II.3.3 of [12]. Finally the fact that the curve is continuously differentiable near the bifurcation point follows from the bifurcation result of Crandall–Rabinowitz for one-dimensional kernel since the operator T is differentiable and the transversality condition holds in ${\mathcal Z}$ because

$$\begin{aligned} \partial _{\alpha z}T(\alpha , 0,0) \begin{pmatrix} 0\\ W_n \end{pmatrix} = - \partial _\alpha \beta (\alpha ) \begin{pmatrix} 0\\ (-\Delta )^{-1} \Bigl ( U^{\frac{4}{N-2}} \, W_n \Bigr ) \end{pmatrix} , \end{aligned}$$

and so

$$\begin{aligned} \biggl ( \begin{pmatrix} 0\\ W_n \end{pmatrix} \biggm | \partial _{\alpha z}T(\alpha ^*_n, 0,0)\begin{pmatrix} 0\\ W_n \end{pmatrix} \biggr )_{(D^{1,2})^2} = - \partial _\alpha \beta (\alpha ^*_n) \int _{{\mathbb R}^N} U^{\frac{4}{N-2}} \, W_n^2 \,\mathrm {d}x \ne 0. \end{aligned}$$

$\square$

Proof of Corollary 2.2

It is easy to check that (F1)–(F5) are satisfied. One readily computes that

$$\begin{aligned} \beta (\alpha )={\left\{ \begin{array}{ll} 2 (2^* - 1 - p) \alpha - (2^* - 1 - 2p)&{}\quad \text {in } (2.2),\\ 2^*\alpha -1&{}\quad \text {in } (1.1),\\ \frac{8}{N-2} \alpha - \frac{6-N}{N-2}&{}\quad \text {in } (1.2). \end{array}\right. } \end{aligned}$$

(4.13)

and so (F6) is also satisfied. Moreover (2.5) holds if and only if $\alpha ^* = \alpha ^*_n$ where $\alpha ^*_n$ is defined by (2.8). Corollary 2.2 immediately follows. $\square$

4.2 The first nonradial case

Let h be the reflection through the hyperplane $x_N=0$, $\mathcal {S}_1 := \langle O(N-1), h \rangle$ be the subgroup generated by $O(N-1)$ and h, and $\sigma _1 : \mathcal {S}_1 \rightarrow \{-1,1\}$ be the group morphism such that $\sigma _1(s) := 1$ if $s \in O(N-1)$ and $\sigma _1(h) := -1$ ($\sigma _1$ is easily seen to be well defined because h commutes with any element of $O(N-1)$). Thus

$$\begin{aligned} X_{\mathcal {S}_1}= & {} \bigl \{ v \in X_k^+ \bigm | \forall x=(x',x_N)\in {\mathbb R}^N,\ v(x', x_N) = v(|x'|, - x_N) \bigr \},\\ {\mathcal Z}\equiv {\mathcal Z}_1^{\pm }= & {} \bigl \{z\in X_k^+ \times X_k^{\pm } \bigm | \forall x=(x',x_N)\in {\mathbb R}^N,\quad z_1(x',x_N) = z_1(|x'|, - x_N) \text { and }\\&\quad z_2(x',x_N) = - z_2(|x'|, - x_N) \bigr \}. \end{aligned}$$

Observe that the odd symmetry helps to kill the radial solution in the kernel of the linearized system while the even symmetries help to avoid the solutions given by the translation invariance of the problem. Indeed since functions in $X_{\mathcal {S}_1}$ are even with respect to each $x_i$, $i = 1,\ldots , N$ and belong to $X_k^+$ from Proposition 3.1, it is easily deduced that the solutions in $X_{\mathcal {S}_1}$ of the first equation of (3.7) (see (3.9)) are the trivial ones. Thus Lemma 4.2 applies and by Proposition 4.5 the bifurcation result can be proved when $\gamma (n)$ is odd.

Proposition 4.6

With this choice of $\mathcal {S}= \mathcal {S}_1$ and $\sigma = \sigma _1$, we have that $\gamma (n)$ is odd if and only if $n = 4\ell + 1$ or $n = 4\ell + 2$ for $\ell = 0,1,\ldots$

Proof

In ${\mathbb R}^N$, we consider the spherical coordinates $(r,\varphi ,\theta _1,\ldots ,\theta _{N-2})$ with $r=|x|\in [0,+\infty )$, $\varphi \in [0,2\pi ]$, and $\theta _i\in [0,\pi ]$ as $i=1,2,\ldots ,N-2$ with

$$\begin{aligned} {\left\{ \begin{array}{ll} x_1 = r\cos \varphi \sin \theta _1\ldots \sin \theta _{N-2}\\ x_2 = r\sin \varphi \sin \theta _1\ldots \sin \theta _{N-2}\\ \quad \vdots \\ x_{N-1} = r\sin \theta _{N-2}\cos \theta _{N-3}\\ x_N = r\cos \theta _{N-2} . \end{array}\right. } \end{aligned}$$

(4.14)

Proposition 3.1 says that the solutions to $-\Delta w = \beta _n \, U^{\frac{4}{N-2}} \, w$ are, in radial coordinates, linear combinations of the $n+1$ functions

$$\begin{aligned}{}[0, +\infty ) \times {\mathbb S}^{N-1} \rightarrow {\mathbb R}: (r, \varphi ,\theta _1,\ldots ,\theta _{N-2}) \mapsto W_{n,k}(r) Y_k(\varphi ,\theta _1,\ldots ,\theta _{N-2}) \end{aligned}$$

(4.15)

for $k=0,\ldots ,n$, where $Y_k(\varphi ,\theta _1,\ldots ,\theta _{N-2})$ are spherical harmonics with eigenvalue $k(k+N-2)$. For any k, there is only a single (up to a scalar multiple) spherical harmonic which is $O(N-1)$-invariant and it is given by the function:

$$\begin{aligned}&Y_k(\varphi ,\theta _1,\ldots ,\theta _{N-2}) = Y_k(\theta _{N-2}) = P_k^{(\frac{N-3}{2}, \frac{N-3}{2})}(\cos \theta _{N-2}) \nonumber \\&\quad \text {where } r \cos \theta _{N-2} = x_N \text { with } \theta _{N-2}\in [0,\pi ], \end{aligned}$$

(4.16)

and $P_k^{(\frac{N-3}{2},\frac{N-3}{2})}$ are the Jacobi Polynomials, see [7] for example. Then, the algebraic multiplicity of the solutions to $-\Delta w = \beta _n \, U^{\frac{4}{N-2}} \, w$ that are $O(N-1)$-invariant is $n+1$. By definition of the space ${\mathcal Z}_1$, the solution $\bigl (0, W_{n,k}(r) Y_k(\theta _{N-2}) \bigr )$ belongs to ${\mathcal Z}_1$ if and only if $Y_k$ is odd with respect to $x_N$, that is iff $Y_h(\theta _{N-2}) = -Y_h(\pi -\theta _{N-2})$. Since the Jacobi Polynomials are even if k is even and odd if k is odd, $Y_k(\theta _{N-2})$ is odd with respect to $x_N$ if and only if k is odd. This implies that to compute $\gamma (n)$ we only have to consider the odd indices k.

The radial part corresponding to the index k is given by

$$\begin{aligned} W_{n,k}(r) = \frac{r^k}{(1+r^2)^{k+\frac{N-2}{2}}} P_{n-k}^{\left( k+\frac{N-2}{2}, k+\frac{N-2}{2}\right) } \left( \frac{1-r^2}{1+r^2}\right) . \end{aligned}$$

If $n=2j$, we consider the operator T defined in $X_k^+\times X_k^-$. In this way, $\frac{1}{|x|^{N-2}} \cdot W_{n,k}\bigl (\frac{x}{|x|^2}\bigr ) = -W_{n,k}(x)$ since $n-k$ is odd for any k odd. Then $\gamma (n)=\sum _{k=0, \ k\text { odd}}^n 1=j$ and it is odd if and only if $j=2\ell +1$, or equivalently $n=4\ell +2$.

If, instead, n is odd, then $n-k$ is even for any k odd and so we consider the operator T defined in $X_k^+\times X_k^+$. Indeed, in this case, $W_{n,k}(r)\in X_k^+$ for every k odd and so $\gamma (n) = j+1$ and it is odd if and only if $j=2\ell$, equivalently $n = 4\ell + 1$ and this concludes the proof. $\square$

Proof of Theorem 2.4

As explained before we are in position to apply Proposition 4.5 using Proposition 4.6. The expansion in (2.11) follows again from Proposition 4.5. Finally let us show that our continuum of solutions contains nonradial functions. If by contradiction we have that $u_1$ and $u_2$ are both radial we get that $z_{2}=u_1-u_2$ is also radial. But $z_{2}$ is odd in the last variable and so we get that $z_{2}\equiv 0$. Then $u_1=u_2$ and by (F3)–(F4) we deduce that $F_i(\alpha ,u_1,u_1)= u_1^{2^*-1}$. This implies that $u_1=u_2=U$, a contradiction. $\square$

4.3 The general case: Proof of Theorem 2.6

Since the general case involves hard notations, for reader’s convenience we consider first the case $m = 2$ and prove Corollary 2.8. The general case does not involve additional difficulties and we just will sketch it.

Let $h_1$ (resp. $h_2$) be the reflection through the hyperplane $x_N = 0$ (resp. $x_{N-1} = 0$), $\mathcal {S}_2 = \langle O(N-2), h_1, h_2 \rangle$ and $\sigma _2 : \mathcal {S}_2 \rightarrow \{-1,1\}$ be the group morphism that satisfies $\sigma _2(s) = 1$ whenever $s \in O(N-2)$ and $\sigma _2(h_1) = \sigma _2(h_2) = -1$. Thus

$$\begin{aligned} X_{\mathcal {S}_2}= & {} \bigl \{ v\in X \bigm | \forall x=(x', x_{N-1}, x_N)\in {\mathbb R}^N , v(x', x_{N-1}, x_N) = v(|x'|, - x_{N-1}, x_N),\\&\quad v(x', x_{N-1}, x_N) = v(|x'|, x_{N-1}, - x_N) \bigr \},\\ {\mathcal Z}\equiv {\mathcal Z}_2= & {} \bigl \{z\in X^+_k \times X^{\pm }_k \bigm | \forall x=(x', x_{N-1}, x_N)\in {\mathbb R}^N,\\&\quad z_1(x', x_{N-1}, x_N) = z_1(|x'|, -x_{N-1}, x_N), \\&\quad z_1(x', x_{N-1}, x_N) = z_1(|x'|, x_{N-1}, - x_N), \\&\quad z_2(x', x_{N-1}, x_N) = - z_2(|x'|, - x_{N-1}, x_N), \text { and}\\&\quad z_2(x', x_{N-1}, x_N) =- z_2(|x'|, x_{N-1}, - x_N) \bigr \}, \end{aligned}$$

With this choice, arguing as in the previous case we have that the only solution in $X_{\mathcal {S}_2}$ to the first equation of (3.7) is the trivial one. As a consequence, Proposition 4.5 applies and a bifurcation occurs when $\gamma (n)$ is odd.

It remains to compute $\gamma (n)$. To do this we will compute the dimension of ${\mathcal Y}_k^{\mathcal {S}_2}({\mathbb R}^N)$, the space of spherical harmonics on ${\mathbb R}^N$ related to the eigenvalue $k(k+N-2)$ which are invariant by the action of $\mathcal {S}$ induced by $\sigma$ (thus, for $\mathcal {S}= \mathcal {S}_2$, we select the spherical harmonics which are invariant under the action of $O(N-2)$ and odd with respect to $x_N$ and $x_{N-1}$).

First, let use prove the following decomposition lemma:

Lemma 4.7

Let ${\mathcal P}^{\mathcal {S}_2}({\mathbb R}^N)$ be the space of the polynomials in N variables which are invariant by the action of $O(N-2)$ and such that $\forall x\in {\mathbb R}^N,\ v(h_i(x))=-v(x)$, for $i=1,2$. Then

$$\begin{aligned} {\mathcal P}^{\mathcal {S}_2}({\mathbb R}^N) = x_N x_{N-1} {\mathbb R}[r^2,x_{N-1}^2,x_N^2] \quad \text {where } r^2=x_1^2+\cdots +x_{N-2}^2 \end{aligned}$$

(4.17)

and ${\mathbb R}[a_1,\ldots ,a_k]$ denotes the space of polynomials in the variables $a_1, \ldots ,a_k$.

Proof

The proof is similar as in Lemma 6.4 in [14]. If p(x) is a polynomial in $x_N x_{N-1}{\mathbb R}[r^2,x_{N-1}^2,x_N^2]$ then it has an odd degree in $x_N$ and $x_{N-1}$ and so it satisfies $p(h_i(x))=-p(x)$ for $i=1,2$. Moreover it depends on even powers of $x_1^2+\cdots +x_{N-2}^2$ and so it is invariant with respect to any $s\in O(N-2)$. Thus $x_N x_{N-1} {\mathbb R}[r^2,x_{N-1}^2,x_N^2] \subseteq {\mathcal P}^{\mathcal {S}_2}({\mathbb R}^N)$.

Conversely, let $p\in {\mathcal P}^{\mathcal {S}_2}({\mathbb R}^N)$. Since $p(h_i(x))=-p(x)$ for $i=1,2$ then each term in p has to contain an odd power of $x_{N-1}$ and $x_N$. We can then define the polynomial $q(x) := \frac{p(x)}{x_{N-1}x_N}$ which is even in $x_{N-1}$ and $x_N$. Now let $s\in O(N-2)$ such that $s(x_1,\ldots ,x_{N-2})=(r,0,\ldots ,0)$ with $r^2=x_1^2+\cdots +x_{N-2}^2$. Then q is invariant so that $q(x_1,\ldots ,x_N) = q\bigr (s(x_1,\ldots ,x_{N-2}),x_{N-1},x_N\bigr ) = q(r,0\cdots ,0,x_{N-1},x_N) = q(-r,0\ldots ,0,x_{N-1},x_N)$ where the last equality comes from the fact that the map $(x_1, x_2,\ldots ,x_{N-2})\mapsto (-x_1, x_2,\ldots ,x_{N-2})$ belongs to $O(N-2)$. Then q has to be even in r and this implies that $q \in {\mathbb R}[r^2,x_{N-1}^2,x_N^2]$. $\square$

Proposition 4.8

With this choice of $\mathcal {S}= \mathcal {S}_2$ and $\sigma = \sigma _2$, $\gamma (n)$ is odd if and only if $n = 8\ell + 2$, $n= 8\ell + 3$, $n= 8\ell + 4$ or $n = 8\ell + 5$ for $\ell = 0,1,\ldots$

Proof

Recall that ${\mathcal Y}_k({\mathbb R}^N)$, the space of spherical harmonics of eigenvalue $k(k+N-2)$ for $-\Delta _{{\mathbb S}^{N-1}}$ consists of harmonic homogeneous polynomials of degree k. As stated in Proposition 5.5 of [2], the space ${\mathcal P}_k$ of homogeneous polynomials of degree k can be decomposed as a direct sum of ${\mathcal Y}_k({\mathbb R}^N)$ with a subspace isomorphic to ${\mathcal P}_{k-2}$. This decomposition still holds when restricted to polynomials that are $O(N-2)$-invariant and odd with respect to $x_N$ and $x_{N-1}$. This follows easily using the formula (5.6) of [2]. As a consequence,

$$\begin{aligned} \dim {\mathcal Y}_k^{\mathcal {S}_2}({\mathbb R}^N) = \dim {\mathcal P}_k^{\mathcal {S}_2}({\mathbb R}^N) - \dim {\mathcal P}_{k-2}^{\mathcal {S}_2}({\mathbb R}^N) \end{aligned}$$

(4.18)

where ${\mathcal P}_k^{\mathcal {S}_2}({\mathbb R}^N)$ is the space of homogeneous polynomials on ${\mathbb R}^N$ of degree k which are $O(N-2)$-invariant and odd with respect to $x_N$ and $x_{N-1}$.

In view of (4.18), we have to compute the dimension of ${\mathcal P}_k^{\mathcal {S}_2}({\mathbb R}^N)$ using the decomposition in Lemma 4.7.

It is not difficult to show that for any $h\in {\mathbb N}$ we have ${\mathcal P}_{2h+1}^{\mathcal {S}_2}({\mathbb R}^N) = \{0\}$ since any polynomial in it must contain $x_{N-1}x_N$ and powers of $x_1^2+\cdots +x_{N-2}^2$ and this is not possible if the degree of the polynomial is odd. So we have proved that $\dim {\mathcal Y}_{2h+1}^{\mathcal {S}_2}({\mathbb R}^N)=0$ for any h and N.

Then let us compute $\dim {\mathcal Y}_{2h}^{\mathcal {S}_2}({\mathbb R}^N)$. Again from Lemma 4.7, we have that ${\mathcal P}_{k}^{\mathcal {S}_2}({\mathbb R}^N) = {{\mathrm{span}}}\bigl \{ x_N^{2h+1} \, x_{N-1}^{k-2\ell -2h-1} \, r^{2\ell } \bigm | h=0, \ldots , \frac{k-2}{2}$ and $\ell = 0, \ldots , \frac{k-2h-2}{2} \bigr \}$ so that

$$\begin{aligned} \dim {\mathcal P}_k^{\mathcal {S}_2}({\mathbb R}^N) = \sum _{h=0}^{\frac{k-2}{2}} \sum _{\ell =0}^{\frac{k-2h-2}{2}} 1 = \frac{k}{4} \left( \frac{k}{2}+1\right) \end{aligned}$$

and using (4.18) we get for k even

$$\begin{aligned} \dim {\mathcal Y}_{k}^{\mathcal {S}_2}({\mathbb R}^N) = \frac{k}{4} \left( \frac{k}{2} + 1\right) - \frac{k-2}{4} \left( \frac{k-2}{2} + 1\right) = \frac{k}{2} . \end{aligned}$$

(4.19)

In this case the unique spherical harmonics which contribute to the computation of $\gamma (n)$ are those of index k even. The corresponding radial part is $W_{n,k}(r)$ which belongs to $X^+$ if n is even and to $X^-$ if n is odd. Then, when n is even we define the operator T in the space $X^+\times X^+$ and we have that

$$\begin{aligned} \gamma (n) = \sum _{k=0}^{n} \dim {\mathcal Y}_{k}^{\mathcal {S}_2}({\mathbb R}^N) = \sum _{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor } \dim {\mathcal Y}_{2j}^{\mathcal {S}_2}({\mathbb R}^N) = \sum _{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor } j = \frac{1}{2} \left\lfloor \frac{n}{2}\right\rfloor \left( \left\lfloor \frac{n}{2}\right\rfloor + 1 \right) \end{aligned}$$

Then $\gamma (n)$ is odd when $n=8j+2$ and $n=8j+4$. When n is odd instead, we define the operator T in the space $X^+\times X^-$ and we have again that

$$\begin{aligned} \gamma (n) = \sum _{k=0}^{n} \dim {\mathcal Y}_{k}^{\mathcal {S}_2}({\mathbb R}^N) = \sum _{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor } \dim {\mathcal Y}_{2j}^{\mathcal {S}_2}({\mathbb R}^N) = \sum _{j=0}^{\left\lfloor \frac{n}{2}\right\rfloor } j = \frac{1}{2} \left\lfloor \frac{n}{2}\right\rfloor \left( \left\lfloor \frac{n}{2}\right\rfloor + 1 \right) \end{aligned}$$

Then $\gamma (n)$ is odd when $n=8j+3$ and $n=8j+5$ concluding the proof. $\square$

Proof of Corollary 2.8

It is the same as the one of Theorem 2.4 (using Proposition 4.8). $\square$

Now we sketch the general case of Theorem 2.6. Let $N\geqslant 3$ and $2\le m\le N-1$. For $i = 1, \ldots , m$ let $h_i$ be the reflection through the hyperplane $x_{N+1-i} = 0$, $\mathcal {S}_m = \langle O(N-m), h_1, \ldots , h_m \rangle$, and $\sigma _m : \mathcal {S}_m \rightarrow \{-1,1\}$ be the group morphism defined by $\sigma _m(s) = 1$ for $s \in O(N-m)$ and $\sigma _m(h_i) = -1$. Thus

$$\begin{aligned} X_{\mathcal {S}_m}&= \bigl \{v\in X_k^+ \bigm | \begin{aligned}&\forall x=(x',x_{N-m+1}, \ldots , x_N) \in {\mathbb R}^{N-m}\times {\mathbb R}^m,\ \forall i_1,\ldots , i_m \in {\mathbb N},\\&\quad v(x) = v(|x'|, (-1)^{i_1} x_{N-m+1}, \ldots , (-1)^{i_m} x_N)\}, \end{aligned} \\ {\mathcal Z}_m&= \bigl \{z\in X_k^+ \times X_k^{\pm } \bigm | \begin{aligned}&\forall x = (x', x_{N-m+1}, \ldots , x_N)\in {\mathbb R}^{N-m}\times {\mathbb R}^m,\ \forall i_1,\ldots , i_m \in {\mathbb N},\\&z_1(x) = z_1(|x'|, (-1)^{i_1} x_{N-m+1}, \ldots , (-1)^{i_m} x_N), \text { and }\\&z_2(x) = (-1)^{i_1+\cdots +i_m}\, z_2(|x'|, (-1)^{i_1} x_{N-m+1}, \ldots , (-1)^{i_m} x_N) \bigr \}. \end{aligned} \end{aligned}$$

As before we have that there is no nontrivial solution in $X_{\mathcal {S}_m}$ to the first equation of (3.7). Hence by Proposition 4.5 we only have to compute $\gamma (n)$. Analogously to the case $m=2$ we use the following decomposition lemma:

Lemma 4.9

Let ${\mathcal P}^{S_m}({\mathbb R}^N)$ be the space of the polynomials in N variables which are invariant under the action of $O(N-m)$ and such that $\forall x\in {\mathbb R}^N,\ v(h_i(x))=-v(x)$ for all $i=1,\ldots ,m$. Then

$$\begin{aligned} {\mathcal P}^{\mathcal {S}_m}({\mathbb R}^N) = x_{N-m+1}\ldots x_{N} \, {\mathbb R}[r^2,x_{N-m+1}^2,\ldots ,x_N^2] \quad \text {where } r^2 = x_1^2 + \cdots + x_{N-m}^2 \end{aligned}$$

(4.20)

and ${\mathbb R}[a_1,\ldots ,a_k]$ denotes the space of polynomials in the variables $a_1, \ldots ,a_k$.

Proposition 4.10

With this choice of $\mathcal {S}= \mathcal {S}_m$ and $\sigma = \sigma _m$, $\gamma (n)$ is odd if and only if

$$\begin{aligned} \left( {\begin{array}{c}m+\left\lfloor \frac{n-m}{2}\right\rfloor \\ m\end{array}}\right) \text { is an odd integer}. \end{aligned}$$

Proof

As in the proof of Proposition 4.8 we have that

$$\begin{aligned} \dim {\mathcal Y}_k^{\mathcal {S}_m}({\mathbb R}^N) = \dim {\mathcal P}_k^{\mathcal {S}_m}({\mathbb R}^N) - \dim {\mathcal P}_{k-2}^{\mathcal {S}_m}({\mathbb R}^N) \end{aligned}$$

(4.21)

where ${\mathcal P}_k^{\mathcal {S}_m}({\mathbb R}^N)$ is the space of homogeneous polynomials on ${\mathbb R}^N$ of degree k which are $O(N-m)$-invariant and odd with respect to $x_{N-m+1},\ldots ,x_{N}$. Because of Lemma 4.9, all non-zero polynomials invariant under the action induced by $\sigma$ on $\mathcal {S}_m$ must have degree at least m and so ${\mathcal P}_k^{\mathcal {S}_m}({\mathbb R}^N) = \{0\}$ for $k=0,\ldots ,m-1$, and $\dim {\mathcal P}_{m}^{\mathcal {S}_m}({\mathbb R}^N) = 1$. Moreover, as in the case $m=2$, ${\mathcal P}_{m+2h+1}^{\mathcal {S}_m}({\mathbb R}^N) = \{0\}$ for any $h \in {\mathbb N}$. For ${\mathcal P}_{m+2h}^{\mathcal {S}_m}({\mathbb R}^N)$, the decomposition in Lemma 4.9 implies that is is isomorphic to ${\mathcal P}_{h}(a_1,\ldots ,a_{m+1})$, the space of homogeneous polynomials of degree h in $m+1$ variables. Thus $\dim {\mathcal P}_{m+2h}^{\mathcal {S}_m}({\mathbb R}^N) = \dim {\mathcal P}_{h}(a_1,\ldots ,a_{m+1}) = \left( {\begin{array}{c}h+m\\ m\end{array}}\right)$. Then, using (4.21), we get

$$\begin{aligned} \dim {\mathcal Y}_{m+2h}^{\mathcal {S}_m}({\mathbb R}^N) = \left( {\begin{array}{c}h+m\\ m\end{array}}\right) - \left( {\begin{array}{c}h-1+m\\ m\end{array}}\right) = \left( {\begin{array}{c}h+m-1\\ m-1\end{array}}\right) , \qquad h \in {\mathbb N}. \end{aligned}$$

This implies that $\gamma (n)=0$ for $n\leqslant m-1$. As when $m=2$ we get

$$\begin{aligned} \gamma (n) = \sum _{h=0}^{\left\lfloor \frac{n-m}{2}\right\rfloor } \dim {\mathcal Y}_{m+2h}^{\mathcal {S}_m}({\mathbb R}^N) = \sum _{h=0}^{\left\lfloor \frac{n-m}{2}\right\rfloor } \left( {\begin{array}{c}h+m-1\\ m-1\end{array}}\right) . \end{aligned}$$

Now we use the so called hockey-stick identity

$$\begin{aligned} \sum _{i=r}^\ell \left( {\begin{array}{c}i\\ r\end{array}}\right) = \left( {\begin{array}{c}\ell +1\\ r+1\end{array}}\right) \end{aligned}$$

which implies

$$\begin{aligned} \gamma (n) = \sum _{h=0}^{\left\lfloor \frac{n-m}{2}\right\rfloor } \left( {\begin{array}{c}h+m-1\\ m-1\end{array}}\right) = \left( {\begin{array}{c}m+\left\lfloor \frac{n-m}{2}\right\rfloor \\ m\end{array}}\right) . \end{aligned}$$

Finally the proof of Theorem 2.6 follows as in Theorem 2.4. $\square$

Proof of Theorem 2.6

From Proposition 4.10 we have that $\gamma (n)$ is odd when $\left( {\begin{array}{c}m+\left\lfloor \frac{n-m}{2}\right\rfloor \\ m\end{array}}\right)$ is odd. Then the proof follows from Proposition 4.5. $\square$

5 Other solutions

The use of other symmetry subgroups of O(N) makes it possible to find different solutions. As an example we give another choice that generates nonradial solutions non equivalent to the previous ones.

For $m \ge 1$, let $R_m$ be the rotation of angle $\frac{2\pi }{m}$ in $\varphi$, $h_i$ the reflection with respect to $x_{i} = 0$, $i=2,\ldots ,N$. Set $\mathcal {S}_m = \langle R_m, h_2, h_3,\ldots ,h_{N} \rangle$, and $\sigma _m : \mathcal {S}_m \rightarrow \{-1,1\}$ be the group morphism defined by $\sigma _m(R_m) = 1$, $\sigma _m(h_2) = -1$, and $\sigma _m(h_i) = 1$ for $i = 3,\ldots ,N$. (One easily checks that $\sigma _m$ is well defined using $R_m h_2 R_m = h_2$.) Thus, using spherical coordinates, see (4.14),

$$\begin{aligned} X_{\mathcal {S}_m}= & {} \Bigl \{v\in X_k^+ \bigm | \forall x=(r,\varphi ,\theta _1, \ldots , \theta _{N-2}) \in {\mathbb R}^{N},\\&\quad v( r,\varphi ,\theta _1, \ldots , \theta _{N-2} ) = v\left( r,2\pi -\varphi , \pi -\theta _1, \ldots , \pi -\theta _{N-2}\right) ,\\&\quad v( r,\varphi ,\theta _1, \ldots , \theta _{N-2} ) = v\Bigl (r,\varphi +\frac{2\pi }{m}, \pi -\theta _1, \ldots , \pi -\theta _{N-2}\Bigr ) \Bigr \} \end{aligned}$$

$$\begin{aligned} {\mathcal Z}\equiv&{\mathcal Z}_m = \Bigl \{z\in X^+_k \times X^+_k \bigm | \forall x = (r,\varphi ,\theta _1, \ldots , \theta _{N-2}) \in {\mathbb R}^{N},\\&\quad z_1(x) = z_1\left( r,2\pi -\varphi , \pi -\theta _1, \ldots , \pi -\theta _{N-2}\right) ,\\&\quad z_1(x) = z_1\Bigl (r,\varphi +\frac{2\pi }{m}, \pi -\theta _1, \ldots , \pi -\theta _{N-2}\Bigr ),\\&\quad z_2(x) = -z_2\left( r,2\pi -\varphi , \pi -\theta _1, \ldots , \pi -\theta _{N-2}\right) ,\\&\quad z_2(x) = z_2\Bigl (r,\varphi +\frac{2\pi }{m}, \pi -\theta _1, \ldots , \pi -\theta _{N-2}\Bigr )\Bigr \}. \end{aligned}$$

Let us show that, for any $m\ge 2$, the first equation in (3.7) admits only the trivial solution. By Proposition 3.1, we have that $w= \sum _{i=1}^N a_i\frac{x_i}{(1+|x|^2)^{N/2}} + bW$. By (4.14) and the definition of $X_{\mathcal {S}_m}$, we get that $a_1=a_2=0$ (using the invariance with respect to $R_m$) and $a_3=\cdots =a_{N-2}=0$ (using that $\cos \theta _i\ne \cos (\pi -\theta _i)$, for any $i=1,\ldots ,\theta _{N-2}$). Finally $b=0$ since $W\not \in X^+_k$. Thus the assumptions of Proposition 4.2 are satisfied. To apply Proposition 4.4, we also need:

Proposition 5.1

Let $m \ge 2$, $n=m$, $\mathcal {S}:= \mathcal {S}_n$ and $\sigma := \sigma _n$. Then

$$\begin{aligned} \gamma (n)=1. \end{aligned}$$

(5.1)

Proof

By Proposition 3.1 all solutions to the second equation of (3.7) corresponding to $\alpha ^*_m$ are given by $\sum _{k=0}^m A_k W_{m,k}(r) Y_k(\varphi ,\theta _1,\ldots ,\theta _{N-2})$. We know from [15] (see also [1] for another use of this expansion in bifurcation theory) that

$$\begin{aligned}&Y_k(\varphi ,\theta _1,\ldots , \theta _{N-2}) \nonumber \\&\quad = \!\! \sum \limits _{\begin{array}{c} j=0,\ldots , k\\ i_0\le i_1\ldots \le i_{N\!-\!2}\\ i_0=j, \ i_{N-2} =k \end{array}} \prod _{\ell =1}^{N-2} G_{i_{\ell }}^{i_{\ell -1}}(\cos \theta _\ell , \ell -1) \left( B_j^{i_1\ldots i_{N\!-\!3}} \cos j\varphi + C^{i_1\ldots i_{N\!-\!3}}_j \sin j\varphi \right) \! , \end{aligned}$$

(5.2)

where $G_i^0(\cdot , \ell )$ are the Gegenbauer polynomials namely,

$$\begin{aligned} \sum \limits _{i=0}^{\infty }G_i^0(\omega ,\ell ) x^i = (1-2x\omega +x^2)^{-(1+\ell )/2} , \end{aligned}$$

while

$$\begin{aligned} G_i^k(\omega ,\ell ) = (1-\omega ^2)^{{k}/{2}} \, \frac{{\mathrm d}^k}{{\mathrm d}\omega ^k} G_i^0(\omega ,\ell ) . \end{aligned}$$

By definition of the space ${\mathcal Z}_m$, the solution $\bigl (0,\sum _{k=0}^m A_j W_{m,k}(r) Y_k( \varphi ,\theta _1,\ldots , \theta _{N-2} ) \bigr )$ belongs to ${\mathcal Z}_m$ if and only if $Y_k( \varphi ,\theta _1,\ldots , \theta _{N-2} )$ is ${2\pi }/{m}$ periodic in $\varphi$, changes sign under the transformation $\varphi \mapsto 2\pi -\varphi$, and is invariant under the transformations $\theta _i \mapsto \pi -\theta _i$. The first two imply that $Y_k(\varphi ,\theta _1,\ldots , \theta _{N-2} )$ must not be constant in $\varphi$ and $k \ge j\ge m$. Thus solutions to the second equation in (3.7) with ${\mathcal Z}_m$-invariance are multiple of $W_{m,m}(r)Y_m(\theta )$. Moreover, the unique nonzero coefficient in (5.2) is $C_m^{m\ldots m}$.

Because $G_i^0(\cdot , \ell )$ is a polynomial of degree $\ell$, $G_m^m(\omega , \ell )$ is a constant multiple of $(1 - \omega ^2)^{m/2}$. A straightforward computation shows that

$$\begin{aligned} Y_m( \varphi ,\theta _1, \ldots , \theta _{N-2}) = (\sin \theta _{N-2})^m \ldots (\sin \theta _1)^m \sin (m\varphi ) = \mathfrak {I}(x_1+ \mathbf {i} x_2)^m. \end{aligned}$$

Observe that $Y_m( \varphi ,\theta _1, \ldots , \theta _{N-2})$ is invariant with respect to the reflection $h_i$ for $i=3,\ldots ,N$, with respect to the rotation $R_m$ and it is odd in $\varphi$ so that $(0,W_{m,m}(r)Y_m(\theta ))$ belongs to $\mathcal {Z}_m$. Recalling that $W_{m,m}(r) = \smash {\frac{r^m}{\left( 1+r^2\right) ^{m+\frac{N-2}{2}}}} \in X^+_k$, we have that $\gamma (m)=1$. $\square$

Proof of Theorem 2.9

From the previous discussion we have that the assumption of Lemma 4.2 are satisfied. Then the proof follows as in the case of Theorem 2.1 since we have a one dimensional kernel. $\square$