Existence of non-radial solutions to semilinear elliptic systems on a unit ball in ${\mathbb {R}}^3$

Abstract

In this paper, we prove the existence of non-radial solutions to the problem $-\triangle u= f(x,u)$, $u|_{\partial \Omega }=0$ on the unit ball $\Omega :=\{x\in {\mathbb {R}}^3: \Vert x\Vert <1\}$ with $u(x)\in {\mathbb {R}}^s$, where f is a sub-linear continuous function, differentiable with respect to u at zero and satisfying $f(gx,u) = f(x,u)$ for all $g\in O(3)$, $f(x,-u)=- f(x,u)$. We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.

Appendices

Appendix A: Notation for subgroups of $O(3)\times {{\mathbb {Z}}}_2$

The subgroups of $O(3)=SO(3)\times {{\mathbb {Z}}}_2$, classified up to conjugacy, are the product subgroups of type $H\times {{\mathbb {Z}}}_2$, where $H\le SO(3)$;

(ii) the twisted subgroups of $SO(3)\times {{\mathbb {Z}}}_2$, which are

$$\begin{aligned} H^\varphi :=\{(g,z)\in SO(3)\times {{\mathbb {Z}}}_2: \varphi (g)=z\}, \end{aligned}$$

where $H\le SO(3)$ and $\varphi :H\rightarrow {{\mathbb {Z}}}_2$ is a homomorphism. In particular, the subgroups H of SO(3), which are SO(3), O(2), SO(2), $D_n$, $n\ge 2$ ($D_2$ denoted by $V_4$ is the Klein group), ${{\mathbb {Z}}}_n$, $n\ge 1$, and the exceptional groups $A_4$ – the tetrahedral group), $S_4$ – the octahedral group) and $A_5$ – the icosahedral group), can be identified with $H^\varphi$ where $\varphi :H\rightarrow {{\mathbb {Z}}}_2$ is a trivial homomorphism. We also denote by $z:D_n\rightarrow {{\mathbb {Z}}}_2$ the epimorphism satisfying $\mathrm{Ker\,}(z)={{\mathbb {Z}}}_n$, by $d:D_{2n}\rightarrow {{\mathbb {Z}}}_2$, the epimorphism with $\mathrm{Ker\,}(d)=D_n$; $d:{{\mathbb {Z}}}_ {2n}\rightarrow {{\mathbb {Z}}}_2$ with $\mathrm{Ker\,}(d)={{\mathbb {Z}}}_n$; and we denote by symbol ${ -}$ the epimorphism $\varphi :S_4\rightarrow {{\mathbb {Z}}}_2$ with $\mathrm{Ker\,}(\varphi )=A_4$; similarly, w denote by − the epimorphisms $\varphi :V_4\rightarrow {{\mathbb {Z}}}_2$ with $\mathrm{Ker\,}(\varphi )={{\mathbb {Z}}}_2$, $\varphi :O(2)\rightarrow {{\mathbb {Z}}}_2$ with $\mathrm{Ker\,}(\varphi )=SO(2)$ and $\varphi :O(3)\rightarrow {{\mathbb {Z}}}_2$ with $\mathrm{Ker\,}(\varphi )=SO(3)$.

Table 5 lists the twisted subgroups $H^\varphi$ in $O(3)=SO(3)\times {{\mathbb {Z}}}_2$ having finite Weyl group $W(H)=N(H)/H$.

Table 5 Twisted subgroups of O(3)

To describe the conjugacy classes of subgroups in $O(3)\times {{\mathbb {Z}}}_2$, we use a special convention that allows us to describe the subgroups of a product group $G_1\times G_2$ (i.e., here, $G_1=O(3)$ and $G_2={{\mathbb {Z}}}_2$). For two groups $G_{1}$ and $G_{2}$, the well-known result of É. Goursat (see [17]) provides the following description of arbitrary subgroup ${\mathscr {H}} \le G_{1}\times G_{2}$; namely, there exist subgroups $H\le G_{1}$ and $K\le G_{2}$, a group L, and two epimorphisms $\varphi :H\rightarrow L$ and $\psi :K\rightarrow L$, such that

$$\begin{aligned} {\mathscr {H}}=\{(h,k)\in H\times K:\varphi (h)=\psi (k)\}. \end{aligned}$$

The notation used to describe ${\mathscr {H}}$ is

$$\begin{aligned} {\mathscr {H}}:=H^{\varphi }{}\times _{L}^{\psi }K, \end{aligned}$$

(11)

in which case $H^{\varphi }{}\times _{L}^{\psi }K$ is called an amalgamated subgroup of $G_{1}\times G_{2}$. In our case, one can identify $\varphi$ and $\psi$ by their kernels $H_o:=\mathrm{Ker\,}(\varphi )$ and $K_o:=\mathrm{Ker\,}(\psi )$ (here, we are trying to describe the conjugacy classes $({\mathscr {H}})$ of ${\mathscr {H}}$), i.e., the group ${\mathscr {H}}$ is the amalgamated notation can be written as

$$\begin{aligned} {\mathscr {H}}=H^{H_o} \times _L^{K_o} K. \end{aligned}$$

Appendix B: Spectral properties of the Laplace operator ${\mathscr {L}}$

1.1 B.1: Spectrum of ${\mathscr {L}}$

Consider first a more general situation where $\Omega \subset {\mathbb {R}}^d$ is the open unit ball in ${\mathbb {R}}^d$ ($d\ge 2$) and consider the eigenvalue equation for the scalar Laplace operator

$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u(x)=\lambda u, \quad x\in \Omega \\ u|_{\partial \Omega }= 0.\end{array} \right. \end{aligned}$$

(12)

for some constant $\lambda$. Then, the problem (12) can be written by the following equation:

$$\begin{aligned} {\mathscr {L}} u=\lambda u, \quad u\in D({\mathscr {L}}), \end{aligned}$$

(13)

where ${\mathscr {L}} u:=-\triangle u$ is considered as an unbounded operator in the space ${\mathscr {E}}=L^2(\Omega ,{\mathbb {R}})$ with the domain $D({\mathscr {L}})=H_o^2(\Omega ,{\mathbb {R}})$. To describe the spectrum of ${\mathscr {L}}$, one can use the spherical coordinates $(r,\varvec{\theta })$ in ${\mathbb {R}}^d$ (here $\varvec{\theta }\in S^{d-1}$). To be more precise, the Laplace operator $-\triangle$ in spherical coordinates can be written as

$$\begin{aligned} \triangle u=\frac{\partial ^2 u}{\partial r^2} +\frac{d-1}{r} \frac{\partial u}{ \partial r}+\frac{1}{r^{2}} \triangle _{S^{d-1}}u, \end{aligned}$$

(14)

where $\triangle _{S^{d-1}}$ is the Laplace–Beltrami operator on $S^{d-1}$ ( $\triangle _{S^{d-1}}$ is also called the spherical Laplacian).

The spectrum of the operator $\triangle _{S^{d-1}}$ is composed of increasing sequence of eigenvalues $\mu _l\ge 0$, $l=0,1,\dots$ (with $\mu _0=0$), each of them of finite multiplicity. To describe the spectrum of the operator ${\mathscr {L}}$, i.e., to find the non-zero solutions to (13), one can use the method of separation of variables, i.e., we consider a particular form of functions u, namely $u(r,\varvec{\theta })=R(r)\cdot T(\varvec{\theta })$, which substituted to the equation $-\triangle u=\lambda u$ leads to

$$\begin{aligned} \left( -R''-\frac{d-1}{r} R'\right) T-\frac{1}{r^2} \left( \triangle _{S^{d-1}}(T)\right) R&=\lambda RT \end{aligned}$$

that can be written as

$$\begin{aligned} \frac{r^2R''+(d-1)rR'}{R}+\lambda r^2=-\frac{\triangle _{S^{d-1}}(T)}{T}. \end{aligned}$$

(15)

Since the functions in (15) standing on the left- and on the right-hand side of equation depend on different variables, there must exist a constant c, such that

$$\begin{aligned} r^2R''+(d-1)rR'+(\lambda r^2-c)R=0, \end{aligned}$$

(16)

$$\begin{aligned} \triangle _{S^{d-1}}(\Theta )= c\Theta . \end{aligned}$$

(17)

The equation (17) implies that c is an eigenvalue of the spherical Laplacian $-\triangle _{S^{d-1}}$. One can express the constant c as $c=k(k+d-1)$. Then, using the substitution $R(r):=r^{-\frac{d-2}{2}}{{\hat{R}}}(r)$, (16) becomes

$$\begin{aligned} {{\hat{r}}}^2R''+r{{\hat{R}}}'+\lambda r^2-\left( k+\frac{d-2}{2}\right) ^2{{\hat{R}}}=0. \end{aligned}$$

(18)

After changing variable using $\widetilde{R}(t)=R(\frac{r}{\sqrt{\lambda }})$, (18) becomes the classical Bessel equation

$$\begin{aligned} t^2\widetilde{R}''(t)+t\widetilde{R}'(t)+\left( t^2-\left( k+\frac{d-2}{2}\right) ^2\right) \widetilde{R}(t)=0. \end{aligned}$$

The bounded at zero solution $\widetilde{R}(t)=J_{k+\frac{d-2}{2}}(t)$, where $J_{k+\frac{d-2}{2}}$ stands for the $k+\frac{d-2}{2}$-th Bessel function of the first kind. Therefore, we obtain that solutions to (16) are constant multiples of $R(r)=r^{-\frac{d-2}{2}}J_{k+\frac{d-2}{2}}(\sqrt{\lambda }r)$ (notice that R(r) is finite at zero). Consequently, $u(r,\theta )=R(r)\Theta (\varvec{\theta })$ satisfies the Dirichlet condition if $R(1)=0$, i.e., $J_{k+\frac{d-2}{2}}(\sqrt{\lambda })=0$. In this way, we obtain that

$$\begin{aligned} \sigma ({\mathscr {L}})=\left\{ s_{km}: s_{km}\ \text {is the}\ m\text {-th positive zero of}\ J_{k+\frac{d-2}{2}}, \; k=1,2,\ldots \right\} . \end{aligned}$$

Moreover, we obtain that corresponding to $\lambda =s_{km}$, eigenspace is generated by the functions

$$\begin{aligned} r^{-\frac{d-2}{2}}J_{k+\frac{d-2}{2}}(s_{km}r)T(\varvec{\theta }), \end{aligned}$$

where T is an eigenfunction of $-\triangle _{S^{d-1}}$ corresponding to the eigenvalue $c =k(k+d-1)$.

We are interested in the cases when $\Omega$ is an open unit ball in ${\mathbb {R}}^3$, i.e., $d=3$.

In the case $d=3$, using the spherical coordinates $(r,\theta ,\phi )$, $r\ge 0$, $\theta \in [0,2\pi ]$, $\varphi \in [0,\pi ]$, we obtain the following formula for the Laplacian $\triangle _{S^{2}}$:

$$\begin{aligned} \triangle _{S^{2}}&=\frac{\partial ^2}{\partial \varphi ^2}+ \frac{\cos \varphi }{\sin \varphi }\frac{\partial }{\partial \varphi }+ \frac{1}{\sin ^2\varphi } \frac{\partial ^2}{\partial \theta ^2}. \end{aligned}$$

(19)

Then, by substituting $u(\theta ,\varphi )=\Theta (\theta )\Phi (\varphi )$ to the equation $-\triangle _{S^{2}}u=\mu u$, we obtain

$$\begin{aligned} \Theta \Phi ''+\frac{\cos \varphi }{\sin \varphi }\Theta \Phi '+\frac{1}{\sin ^2 \varphi } \Theta ''\Phi =-\mu \Theta \Phi . \end{aligned}$$

(20)

Then, by separating the variables in (20), we obtain

$$\begin{aligned} \frac{\sin ^2 \varphi \Phi ''+\cos \varphi \sin \varphi \Phi '}{\Phi }+\mu \sin ^2\varphi =-\frac{\Theta ''}{\Theta }=n^2, \end{aligned}$$

where $n=0,1,\dots$. Then, clearly, $\Theta (\theta )=a\cos (n\theta )+b\sin (n\theta )$, and therefore

$$\begin{aligned} \sin ^2 \varphi \Phi ''+\cos \varphi \sin \varphi \Phi '+\Big (k(k+1)\sin ^2\varphi -n^2\Big )\Phi =0, \end{aligned}$$

(21)

where $\mu =k(k+1)$. Equation (21), using the change of variables $s=\cos \varphi$, and $P(s)=P(\cos \varphi )=\Phi (\phi )$, is transformed into the classical Legendre equation

$$\begin{aligned} (1-s^2)P''-2sP'+\left( k(k+1)-\frac{n^2}{1-s^2}\right) P=0, \end{aligned}$$

(22)

which, for $k\in {{\mathbb {Z}}}$, admits a bounded solution $P_k^n$, the so-called called Legendre function) on $[-1,1]$. One can use the following well-known formula for $P_k^n$:

$$\begin{aligned} P_k^n(s)=\frac{(1-s^2)^{\frac{n}{2}}}{2^kk!}\frac{d^{k+n}}{ds^{k+n}}(s^2-1)^k, \end{aligned}$$

to conclude that $P_k^n\equiv 0$ for $n>k$, i.e., for $n=1,2,3,\dots$, and $k\ge n$, $P_k^n$ is a non-constant solution to (22) satisfying the boundary condition $P(-1)=P(1)=0$.

Consequently, we obtain that the eigenfunctions of $-\triangle _{S^{2}}$, corresponding to the eigenvalue $\mu _k=k(k+1)$, $k\ge 0$, are

$$\begin{aligned} T(\theta ,\varphi ):=P(\cos \varphi )\Big (a\cos (n\theta )+b\sin (n\theta )\Big )\quad a,\, b\in {\mathbb {R}},\; 0\le n\le k, \end{aligned}$$

where P is a solution to (21). Put

$$\begin{aligned} T^n_{k}(\theta ,\varphi ):=P_k^n(\cos \varphi )\Big (a\cos (n\theta )+b\sin (n\theta )\Big )\quad a,\, b\in {\mathbb {R}},\; 0\le n\le k.\nonumber \\ \end{aligned}$$

(23)

The function $T^n_{k}$ is called surface harmonics of degree k. Then, the eigenspace ${\mathbb {V}}_k$ of $-\triangle _{S^{2}}$ (considered as an unbounded operator in $L^2(S^2,{\mathbb {R}})$) is generated by the functions $T^n_{k}$, that is

$$\begin{aligned} {\mathbb {V}}_{k}=\text {span}\{ T^n_{k}: 0\le n\le k\}. \end{aligned}$$

The orthogonality properties of Legendre functions imply that

$$\begin{aligned} \text {dim} {\mathbb {V}}_k=1+2k. \end{aligned}$$

Consequently, we obtain that the spectrum of ${\mathscr {L}}$ for $d=3$ is given by

$$\begin{aligned} \sigma ({\mathscr {L}})=\left\{ s_{km}: s_{km}\ \text {is the}\ m\text {-th positive zero of}\ J_{k+\frac{1}{2}}\right\} . \end{aligned}$$

(24)

Moreover, we have the following eigenfunctions corresponding to $s_{km}$:

$$\begin{aligned} u(r,\theta ,\varphi )=r^{-\frac{1}{2}}J_{k+\frac{1}{2}}(s_{km}r)P_k^n(\cos \varphi )\Big ( a\cos (n\theta )+b\sin (n\theta )\Big ), \end{aligned}$$

where $a,\, b\in {\mathbb {R}}$ and $0\le n\le k$. Consequently, the eigenspace ${\mathscr {E}}(s_{km})$ of ${\mathscr {L}}$ corresponding to $s_{km}$ is given by

$$\begin{aligned} {\mathscr {E}}(s_{km})=\text {span}\,\Big \{r^{-\frac{1}{2}}J_{k+\frac{1}{2}}(s_{km}r)T^n_k(\theta ,\varphi ): a, \, b\in {\mathbb {R}}, \; 0\le n \le k \Big \}\nonumber \\ \end{aligned}$$

(25)

1.2 B.2: Irreducible $O(3)\times {{\mathbb {Z}}}_2$-representations and eigenspaces of ${\mathscr {L}}$

The irreducible O(3)-representations can be described using homogeneous polynomials (for more details, we refer to [16]). We denote by $W_k$, $k=0,1,2,3,\dots$, the space of homogeneous polynomials $p:{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ of degree k and $W_k:=\{0\}$ for $k<0$. Clearly, $g\in O(3)$ acts on $W_k$ by

$$\begin{aligned} (gp)(v)=p\left( g^{-1}v\right) ,\quad v=(x,y,z)^T\in {\mathbb {R}}^3. \end{aligned}$$

In addition, define $\rho :{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ by $\rho (x,y,z)=x^2+y^2+z^2$, and put

$$\begin{aligned} U_k:=\{\rho p: p\in W_{k-2}\}. \end{aligned}$$

Then, the space $W_k$, $k=0,1,2,\dots$, admits an O(3)-invariant direct sum decomposition

$$\begin{aligned} W_k=U_k\oplus {\mathcal {V}}_k, \end{aligned}$$

which leads the following classical result

Proposition B.1

The O(3)-representations ${\mathcal {V}}_k$, $k=0,1,2,\dots$, which are called the natural irreducible O(3)-representations, are absolutely irreducible and $\textrm{dim} {\mathcal {V}}_k=1+2k$. Then, the $O(3)\times {{\mathbb {Z}}}_2$-representation ${\mathcal {V}}_k^-={\mathcal {V}}_k$ (equipped with the antipodal ${{\mathbb {Z}}}_2$-action) is also absolutely irreducible. In addition, the eigenspace ${\mathscr {E}}(s_{km})$ is $O(3)\times {{\mathbb {Z}}}_2$-equivalent to ${\mathcal {V}}_k^-$ (for every $m\in {{\mathbb {N}}}$).

Remark B.1

Let us point out that there is no way to be assured that the eigenvalues $s_{km}$ do not repeat; however, for a fixed k, the consecutive zeros $s_{km}$ of the Bessel functions $J_{k+\frac{1}{2}}$ are different; therefore, ${\mathscr {E}}(s_{km})$, which is $O(3)\times {\mathbb {Z}}_2$-equivariantly equivalent to the irreducible G-representation ${\mathcal {V}}_k$, are in fact the ${\mathcal {V}}_k$-isotypic component of the eigenspace corresponding to the eigenvalue $s_{km}$.