A monotonicity result under symmetry and Morse index constraints in the plane

TL;DR

This paper proves a monotonicity and symmetry result for solutions of semilinear elliptic equations in symmetric plane domains, linking Morse index constraints to solution symmetry or monotonicity properties.

Contribution

It establishes a new monotonicity and symmetry theorem for solutions with bounded Morse index under rotational symmetry constraints in the plane.

Findings

Solutions are either radial or symmetric with respect to a direction and monotone in an angular sector.

The result applies to least-energy and nodal least-energy solutions.

Produces multiplicity results for solutions under symmetry constraints.

Abstract

This paper deals with solutions of semilinear elliptic equations of the type \[ \left\{\begin{array}{ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \] where $\Omega$ is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions $u$ that are invariant by rotations of a certain angle $\theta$ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or $u$ is radial, or, else, there exists a direction $e\in \mathcal S$ such that $u$ is symmetric with respect to $e$ and it is strictly monotone in the angular variable in a sector of angle $\frac{\theta}2$. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

Figures

Equations

$$\left\{\begin{array}[]{ll}-\Delta u=f(|x|,u)&\text{ in }\Omega,\\ u=0&\text{ on }\partial\Omega,\end{array}\right.$$

$$-\Delta u=f(|x|,u)\text{ in }\Omega ,$$

$$u=0\text{ on }\partial \Omega .$$

$$Q_{u,\Omega }(v,v):={\int }_{\Omega }|\nabla v{|}^2-f^{\prime }(|x|,u)v^{2}dx$$

$$L_u(v):=-\Delta v-f^{\prime }(|x|,u)v$$

$$u(x)=u\left(g(x)\right)\text{ for all }x\in \Omega ,\text{ for every }g\in {\mathcal{G}}_{\frac{2\pi }{k}}$$

$$m_k(u)\le 2.$$

$$w_e(x):=u({\sigma }_e(x))-u(x)$$

$$v(x)=v\left(g(x)\right)\text{ for all }x\in \Omega ,\text{ for every }g\in {\mathcal{G}}_{\frac{2\pi }{k}}$$

$$S_{k,e}^{+}:=\{(r,\theta )\in S_{2k,e}:\psi <\theta <\psi +\frac{\pi }{k}\}$$

$$S_{k,e}^{-}:\{(r,\theta )\in S_{2k,e}:\psi -\frac{\pi }{k}<\theta <\psi \}$$

$$L_u:=-\Delta -f^{\prime }(|x|,u)I$$

$$Q_{u,D}(v,w):={\int }_D\nabla v\nabla w-f^{\prime }(|x|,u)vw$$

$$V_e(x):={\int }_0^1{f}^{\prime }(|x|,tu({\sigma }_e(x))+(1-t)u(x))dt,$$

$$V_{es}(x)=\frac{1}{2}\left(f^{\prime }(|x|,u(x))+f^{\prime }(|x|,u({\sigma }_e(x)))\right)$$

$$L_e:=-\Delta -V_e(x)I{L}_{es}:=-\Delta -V_{es}(x)I$$

$$Q_{e,D}(v,w):={\int }_D\nabla v\nabla w-V_e(x)vw{Q}_{es,D}(v,w):={\int }_D\nabla v\nabla w-V_{es}(x)vw.$$

$$L_e{w}_e=0.$$

$$w_e>0in{S}_{k,e}^{+}or{w}_e<0in{S}_{k,e}^{+}.$$

$${\lambda }_1(L_u,S_{k,e^{\prime }}^{+})\ge 0.$$

$$\tilde{\psi }=\sup \{\psi \in [0,\frac{\pi }{k}):s.t.w_{{\psi }^{\prime }}\ge 0in{S}_{k,{\psi }^{\prime }}^{+}\text{ for all }{\psi }^{\prime }\in [0,\psi )\}$$

$$w_{\tilde{\psi }+\epsilon }>\frac{\eta }{2}>0inK\subset S_{k,\tilde{\psi }+\epsilon }^{+}$$

$$w_{\tilde{\psi }+\epsilon }>0in{S}_{k,\tilde{\psi }+\epsilon }^{+}\setminus K$$

$${\lambda }_1(L_u,S_{k,e}^{+})\ge 0.$$

$$-\Delta u_{\theta }=f^{\prime }(|x|,u)u_{\theta }in{S}_{k,e}^{+}$$

$$f(|x|,u({\sigma }_e(x)))-f(|x|,u(x))\ge f^{\prime }(|x|,u(x))w_e(x)$$

$$-\Delta w_e-f^{\prime }(|x|,u)w_e\ge 0in{S}_{k,e}^{+}$$

$$Q_{u,S_{k,e}^{+}}(w_e^{-},w_e^{-})\le 0$$

Full text

[TABLE]

where $\Omega$ is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions $u$ that are invariant by rotations of a certain angle $\theta$ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or $u$ is radial, or, else, there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $e$ and it is strictly monotone in the angular variable in a sector of angle $\frac{\theta}{2}$.

The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.

This work was supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), by Prin-2015KB9WPT and by Fabbr.

Keywords: semilinear elliptic equations, symmetry and monotonicity results, bounded and unbounded domains, linearized equation, first eigenvalue and first eigenfunction, Morse index.

1. Introduction

In this paper we study symmetry and monotonicity properties of classical solutions of semilinear elliptic problems of the type

[TABLE]

where $\Omega$ is a radially symmetric domain of $\mathbb{R}^{2}$ which can be bounded, in which case $\Omega$ is either a ball $B$ or an annulus centered at the origin, or can be unbounded in which case either $\Omega=\mathbb{R}^{2}$ or $\Omega=\mathbb{R}^{2}\setminus B$. When $\Omega$ is a ball $B$, an annulus or an exterior domain $\mathbb{R}^{2}\setminus B$ we also assume that $u$ satisfies Dirichlet boundary conditions

[TABLE]

Throughout the paper $f:\bar{\Omega}\times\mathbb{R}\to\mathbb{R}$ is (locally) a $C^{0,{\alpha}}$-function whose first derivative with respect the second variable, that we denote hereafter $f^{\prime}(|x|,s):=\frac{\partial f}{\partial s}(|x|,s)$, belongs to $C^{0,{\alpha}}$.

When $\Omega$ is a ball and $f$ is nonicreasing in the radial variable, positive solutions to (1.1) and (1.2) are radially symmetric by the well known results of [GNN] where the moving plane method has been employed. Similar results hold also when $\Omega=\mathbb{R}^{2}$ at least under some decay assumption at infinity, see [GNN2] or under some summability conditions as in [CL]. However, when $\Omega$ is not convex or when $f$ depends increasingly on the radial variable or when $u$ is a sign changing solution, the symmetry of all solutions does not hold anymore and indeed, in each of these cases, there are examples where radial and nonradial solutions coexist. We quote here the seminal paper [SSW] dealing with positive solutions of the Hénon problem, where $f(|x|,s)=|x|^{\alpha}s^{p}$ for ${\alpha}>0$ and $p>1$ in a ball, and it is proved that least energy solutions, namely solutions which minimize the energy functional, are nonradial provided ${\alpha}$ is large enough. This symmetry breaking result is stated in dimension $N\geq 3$, but it holds also in the plane as one can see in the examples in Section 6. Nonradial positive solutions have been found also in an annulus and one can see the papers [BN], [C], [L] and [GGPS]. Also least energy nodal solutions are nonradial, when $\Omega$ is bounded as proved in [AP].

Nevertheless in some situations, or for a certain class of solutions, it is natural to expect that solutions inherit some of the symmetry of the domain, even if $\Omega$ is not convex, $u$ changes sign and $f$ is increasing in the radial variable. This is indeed the case for solutions of low Morse index, under some convexity assumption on $f$ that we shall make clear very soon, and it has been proved in [P] and [PW] when $\Omega$ is bounded, and in [GPW] when $\Omega$ is unbounded, that every solutions to (1.1) (and possibly (1.2)) of low Morse index is axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis, i.e. only depends on $r=|x|$ and $\theta=\arccos(x\cdot p)$, for a certain unit vector $p$, and $u$ is nonincreasing in $\theta$. This kind of symmetry is often called foliated Schwarz symmetry. See also the papers [DP], [DGP1] and [DGP2] where some extensions to the case of systems are considered.

The foliated Schwarz symmetry for minimizers of certain variational problems has been obtained in [SW] for positive solutions and in [BWW] in the case on nodal solutions, using a completely different method based on symmetrization techniques.

Let us recall that the Morse index of a solution $u$ to (1.1) is the maximal dimension of a subspace of $C^{1}_{0}(\Omega)$ in which the quadratic form

[TABLE]

associated to the linearized operator

[TABLE]

is negative defined. Here and in the following by $f^{\prime}(|x|,u)$ we mean $\frac{\partial f}{\partial s}(|x|,s)$.

In this paper we are interested in solutions which admit some invariances, namely they belong to suitable symmetric spaces, in the case when $\Omega$ is contained in the plane. Inspiring to the previous depicted papers [PW] and [GPW] we consider solutions which have low Morse index in these symmetric spaces and we prove, that under some convexity assumptions on $f$ or they are radial or they inherit only one extra-symmetry among the ones they can possess. This extra-symmetry of low Morse index solutions is what we think is the right generalization to the foliated Schwarz symmetry to this symmetric setting.

Entering the details let us explain which type of symmetric spaces we are concern with. To this end, for any angle $\psi$ we denote by $R_{\psi}$ the rotation of angle $\psi$ in counterclockwise direction centered at the origin and by $\mathcal{G}_{\psi}$ the subgroup of $SO(2)$ generated by $R_{\psi}$. In particular we consider angles $\psi=\frac{2\pi}{k}$ with $k\in\mathbb{N}$, $k\geq 1$, so that $\mathcal{G}_{\frac{2\pi}{k}}$ is a proper subgroup of $SO(2)$. We say that a function $u$ defined in $\Omega$ is $k$-invariant if it satisfies

[TABLE]

Next we denote by $m_{k}(u)$ the $k$-invariant Morse index of $u$, namely the maximal dimension of a subspace of $C^{1}_{0,k}$, i.e. the subspace of $C^{1}_{0}(\Omega)$ given by functions that satisfy (1.3), in which the quadratic form $Q_{u,\Omega}$ is negative defined.

Since we are considering functions that satisfy (1.3) it is enough to know them in any sector of $\Omega$ of angle $\frac{2\pi}{k}$. Then, for any direction $e\in\mathcal{S}$, $\mathcal{S}$ being the unit sphere, we denote by $S_{2k,e}$ the sector of $\Omega$ of angle $\frac{2\pi}{k}$, centered in the origin which has the straight line of direction $e$ passing through the origin, that we call $r_{e}$, as symmetry axis and lies in the halfplane $x\cdot e>0$, see Fig. 1.

In order to state our results we also need to introduce the two semisectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$ that cover $S_{2k,e}$ and which are the part of $S_{2k,e}$ on one side of $r_{e}$ and on the other side respectively, see Fig. 2 and see Section 2 where they are defined in a rigorous way.

Now we can state the main results of this paper. In particular we can prove the following result for solutions of low Morse index, in the case when the nonlinear term $f$ is convex in the second variable:

Theorem 1.1**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3) and such that

[TABLE]

Suppose $f(|x|,s)$ is convex in the $s$-variable. Then, or $u$ is radial or else there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $e$ in the sector $S_{2k,e}$ and it is strictly monotone in the angular variable in the sectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$.

Also in the case when the nonlinear term $f$ has a first derivative convex in the second variable we can prove an analogous result:

Theorem 1.2**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3) and such that

[TABLE]

Suppose $f(|x|,s)$ has a convex derivative $f^{\prime}(|x|,s)=\frac{\partial f}{\partial s}(|x|,s)$. Then, or $u$ is radial or else there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $e$ in the sector $S_{2k,e}$ and it is strictly monotone in the angular variable in the sectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$.

To our knowledge this is the first general result in this direction, i.e. in showing that solutions, constraint with some invariances, when they are nonradial, inherit a symmetry axis in their free sector $S_{2k,e}$, but they repulse additional symmetries, favoring the strict angular monotonicity in their free semisectors $S_{k,e}^{\pm}$. It seems that this symmetry in $S_{2k,e}$ and monotonicity in $S_{k,e}^{\pm}$ is the exact generalization of the foliated Schwarz symmetry to the case of $k$-invariant functions or to the case of sectors.

Let us explain the difficulties of the achievements. As in the previous papers the convexity assumptions on $f$ are needed in order to compare the quadratic form $Q_{u}$ with the quadratic form associated with the equation satisfied by the difference of two solutions to (1.1). In particular we compare a solution $u$ with its reflection with respect to $r_{e}$ in the sector $S_{2k,e}$ and we move the direction $e$.

Next, the assumption on the $k$-Morse index of $u$ allows to say that at least in one of the semisectors $S_{k,e}^{+}$, for a suitable direction $e$, the first eigenvalue of the linearized operator $L_{u}$ is nonnengative. This implies that the angular derivative of $u$ in this sector $S_{k,e}^{+}$ or it is zero or it has a sign. However it is possibile to prove that the first eigenvalue is nonnegative only when $u$ is symmetric with respect to $r_{e}$ in $S_{2k,e}$ so that the solution $u$ when nonradial admits this extra-symmetry.

Let us note in particular that the proof of Theorem 1.2 is technically complicated when $m_{k}(u)=2$ and also the sectors of amplitude $\frac{\pi}{2k}$ play an important role.

As one can see in the applications in Section 6 the assumption $m_{k}(u)\leq 2$ is not very restrictive and allows to treat the case of solutions of variational problems which minimizes the energy functional associated with (1.1) constraint either on the Nehari manifold or on the nodal Nehari manifold in the case of solutions that change sign. So also nodal solutions can be considered. According to the previous results in [PW] and [GPW] we believe moreover that this assumption should be optimal and that allowing a higher $k$-Morse index would produce more symmetries in the sectors $S_{2k,e}$.

Note that the convexity assumption of Theorem 1.1 is satisfied by the exponential nonlinearities of Gelfand type, i.e., $f(r,s)={\lambda}V(r)e^{s}$ and, for $s>0$ by the Lane-Emden type, i.e. $f(r,s)=|s|^{p-1}s$ for $p>1$ and by their extension of Hénon type, namely $f(r,s)=r^{\alpha}e^{s}$ and $f(r,s)=r^{\alpha}|s|^{p-1}s$ for ${\alpha}>0$.

The convexity assumption in Theorem 1.2, instead, allows to deal with nodal solutions to the Lane-Emden problem, for $p\geq 2$ and with positive or nodal solutions to the sinh-Poisson problem, namely when $f(r,s)={\varepsilon}(e^{s}-e^{-s})$ as well as their extensions of Hénon type. Section 6 provides a broad range of problems that satisfy the convexity assumptions and to which these topics can be applied and some multiplicity results are produced.

Finally in the case when the domain $\Omega$ is not bounded we can prove the same type of monotonicity results.

Theorem 1.3**.**

Assume $\Omega$ is unbounded. Let $u$ be a solution to (1.1) and possibly (1.2) that satisfies (1.3) and such that $|\nabla u|\in L^{2}(\Omega)$ and

[TABLE]

Assume further that $f$ or $f^{\prime}$ are convex in the second variable. Then, or $u$ is radial or else there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $e$ in $S_{2k,e}$ and it is strictly monotone in the angular variable in the sectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$.

Let us remark that very few results are available when $f$ depends on the radial varaibale, or when the solutions change sign or when the underlying domain is $\mathbb{R}^{2}\setminus B$ and we think that Theorem 1.3 is a first step in this direction.

When passing from the bounded to the unbounded case we have to take into account either the fact that the first eigenvalue and the first eigenfunction is not defined anymore in an unbounded domain either that some of the functions constructed do not have the right sommability. Then we have to divide the problem considering first a bounded section of the sectors $S_{2k,e}$ and then looking at the unbounded part of the sectors. In this last issue it is important that, in a certain sense, the bound on the $k$-Morse index, means that the maximum principle should hold in the unbounded part.

Finally let us observe that we are confident that similar results should hold also in higher dimensions. Nevertheless while the $k$-invariance in (1.3) is very natural for solutions in a radially symmetric domain of the plane, since they are the same invariances of the Spherical Harmonics, in higher dimension the Spherical Harmonics are far more complicated and many different invariances should be taken into account.

2. Notations and preliminary results

In this section we introduce all the notations we need to prove the main Theorems and we give some preliminary results in the case when $\Omega$ is bounded. Let us denote by $e\in\mathcal{S}$ any direction, $e=e_{\psi}=(\cos\psi,\sin\psi)$ with $\psi\in[0,2\pi)$ whose orthogonal vector is given by $e_{\Tiny\perp}:=(-\sin\psi,\cos\psi)$, and by $r_{e}$ or $r_{\psi}$ the straight line passing through the origin of direction $e$, namely $r_{e}:=x\cdot e_{\Tiny\perp}=0$.

Let $\sigma_{e}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the reflection with respect the line $r_{e}$, i.e. $\sigma_{e}(x):=x-2(x\cdot e_{\Tiny\perp})e_{\Tiny\perp}$ for every $x\in\Omega$ and, if $u$ is any solution to (1.1) we let

[TABLE]

the difference between the reflection of $u$ with respect to $r_{e}$ and $u$.

Since we are interested in solutions with some invariances, as explained in the Introduction, for any angle $\psi$ we denote by $R_{\psi}$ the rotation of angle $\psi$ in counterclockwise direction centered at the origin and by $\mathcal{G}_{\psi}$ the subgroup of $SO(2)$ generated by $R_{\psi}$. In particular we consider angles $\psi=\frac{2\pi}{k}$ with $k\in\mathbb{N}$, $k\geq 1$, so that $\mathcal{G}_{\frac{2\pi}{k}}$ is a proper subgroup of $SO(2)$. Note that for $k=1$, $\mathcal{G}_{2\pi}=\{I\}$ is the trivial subgroup.

We say that a function $u$ defined in $\Omega$ is $k$-invariant if it satisfies

[TABLE]

Since we will consider functions in that satisfy (1.3) it is enough to know them in any sector of $\Omega$ of angle $\frac{2\pi}{k}$. Then, for any direction $e\in\mathcal{S}$ we denote by $S_{2k,e}$ the circular sector of $\Omega$ of angle $\frac{2\pi}{k}$, centered in the origin which has $r_{e}$ has symmetry line and lies in the halfplane $x\cdot e>0$, see Fig. 1 in the Introduction.

In order to understand the behavior, and the symmetries of the solutions we need to work also in sectors of amplitude $\frac{\pi}{k}$ and so, for any $e\in\mathcal{S}$, such that $e=(\cos\psi,\sin\psi)$ we let

[TABLE]

and by

[TABLE]

see Fig. 2 in the Introduction. Observe that $\partial S_{k,e}^{+}=\Gamma_{1,e}\cup\Gamma_{2,e}\cup\Gamma_{3,e}$ where $\Gamma_{1,e}$ is contained in $\partial\Omega$, $\Gamma_{2,e}$ is contained in $r_{e}$ while $\Gamma_{3,e}$ is contained in the line $r_{\psi+\frac{\pi}{k}}$ if $e=(\cos\psi,\sin\psi)$, see Fig. 3.

When $e=e_{\psi}=(\cos\psi,\sin\psi)$ we will use also the notation $S_{k,\psi}^{+}$, $S_{k,\psi}^{-}$, $S_{2k,\psi}$ and $w_{\psi}$ to denote respectively $S_{k,e_{\psi}}^{+}$, $S_{k,e_{\psi}}^{-}$, $S_{2k,e_{\psi}}$ and $w_{e_{\psi}}$.

By elliptic regularity theory $u\in C^{3,\beta}(\bar{\Omega})$ for some $\beta>0$. In particular $f^{\prime}(|x|,u)=\frac{\partial f}{\partial u}(|x|,u)\in C^{0,\alpha}(\bar{\Omega})$ for some $\alpha>0$. We can then define the linearized operator at a solution $u$

[TABLE]

and, for any $D\subseteq\Omega$ the quadratic form associated with $L_{u}$ in $D$, namely

[TABLE]

which is defined for $v,w\in H^{1}_{0}(D)$. Next, for any direction $e\in\mathcal{S}$ and for any $x\in\Omega$ we denote by

[TABLE]

[TABLE]

and by

[TABLE]

the corresponding linear operators, which are associated with the quadratic forms

[TABLE]

For any $e\in\mathcal{S}$, the function $w_{e}$ satisfies

[TABLE]

Moreover $w_{e}=0$ on $\partial\Omega$ and on $r_{e}$ by construction. Further, if $u$ satisfies (1.3) then $w_{e}=0$ also on $r_{\psi+\frac{\pi}{k}}$ by the rotation invariance of $u$, showing that for any $e\in\mathcal{S}$, $w_{e}=0$ on $\partial S_{k,e}^{+}$ and on $\partial S_{k,e}^{-}$.

Observe also that, if $w_{e}\equiv 0$ in $D$ then $L_{e}=L_{es}=L_{u}$ in $D$.

Next, for any $D\subseteq\Omega$ and for any linear operator $L$ we denote by ${\lambda}_{1}(L,D)$ the first eigenvalue of the linear operator $L$ in $D$ with Dirichlet boundary conditions.

First we can prove the following

Proposition 2.1**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3). Suppose further that there exists a direction $e\in\mathcal{S}$ such that

[TABLE]

Then there exists another direction $e^{\prime}\in\mathcal{S}$ such that $w_{e^{\prime}}\equiv 0$ in $S_{k,e^{\prime}}^{+}$ (i.e. $u(\sigma_{e^{\prime}}(x))=u(x)$ in $S_{k,e^{\prime}}^{+}$) and

[TABLE]

Proof.

We can assume w.l.o.g. that $w_{e}>0$ in $S_{k,e}^{+}$ and that, up to a rotation, $e=(1,0)=(\cos\psi_{0},\sin\psi_{0})$ with $\psi_{0}=0$. Now we consider other directions $e_{\psi}:=(\cos\psi,\sin\psi)$ for $\psi>0$. Note that, if $w_{\psi}>0$ in $S_{k,\psi}^{+}$ for some $\psi>0$ then ${\lambda}_{1}(L_{e_{\psi}},S_{k,\psi}^{+})=0$, since $w_{\psi}$ solves $L_{e_{\psi}}(w_{\psi})=0$ in $S_{k,\psi}^{+}$ and $w_{\psi}=0$ on $\partial S_{k,\psi}^{+}$ by previous considerations. Next we define

[TABLE]

By continuity ${\lambda}_{1}(L_{e_{\tilde{\psi}}},S_{k,\tilde{\psi}}^{+})=0$ and $w_{\tilde{\psi}}\geq 0$ in $S_{k,\tilde{\psi}}^{+}$. This implies that $\tilde{\psi}<\frac{\pi}{k}$, because, by the rotation invariance of $u$, we have $w_{\frac{\pi}{k}}=-w_{e}<0$ in $S_{k,\frac{\pi}{k}}^{+}$.

We would like to prove that $w_{\tilde{\psi}}\equiv 0$ in $S_{k,\tilde{\psi}}^{+}$.

Assume, by contradiction, that $w_{\tilde{\psi}}\neq 0$. Then, necessarely $w_{\tilde{\psi}}>0$ in $S_{k,\tilde{\psi}}^{+}$. Indeed, by definition of $\tilde{\psi}$ we have $w_{\tilde{\psi}}\geq 0$, and since $w_{\tilde{\psi}}$ satisfies $L_{e_{\psi}}w_{\psi}=0$ then the strong maximum principle implies the assertion.

Now let $K\subset S_{k,\tilde{\psi}}^{+}$ be a compact set such that $S_{k,\tilde{\psi}}^{+}\setminus K$ has small measure to allow the weak maximum principle hold for the operator $L_{e_{\tilde{\psi}+{\varepsilon}}}$ in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus K$ for sufficiently small ${\varepsilon}$.

Since $w_{\tilde{\psi}}>\eta>0$ in $K$ for some $\eta>0$, then

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for sufficiently small ${\varepsilon}>0$, while

[TABLE]

by the weak and the strong maximum principle in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus K$.

Therefore $w_{\tilde{\psi}+{\varepsilon}}>0$ in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}$ contradicting the definition of $\tilde{\psi}$. Hence $w_{\tilde{\psi}}$ must be identically zero in $S_{k,\tilde{\psi}}^{+}$. This implies that $0={\lambda}_{1}(L_{e_{\tilde{\psi}}},S_{k,\tilde{\psi}}^{+})={\lambda}_{1}(L_{u},S_{k,\tilde{\psi}}^{+})$ concluding the proof. ∎

Next we have

Proposition 2.2**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3). Suppose further that there exists a direction $e\in\mathcal{S}$ such that $w_{e}\equiv 0$ in $S_{k,e}^{+}$ and

[TABLE]

Then or $u$ is radial or it is strictly monotone in the angular variable in $S_{k,e}^{+}$.

Proof.

We can assume w.l.o.g. that $e=(1,0)$ and we use the polar coordinates $(x_{1},x_{2})=(r\cos\theta,r\sin\theta)$. As said before, if $u$ solves (1.1) then $u\in C^{3,\beta}(\bar{\Omega})$ and the derivative of $u$ with respect to the angular variable $\theta$, that we denote $u_{\theta}$, satisfies

[TABLE]

Moreover $u_{\theta}=0$ on $\partial\Omega$, since $u$ satisfies zero Dirichlet boundary conditions on $\partial\Omega$, and $u_{\theta}=0$ on the $\Gamma_{2,e}$ by the symmetry of $u$ with respect to $r_{e}$. Further, the symmetry of $u$ with respect to $r_{e}$ together with the invariance by rotations of angle $\frac{2\pi}{k}$ implies also that $u$ is symmetric with respect to the line $\theta={\frac{\pi}{k}}$ which is part of the boundary of $S_{k,e}^{+}$. Hence $u_{\theta}=0$ also on $\theta={\frac{\pi}{k}}$ meaning that $u_{\theta}=0$ on $\partial S_{k,e}^{+}$.

Now, since ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$ we have or that $u_{\theta}=0$ in $S_{k,e}^{+}$, meaning that $u$ is radial, or, else that $u_{\theta}$, which is nonzero, is a first eigenfunction of $L_{u}$ in $S_{k,e}^{+}$ and so it has one sign in $S_{k,e}^{+}$, namely it is positive or negative in $S_{k,e}^{+}$. This shows that if $u$ is not radial it is strictly monotone in the polar angle $\theta$ and concludes the proof. ∎

By the symmetry assumption on $u$, whenever $u$ is nonradial then it is strictly monotone in the polar angle $\theta$ also in the sector $S_{k,e}^{-}$.

Next, for every real valued function $g$, we let $g^{+}(x)=\max\{g(x),0\}$ and $g^{-}(x)=\min\{g(x),0\}$ denote the positive and negative part of $g$, respectively. For every $D\subseteq\Omega$, $\chi_{D}$ stands for the characteristic function of $D$. If $D$ is a domain, we let $C^{1}_{0}(D)$ denote the space of all $C^{1}$-functions on $D$ with compact support strictly contained in $D$.

Finally, as explained in the Introduction, we denote by $m_{k}(u)$ the $k$-Morse index of $u$, namely the maximal dimension of a subspace of $C^{1}_{0,k}$ in which the quadratic form $Q_{u,\Omega}$ is negative defined. Equivalently, when $\Omega$ is bounded, it is the number, counted with multiplicity, of the negative eigenvalues of $L_{u}$ with corresponding eigenfunction $k$-invariant.

3. The case of $f$ convex

In this section we prove the monotonicity and symmetry results in the case when $\Omega$ is bounded and $f$ is convex with respect to the second variable. Under this assumption

[TABLE]

and, for any direction $e\in\mathcal{S}$, the function $w_{e}$ satisfies the inequality

[TABLE]

and also the boundary conditions $w_{e}=0$ on $\partial S_{k,e}^{+}$. First we prove the following

Lemma 3.1**.**

Assume $\Omega$ is bounded. Assume further that one among ${\lambda}_{1}(L_{u},S_{k,e}^{+})$ and ${\lambda}_{1}(L_{u},S_{k,e}^{-})$ is nonnegative. Then or $w_{e}\equiv 0$ in $S_{k,e}^{+}$ and ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$ or $w_{e}$ has a sign in $S_{k,e}^{+}$.

Proof.

We can assume w.l.o.g. that ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$.

If $w_{e}\equiv 0$ in $S_{k,e}^{+}$ then we are done. Assume else that $w_{e}\neq 0$ in $S_{k,e}^{+}$. We want to prove that $w_{e}$ has a sign in $S_{k,e}^{+}$. Assume not, then $w_{e}^{-}\neq 0$, and $w_{e}^{-}=0$ on $\partial S_{k,e}^{+}$, since $w_{e}=0$ on $\partial S_{k,e}^{+}$ by previous considerations. Multiplying (3.1) by $w_{e}^{-}$ and integrating over $S_{k,e}^{+}$ we have

[TABLE]

which, together with the relation ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$ implies

[TABLE]

meaning that $w_{e}^{-}$ reaches the infimum of $Q_{u,S_{k,e}^{+}}(v,v)$ in $H^{1}_{0}(S_{k,e}^{+})$ and hence it solves

[TABLE]

Now, since $w_{e}^{-}\leq 0$ the strong maximum principle implies either that $w_{e}^{-}\equiv 0$ in $S_{k,e}^{+}$, which is not possible by assumption, or that $w_{e}^{-}<0$ in $S_{k,e}^{+}$ which implies $w_{e}<0$. This concludes the proof. ∎

We are now in position to prove the main point:

Proposition 3.2**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3) and such that $m_{k}(u)\leq 2$. Then there exists a direction $e\in\mathcal{S}$ such that, or $w_{e}\equiv 0$ in $S_{k,e}^{+}$ and ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$ or $w_{e}$ has a sign in $S_{k,e}^{+}$.

This proposition, together with Propositions 2.1 and 2.2 then implies Theorem 1.1 and concludes this case.

Proof.

Let us denote by $\varphi_{1}$ and $\varphi_{2}$ two $k$-invariant eigenfunctions of the linearized operator $L_{u}$ orthogonal in $L^{2}(\Omega)$ corresponding to the two negative eigenvalues.

For any direction $e\in\mathcal{S}$ let us denote by $\varphi_{e}^{+}$ the unique first positive eigenfunction of $L_{u}$ in $S_{k,e}^{+}$ with Dirichlet boundary conditions, which corresponds to ${\lambda}_{1}(L_{u},S_{k,e}^{+})$ and by $\varphi_{e}^{-}$ the unique first positive eigenfunction of $L_{u}$ in $S_{k,e}^{-}$ with Dirichlet boundary conditions, which corresponds to ${\lambda}_{1}(L_{u},S_{k,e}^{-})$, normalized such that $\int_{S_{k,e}^{+}}\left(\varphi_{e}^{+}\right)^{2}\ dx=\int_{S_{k,e}^{-}}\left(\varphi_{e}^{-}\right)^{2}\ dx=1$ and by $\tilde{\varphi}_{e}^{\pm}$ their extension in all of $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. We want to prove that there exists a direction $e\in\mathcal{S}$ for which at least one, among ${\lambda}_{1}(L_{u},S_{k,e}^{+})$ and ${\lambda}_{1}(L_{u},S_{k,e}^{-})$ is nonnegative, so that the result follows by Lemma 3.1.

Define the function

[TABLE]

which is supported in $S_{2k,e}$, and denote by $\tilde{\xi}_{e}$ its extension in all of $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. Then, by construction

[TABLE]

which means that $\tilde{\xi}_{e}$ is orthogonal in $L^{2}(\Omega)$ to $\varphi_{1}$ for every $e\in\mathcal{S}$. Assume $e=e_{\psi}=(\cos\psi,\sin\psi)$, with $\psi\in[0,2\pi)$. Let us consider the map

[TABLE]

which is continuous with respect to the angle $\psi$. Here $\xi_{\psi}$ stands for $\xi_{e_{\psi}}$. We want to prove that there exists $\psi\in[0,\frac{\pi}{k})$ such that $h(\psi)=0$. If $h(0)=0$ then we are done. If else $h(0)\neq 0$, observe that $S_{k,\frac{\pi}{k}}^{-}=S_{k,0}^{+}$ and that $S_{k,\frac{\pi}{k}}^{+}$ can be obtained rotating the sector $S_{k,0}^{-}$ by $\frac{2\pi}{k}$, namely $S_{k,\frac{\pi}{k}}^{+}=R_{\frac{2\pi}{k}}(S_{k,0}^{-})$. Then $\varphi_{\frac{\pi}{k}}^{+}=\tilde{\varphi}_{0}^{-}$ by the uniqueness of the first positive eigenfunction of norm $1$ and $\varphi_{\frac{\pi}{k}}^{-}=\varphi_{0}^{+}$.

Moreover

[TABLE]

where we used the fact that $\varphi_{1}$ is $k$-invariant. In the same manner

[TABLE]

It is then easy to see that $\tilde{\xi}_{\frac{\pi}{k}}=-\tilde{\xi}_{0}$, that implies $h(\frac{\pi}{k})=-h(0)$ since $\varphi_{2}$ is $k$-invariant. Since $h$ is a continuous function then it has one zero between [math] and $\frac{\pi}{k}$. This means that there exists a direction $e^{\prime}\in\mathcal{S}$ such that $\tilde{\xi}_{e^{\prime}}$ is orthogonal to $\varphi_{2}$ in $L^{2}(\Omega)$, since $\int_{\Omega}\tilde{\xi}_{e^{\prime}}\varphi_{2}=k\int_{S_{2k,e^{\prime}}}\xi_{e^{\prime}}\varphi_{2}=0$. Since $\varphi_{1},\varphi_{2}$ and $\tilde{\xi}_{e^{\prime}}$ are $k$-invariant and $m_{k}(u)=2$ then

[TABLE]

Finally by the definition of $\tilde{\xi}_{e^{\prime}}$ we have

[TABLE]

from which it follows that al least one among ${\lambda}_{1}(L_{u},S_{k,e^{\prime}}^{+})$ and ${\lambda}_{1}(L_{u},S_{k,e^{\prime}}^{-})$ is nonnegative. ∎

Remark 3.3**.**

*When a $k$-invariant solution $u$ is not radial, then, by Theorem 1.1 it is symmetric with respect to a direction $e=e_{\psi}$ in $S_{2k,e}$ and by the invariance (1.3) it is symmetric also with respect to the directions $e_{\psi+\frac{\pi}{k}}$ and also $e_{\psi+\frac{h\pi}{k}}$ for $h=1,\dots,2k$. Moreover in each sector $S_{k,\psi+\frac{h\pi}{k}}^{\pm}$ it is strictly monotone with respect to the angular variable and consecutive sectors see the angular derivative change sign.

Then, the maxima and minima of $u$ either belong to the symmetry axes or they are placed in the origin. Both this configurations can appear, in particular for sign changing solutions. Indeed a nodal $k$-invariant solution can have, as an example, either $k$ maxima and $k$ minima placed alternately along the directions $(\cos(\psi+\frac{h\pi}{k}),\sin(\psi+\frac{h\pi}{k}))$ at an angular distance of $\frac{\pi}{k}$, or else only $1$ maximum in the origin and $k$ minima placed along the directions $(\cos(\psi+\frac{2h\pi}{k}),\sin(\psi+\frac{2h\pi}{k}))$ at an angular distance of $\frac{2\pi}{k}$.*

4. the case of $f^{\prime}$ convex

In this section we consider the case when $f^{\prime}(|x|,s)=\frac{\partial f}{\partial s}(|x|,s)$ is convex with respect to the second variable and we prove Theorem 1.2. Under this assumption we have for any $x\in\Omega$

[TABLE]

First, we can prove the following:

Proposition 4.1**.**

Assume $\Omega$ is bounded. Let $u$ be a solution to (1.1) and (1.2) that satisfies (1.3) and such that $m_{k}(u)\leq 2$. Suppose further that $f^{\prime}$ is convex with respect to the second variable. Then there exists a direction $e\in\mathcal{S}$ such that, or $w_{e}\equiv 0$ in $S_{k,e}^{+}$ and ${\lambda}_{1}(L_{u},S_{k,e}^{+})\geq 0$ or $w_{e}$ has one sign in $S_{k,e}^{+}$.

This proposition, together with Propositions 2.1 and 2.2 then implies Theorem 1.2 and concludes this case.

Proof.

Let us denote by $\varphi_{1}$ and $\varphi_{2}$ two k-invariant eigenfunctions of the linearized operator $L_{u}$, orthogonal in $L^{2}(\Omega)$, corresponding to the two negative eigenvalues.

Define

[TABLE]

Case 1: $S_{*}=\emptyset$

Assume by contradiction that for every $e\in\mathcal{S}$ the function $w_{e}$ changes sign in $S_{k,e}^{+}$. In this case for every $e\in\mathcal{S}$ we can define the functions

[TABLE]

which are supported in $S_{2k,e}$.

Observe that $w^{1}_{e}$ and $w^{2}_{e}$ are both nonnegative in $S_{2k,e}$, have disjoint supports and satisfy $w_{e}^{i}=0$ on $\partial S_{k,e}^{\pm}$ and also on $\partial S_{2k,e}$. Moreover $w_{e}^{i}(\sigma_{e}(x))=w_{e}^{i}(x)$ for $x\in S_{k,e}^{+}$.

Further, since $w_{e}$ satisfies $L_{e}w_{e}=0$ in $S_{2k,e}$ and $w_{e}=0$ on $\partial S_{2k,e}$, multiplying by $w_{e}^{+}\chi_{S_{k,e}^{+}}+w_{e}^{-}\chi_{S_{k,e}^{-}}$ and integrating over $S_{2k,e}$ we obtain

[TABLE]

Then, (4.1), together with $w_{e}^{i}(\sigma_{e}(x))=w_{e}^{i}(x)$, gives

[TABLE]

where we are denoting by $Q_{es,D}$ the quadratic form associated with the operator $L_{es}$ in $D$. Similarly we can show

[TABLE]

Let us denote by $\tilde{w}_{e}^{i}$ the extension of $w_{e}^{i}$ in all $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. For every $e\in\mathcal{S}$ we can define the function

[TABLE]

which is supported in $S_{2k,e}$ and denote by $\tilde{\xi}_{e}$ its extension in all of $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. Then, by construction

[TABLE]

and

[TABLE]

since $w_{e}^{1}$ and $w_{e}^{2}$ have disjoint supports. Assume $e=e_{\psi}=(\cos\psi,\sin\psi)$, with $\psi\in[0,2\pi)$. Let us consider the map

[TABLE]

which is continuous with respect to the angle $\psi$. Here $\xi_{\psi}$ stands for $\xi_{e_{\psi}}$.

We want to prove that there exists $\psi\in[0,\frac{\pi}{k})$ such that $h(\psi)=0$. If $h(0)=0$ then we are done. If else $h(0)\neq 0$ as in the proof of Proposition 3.2 we have $S_{k,\frac{\pi}{k}}^{-}=S_{k,0}^{+}$ and $S_{k,\frac{\pi}{k}}^{+}=R_{\frac{2\pi}{k}}(S_{k,0}^{-})$. Observe that, for $x\in S_{k,\frac{\pi}{k}}^{-}$,

[TABLE]

since $u$ is $k$-invariant, while, for $x\in S_{k,\frac{\pi}{k}}^{+}$

[TABLE]

for $y\in S_{k,0}^{-}$. From this it follows that

[TABLE]

and

[TABLE]

This implies that

[TABLE]

where we used the $k$-invariance of $\varphi_{1}$, and, in the same manner

[TABLE]

It is then easy to see that $\tilde{\xi}_{\frac{\pi}{k}}=-\tilde{\xi}_{0}$, that implies, together with $\varphi_{1}(R_{\frac{2\pi}{k}}(x))=\varphi_{1}(x)$, that $h(\frac{\pi}{k})=-h(0)$.

Since $h$ is a continuous function then it has one zero in $(0,\frac{\pi}{k})$. This means that there exists a direction $e^{\prime}\in\mathcal{S}$ such that $\tilde{\xi}_{e^{\prime}}$ is orthogonal to $\varphi_{2}$ in $L^{2}(\Omega)$, since $\int_{\Omega}\tilde{\xi}_{e^{\prime}}\varphi_{2}=k\int_{S_{2k,e^{\prime}}}\xi_{e^{\prime}}\varphi_{2}=0$. Since $\varphi_{1},\varphi_{2}$ and $\tilde{\xi}_{e^{\prime}}$ are $k$-invariant and $m_{k}(u)\leq 2$ then

[TABLE]

which, together with (4.4) implies that $\tilde{\xi}_{e^{\prime}}$ is a minimizer for $Q_{u,\Omega}$ in the space of functions $k$-invariant and hence, by the principle of symmetric criticality of [Pa], $\tilde{\xi}_{e^{\prime}}$ weakly solves $L_{u}\tilde{\xi}_{e^{\prime}}=0$ in $\Omega$ and also $L_{u}\xi_{e^{\prime}}=0$ in $S_{2k,e^{\prime}}$. Recall that $\xi_{e^{\prime}}=0$ on $\partial S_{2k,e^{\prime}}$ and since $\xi_{e^{\prime}}(\sigma_{e^{\prime}}(x))=\xi_{e^{\prime}}(x)$ for any $x\in S_{2k,e^{\prime}}$ (by the symmetry of $w_{e^{\prime}}^{i}$), then $\frac{\partial\xi_{e^{\prime}}}{\partial\nu}=0$ on $\Gamma_{2,e^{\prime}}$, where $\frac{\partial}{\partial\nu}$ denotes the inner normal derivative on $\Gamma_{2,e^{\prime}}$. But, this implies that the function

[TABLE]

is also a weak solution to $L_{u}\widehat{\xi}_{e}=0$ in $S_{2k,e^{\prime}}$, contradicting the unique continuation principle for this equation. The only possibility is that $\xi_{e^{\prime}}\equiv 0$ which is not possible since we are assuming that $w_{e^{\prime}}$ changes sign in $S_{k,e^{\prime}}^{+}$.

Case 2: $S_{*}\neq\emptyset$.

Let us assume that $e=e_{\psi}\in S_{*}$, so that $w_{\psi}\equiv 0$ in $S_{k,\psi}^{+}$. By the symmetry of $u$ we also have that $\varphi_{1}(\sigma_{\psi}(x))=\varphi_{1}(x)$.

Let us denote by $g_{\psi}$ the unique first positive eigenfunction of $L_{u}$ in $S_{k,\psi}^{+}$ with Dirichlet boundary conditions, which corresponds to ${\lambda}_{1}(L_{u},S_{k,\psi}^{+})$ normalized such that $\int_{S_{k,\psi}^{+}}g_{\psi}^{2}=1$, by $\widehat{g}_{\psi}$ its odd extension on $S_{2k,\psi}$, and by $\tilde{g}_{\psi}$ its extension in $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$.

It is easy to see that

[TABLE]

Moreover, by the symmetry of $u$ in $S_{k,\psi}^{+}$ we have

[TABLE]

since we are assuming that $e_{\psi}\in S_{*}$.

To conclude we want to show that, when we consider the direction $e_{\psi+\frac{\pi}{2k}}$ (which divides $S_{k,\psi}^{+}$ into two equal parts), then we must have either that $w_{\psi+\frac{\pi}{2k}}>0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ either $w_{\psi+\frac{\pi}{2k}}<0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ or, else, that $w_{\psi+\frac{\pi}{2k}}\equiv 0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ and ${\lambda}_{1}(L_{u},S_{k,\psi+\frac{\pi}{2k}}^{+})\geq 0$.

i) First we show that whenever $w_{\psi+\frac{\pi}{2k}}\equiv 0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ then ${\lambda}_{1}(L_{u},S_{k,\psi+\frac{\pi}{2k}}^{+})\geq 0$.

Suppose by contradiction that ${\lambda}_{1}(L_{u},S_{k,\psi+\frac{\pi}{2k}}^{+})<0$. Since $u$ is symmetric with respect to $r_{\psi+\frac{\pi}{2k}}$, namely $u(x)=u(\sigma_{\psi+\frac{\pi}{2k}})$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$, and since $r_{\psi+\frac{\pi}{2k}}$ divides the sector $S_{k,\psi}^{+}$ into two equal parts, then the first eigenfunction $g_{\psi}$ in $S_{k,\psi}^{+}$ is symmetric with respect $r_{\psi+\frac{\pi}{2k}}$.

Next we let $g_{\psi+\frac{\pi}{2k}}$ be the first positive eigenfunction of the linear operator $L_{u}$ in the sector $S_{k,\psi+\frac{\pi}{2k}}^{+}$ with Dirichlet boundary conditions, which corresponds to ${\lambda}_{1}(L_{u},S_{k,\psi+\frac{\pi}{2k}}^{+})$ normalized such that $\int_{S_{k,\psi+\frac{\pi}{2k}}^{+}}g_{\psi+\frac{\pi}{2k}}^{2}=1$, by $\widehat{g}_{\psi+\frac{\pi}{2k}}$ its odd extension on $S_{2k,\psi+\frac{\pi}{2k}}$, and by $\tilde{g}_{\psi+\frac{\pi}{2k}}$ its extension in $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. Again by the symmetry of $u$ we have that $g_{\psi+\frac{\pi}{2k}}(\sigma_{\psi+\frac{\pi}{k}}(x))=g_{\psi+\frac{\pi}{2k}}(x)$ in $S_{k,\psi+\frac{2\pi}{k}}^{+}$. Then

[TABLE]

where

[TABLE]

since $g_{\psi}$ is odd with respect $r_{\psi}$ while $\widehat{g}_{\psi+\frac{\pi}{2k}}$ is even with respect to $r_{\psi}$. Next

[TABLE]

since $\tilde{g}_{\psi}(x)=\widehat{g}(R_{-\frac{2\pi}{k}}(x))=-g_{\psi}(\sigma_{\psi+\frac{\pi}{k}}(x))$ for $x\in S_{k,\psi+\frac{\pi}{2k}}^{+}\setminus S_{k,\psi}^{+}$. Similarly

[TABLE]

so that $\tilde{g}_{\psi+\frac{\pi}{2k}}$ is orthogonal to $\varphi_{1}$ and to $\tilde{g}_{\psi}$ in $L^{2}(\Omega)$. By assumption then we have

[TABLE]

which contradicts the bound on the $k$-Morse index. This shows that whenever $w_{\psi+\frac{\pi}{2k}}\equiv 0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$, then ${\lambda}_{1}(L_{u},S_{k,\psi+\frac{\pi}{2k}}^{+})\geq 0$ and concludes this part.

ii) It lasts to prove that whenever $w_{\psi+\frac{\pi}{2k}}\neq 0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$, then the function $w_{\psi+\frac{\pi}{2k}}$ cannot change sign in $S_{k,\psi+\frac{\pi}{2k}}^{+}$. Assume by contradiction that it changes sign. We can then define the functions $w_{e}^{1}$ and $w_{e}^{2}$ as in (4.3) relative to the direction $e=e_{\psi+\frac{\pi}{2k}}$, which are nonnegative, symmetric with respect to $r_{\psi+\frac{\pi}{2k}}$ and satisfies

[TABLE]

Moreover, by the symmetry of $u$ we have that $w_{\psi+\frac{\pi}{2k}}(\sigma_{\psi}(x))=w_{\psi+\frac{\pi}{2k}}(x)$ which implies that $w_{e}^{i}(\sigma_{\psi}(x))=w_{e}^{i}(x)$. Next we construct as before the function $\xi_{e}$, and $\tilde{\xi}_{e}$ which, again satisfies by construction $\xi_{e}(\sigma_{\psi}(x))=\xi_{e}(x)$ and also

[TABLE]

Further $\tilde{\xi}_{e}$ is orthogonal to $\varphi_{1}$ by construction and $\tilde{\xi}_{e}$ is orthogonal to $\tilde{g}_{\psi}$ since the first is even with respect to $r_{\psi}$ while the second is odd. Since $\varphi_{1}$ and $\tilde{g}_{\psi}$ and $\tilde{\xi}_{e}$ are $k$-invariant and $m_{k}(u)\leq 2$ then

[TABLE]

which, together with (4.7) implies that $\tilde{\xi}_{e}$ is a minimizer for $Q_{u,\Omega}$ in the space of $k$-invariant functions, and, as before, $\tilde{\xi}_{e}$ solves $L_{u}\tilde{\xi}_{e}=0$ in $\Omega$ and also $L_{u}\xi_{e}=0$ in $S_{2k,e}$. Then we can conclude defining the function $\widehat{\xi}$ as in (4.6) as in the end of the proof of case 1 showing that it is not possible that $w_{\psi+\frac{\pi}{2k}}$ changes sign in $S_{k,\psi+\frac{\pi}{2k}}$. ∎

Of course Remark 3.3 holds also in this case.

5. Unbounded domains

In this section we extend the previous results to the case of an unbounded radial domain $\Omega$, where $\Omega$ either coincides with $\mathbb{R}^{2}$ or $\Omega=\mathbb{R}^{2}\setminus B$, extending some ideas in [GPW] to the case of functions which are $k$-invariant.

As before we denote by $C^{1}_{0,k}$ the subset of $C^{1}_{0}(\Omega)$ of functions that satisfy (1.3).

To prove the results we only highlight the difference between the bounded case. First we can prove an extension of Proposition 2.2, namely

Proposition 5.1**.**

Assume $u$ is a solution to (1.1) and possibly (1.2) that satisfies (1.3) and such that $|\nabla u|\in L^{2}(\Omega)$. Suppose further that there exists $e\in\mathcal{S}$ such that $w_{e}\equiv 0$ in $S_{k,e}^{+}$ and

[TABLE]

Then or $u$ is radial or it is strictly monotone in the angular variable in $S_{k,e}^{+}$.

Proof.

As in the proof of Proposition 2.2 we have that the angular derivative $u_{\theta}$ satisfies

[TABLE]

We claim that either $u_{\theta}\equiv 0$ in $S_{k,e}^{+}$ showing that $u$ is radial or, else, $u_{\theta}$ has a sign in $S_{k,e}^{+}$.

Using a radial cut-off function $\xi_{R}$ supported in $\Omega\cap B_{R}(0)$ and reasoning exactly as in the proof of Proposition 2.5 of [GPW] we can prove that in this case the function $u_{\theta}^{+}$ satisfies

[TABLE]

for any $\phi\in C^{\infty}_{0}(S_{k,e}^{+})$, meaning that $u_{\theta}^{+}$, if it is nonzero, solves

[TABLE]

and by the unique continuation principle or $u_{\theta}^{+}\equiv 0$ or $u_{\theta}^{+}>0$ in $S_{k,e}^{+}$ concluding the proof. ∎

By the symmetry assumption on $u$, whenever $u$ is nonradial then it is strictly monotone in the polar angle $\theta$ also in the sector $S_{k,e}^{-}$.

Moreover, when $f$ or $f^{\prime}$ are convex in the second variable we can extend also Proposition 2.1, getting the following result:

Proposition 5.2**.**

Assume $u$ is a solution to (1.1) and possibly (1.2) that satisfies (1.3) and such that $|\nabla u|\in L^{2}(\Omega)$ and $m_{k}(u)<\infty$. Suppose furthermore that $f$ or $f^{\prime}$ are convex in the second variable and that there exists a direction $e\in\mathcal{S}$ s.t.

[TABLE]

Then there exists another direction $e^{\prime}\in\mathcal{S}$ s.t. $w_{e}\equiv 0$ and

[TABLE]

Proof.

We define $\tilde{\psi}$ as in (2.4) and we want to prove that $w_{\tilde{\psi}}\equiv 0$ in $S_{k,\tilde{\psi}}^{+}$. Indeed (5.1) follows as in Proposition 2.1 using Lemma 2.1 in [GPW]. The proof follows very closely the one in Proposition 2.8 of [GPW], adapted to the case of the sector $S_{k}$ as in Proposition 2.1.

Assume by contradiction that $w_{\tilde{\psi}}\neq 0$ in $S_{k,\tilde{\psi}}^{+}$. Then necessarily $w_{\tilde{\psi}}>0$ in $S_{k,\tilde{\psi}}^{+}$ by the strong maximum principle. Moreover applying the Hopf Lemma on the straight lines $\Gamma_{2,\tilde{\psi}}$ and $\Gamma_{3,\tilde{\psi}}$, where $w_{\tilde{\psi}}$ vanishes we have:

[TABLE]

where $\theta$ denotes the angular variable in polar coordinates. Since by hypothesis $m_{k}(u)<\infty$, there exists $R_{0}>0$ such that $Q_{u,\Omega\setminus B_{R}}(\phi,\phi)\geq 0$ for every $\phi\in C^{1}_{0}(\Omega\setminus B_{R})$ that satisfies (1.3) and every $R>R_{0}$. Then also $Q_{u,S_{2k,\tilde{\psi}}\setminus B_{R}}(\phi,\phi)\geq 0$ for every $\phi\in C^{1}_{0}(S_{2k,\tilde{\psi}}\setminus B_{R})$ and every $R>R_{0}$ which implies that $Q_{u,S_{k,\tilde{\psi}}^{+}\setminus B_{R}}(\phi,\phi)\geq 0$ for every $\phi\in C^{1}_{0}(S_{k,\tilde{\psi}}^{+}\setminus B_{R})$ and every $R>R_{0}$.

We fix $R_{1}>R_{0}$ and we claim that there exists ${\varepsilon}_{0}>0$ such that

[TABLE]

In the case $\Omega=\mathbb{R}^{2}\setminus B$ let $B_{\delta}$ be a neighborhood of $\partial B$ in $\Omega$ of small measure, in case of $\Omega=\mathbb{R}^{2}$ instead let $B_{\delta}$ be a neighborhood of the origin of small measure. The measure of $B_{\delta}$ in both cases is so small to allow the strong maximum principle to hold in $B_{\delta}\cap S_{k,\tilde{\psi}+{\varepsilon}}^{+}$ for the operator $L_{\tilde{\psi}+{\varepsilon}}$ for sufficiently small ${\varepsilon}>0$. We first show that

[TABLE]

If (5.4) is not true we have sequences ${\varepsilon}_{n}\to 0$, $x_{n}\in(B_{R_{1}}\cap S_{k,\tilde{\psi}+{\varepsilon}_{n}}^{+})\setminus B_{\delta}$ such that $w_{\tilde{\psi}+{\varepsilon}_{n}}(x_{n})<0$. After passing to a subsequence, $x_{n}\to x_{0}\in\overline{(B_{R_{1}}\cap S_{k,\tilde{\psi}}^{+})\setminus B_{\delta}}$ and $w_{\tilde{\psi}}(x_{0})=0$, hence $x_{0}\in(\Gamma_{2,\tilde{\psi}}\cup\Gamma_{3,\tilde{\psi}})\cap(\overline{B_{R_{1}}\setminus B_{\delta}})$. Assume, without l.o.g. $x_{0}\in\Gamma_{2,\tilde{\psi}}$. Since $w_{\tilde{\psi}+{\varepsilon}_{n}}(x)=0$ on $\Gamma_{2,\tilde{\psi}+{\varepsilon}_{n}}$ and $w_{\tilde{\psi}+{\varepsilon}_{n}}(x_{n})<0$, there should be points $\xi_{n}$ on the line segment joining $x_{n}$ with $\Gamma_{2,\tilde{\psi}+{\varepsilon}_{n}}$ and perpendicular to $\Gamma_{2,\tilde{\psi}+{\varepsilon}_{n}}$, such that $\frac{\partial w_{\tilde{\psi}+{\varepsilon}_{n}}}{\partial\theta}(\xi_{n})<0$. Passing to the limit we get $\frac{\partial w_{\tilde{\psi}}}{\partial\theta}(x_{0})\leq 0$ in contradiction with (5.2). So we get (5.4).

By the strong maximum principle, the definition of $B_{\delta}$ and (5.4) we get $w_{\tilde{\theta}+{\varepsilon}_{n}}\leq 0$ also in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}\cap B_{\delta}$ and hence (5.3) holds.

It lasts to prove that $w_{\tilde{\psi}+{\varepsilon}_{n}}\geq 0$ also in $S_{k,\tilde{\psi}+{\varepsilon}_{n}}^{+}\setminus B_{R_{1}}$ for all ${\varepsilon}\in[0,{\varepsilon}_{0})$. We recall that $w_{\tilde{\psi}+{\varepsilon}}$ satisfies

[TABLE]

By (5.3) the function $v:=w_{\tilde{\psi}+\epsilon}^{-}\>\chi_{\stackrel{{\scriptstyle}}{{S_{k,\tilde{\psi}+{\varepsilon}}^{+}}}}$ has its support strictly contained in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}}$. We claim that

[TABLE]

We first consider the case where $f$ is convex in the second variable, and we let $\phi\in C_{0}^{\infty}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$. By previous considerations $Q_{u,S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}}}$ defines a (semidefinite) scalar product on $C^{1}_{0}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}}$), and the corresponding Cauchy-Schwarz-inequality yields

[TABLE]

for all $\psi,\rho\in C^{1}_{0}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$. Consequently, we obtain

[TABLE]

for $R>0$, where $\xi_{R}$ is a cut-off function supported in $B_{2R}$. By Lemma 2.3 (ii) in [GPW], we have

[TABLE]

so that

[TABLE]

Since $\phi\in C_{0}^{\infty}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$ was chosen arbitrarily, we conclude that $v$ is a solution of

[TABLE]

Then however $v=w_{\tilde{\psi}+\epsilon}^{-}\chi_{S_{k,\tilde{\psi}+\epsilon}^{+}}\equiv 0$ by the unique continuation principle, since $w_{\tilde{\psi}+\epsilon}^{-}(x)\equiv 0$ in $B_{R_{1}}\cap S_{k,\tilde{\psi}+{\varepsilon}}^{+}$ by (5.3) and $R_{1}>R_{0}$. Hence (5.6) holds.

Next we consider the case where $f^{\prime}$ is convex in the second variable. Since every function $\tau\in C^{1}_{0}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$ can be extended to an odd function $\tilde{\tau}\in C^{1}_{0}(S_{2k,\tilde{\psi}+{\varepsilon}}\setminus B_{R_{0}})$ with respect to the reflection at $r_{\tilde{\psi}+\epsilon}$, we have

[TABLE]

for all $\tau\in C^{1}_{0}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$. Hence $Q_{{\tilde{\psi}+{\varepsilon}}}(\tau,\tau):=\int_{S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}}}|\nabla\tau|^{2}-V_{\tilde{\psi}+{\varepsilon}}(x)\tau^{2}$ defines a (semidefinite) scalar product on $C^{1}_{0}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$, and the corresponding Cauchy-Schwarz-inequality reads

[TABLE]

which by density holds for all $\psi,\rho\in H^{1}_{0}(S_{k,e_{\tilde{\psi}+{\varepsilon}}}^{+}\setminus B_{R_{0}})$ vanishing a.e. outside a bounded set. Now we let again $\phi\in C_{0}^{\infty}(S_{k,\tilde{\psi}+{\varepsilon}}^{+}\setminus B_{R_{0}})$. By Lemma 2.3(i) in [GPW], we have

[TABLE]

Combining this with (5.7), we find that

[TABLE]

Since $\phi\in C_{0}^{\infty}(\Sigma_{\tilde{\psi}+{\varepsilon}}\setminus B_{R_{0}})$ was chosen arbitrarily, we conclude that $v$ is a solution of

[TABLE]

As above, this implies (5.6) by the unique continuation principle.

As a consequence of (5.6), we have got $w_{\tilde{\psi}+{\varepsilon}}\geq 0$ in $S_{k,\tilde{\psi}+{\varepsilon}}^{+}$ contradicting the definition of $\tilde{\psi}$. Then the definition of $\tilde{\psi}$ implies that $w_{\tilde{\psi}}\equiv 0$.

∎

In the case where $f$ is convex in the second variable then we have:

Proposition 5.3**.**

Assume $u$ is a solution to (1.1) and possibly (1.2) that satisfies (1.3) and such that $|\nabla u|\in L^{2}(\Omega)$ and $m_{k}(u)\leq 2$. Suppose furthermore that $f$ is convex in the second variable. Then, or $u$ is radial or, else, there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $r_{e}$ in $S_{2k,e}$ and it is strictly monotone in the angular variable in the sectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$.

Proof.

We want to prove that there exists a direction $e\in\mathcal{S}$ such that

[TABLE]

With respect to this direction then, either $w_{e}\equiv 0$, and then the monotonicity of $u$ follows by Proposition 5.1, or else it can be proved, as in the proof of Theorem 1.4 in [GPW] and as in Proposition 5.2, using the function $v_{R}=w_{e}^{+}\chi_{S_{k,e}^{+}}\xi_{R}$ for a cut off $\xi_{R}$, that $w_{e}$ has one sign in $S_{k,e}^{+}$ so that the monotonicity follows by Propositions 5.2 and 5.1.

Suppose that (5.8) does not hold. Then, by Lemma 2.9 of [GPW] there exists $\bar{R}>0$ such that for any $e\in\mathcal{S}$ and for every $R>\bar{R}$ either ${\lambda}_{1}(L_{u},S_{k,e}^{+}\cap B_{R})<0$ or ${\lambda}_{1}(L_{u},S_{k,e}^{-}\cap B_{R})<0$. By definition of $m_{k}(u)$, $2$ is the maximal dimension of a subspace $X=\mathit{span}\{\psi_{1},\psi_{2}\}$ of $C^{1}_{0,k}$ such that $Q_{u,\Omega}(\psi,\psi)<0$ for every $\psi\in X\setminus\{0\}$.

We take a ball $B_{R}$ with radius $R$ sufficiently large to contain the supports of $\psi_{1}$ and $\psi_{2}$ and also $B_{\bar{R}}$. We deduce that in $B_{R}\cap\Omega$ the operator $L_{u}$ admits precisely $2$ negative eigenvalues in $H^{1}_{0,k}$ and the third is nonnengative. Let us denote by $\varphi_{1}$ and $\varphi_{2}$ two eigenfunctions of $L_{u}$, orthogonal in $L^{2}(\Omega)$, corresponding to the negative eigenvalues in $\Omega\cap B_{R}$.

We then follow exactly the proof of Proposition 3.2 in $\Omega\cap B_{R}$ and we obtain that there exists a direction $e\in\mathcal{S}$ for which at least one among ${\lambda}_{1}(L_{u},S_{k,e}^{+}\cap B_{R})$ and ${\lambda}_{1}(L_{u},S_{k,e}^{-}\cap B_{R})$ is nonnegative obtaining a contradiction. This proves (5.8) and concludes the proof. ∎

In the case when $f^{\prime}$ is convex instead we have:

Proposition 5.4**.**

Assume $u$ is a solution to (1.1) and possibly (1.2) that satisfies (1.3) and such that $|\nabla u|\in L^{2}(\Omega)$ and $m_{k}(u)\leq 2$. Suppose furthermore that $f^{\prime}$ is convex in the second variable. Then, or $u$ is radial or, else, there exists a direction $e\in\mathcal{S}$ such that $u$ is symmetric with respect to $r_{e}$ in $S_{2k,e}$ and it is strictly monotone in the angular variable in the sectors $S_{k,e}^{+}$ and $S_{k,e}^{-}$.

Proof.

First we let $S_{*}$ as in (4.2). Next, as in the proof of the previous Proposition, from $m_{k}(u)\leq 2$, we deduce that there exists $R_{0}>0$ such that $L_{u}$ admits exactly two negative eigenvalues in $\Omega\cap B_{R}$, for every $R\geq R_{0}$, with eigenfunctions $k$-invariant while the third eigenvalue is nonnengative. Let $\varphi_{1}$ and $\varphi_{2}$ two eigenfunctions $k$-invariant of $L_{u}$, orthogonal in $L^{2}(\Omega)$ corresponding to the negative eigenvalues in $\Omega\cap B_{R}$, for some $R\geq R_{0}$. Hence $\varphi_{1},\varphi_{2}\in H^{1}_{0,k}(B_{R})$ and, extending these functions by zero in $\Omega\setminus B_{R}$

[TABLE]

where the second equality follows by the orthogonality of $\varphi_{1}$ and $\varphi_{2}$ in $L^{2}(B_{R})$ and the fact that $\varphi_{i}\in H^{1}_{0,k}(B_{R})$ solves $-\Delta\varphi_{i}-f^{\prime}(|x|,u)\varphi_{i}={\lambda}_{i}\varphi_{i}$ in $\Omega\cap B_{R}$.

Case 1: $S_{*}=\emptyset$

Assume by contradiction that for every $e\in\mathcal{S}$ the function $w_{e}$ changes sign in $S_{k,e}^{+}$. We define the functions $w_{e}^{1}$ and $w_{e}^{2}$ as in (4.3). They belong to $H^{1}_{0,k,loc}(\Omega)$. Next we need to change a little the proof of Proposition 4.1 in order to deal with the unboundedness of $\Omega$ following the proof of Theorem 1.3 in [GPW]. For every $e\in\mathcal{S}$ we define the function

[TABLE]

which is supported in $S_{2k,e}$, and denote by $\tilde{\xi}_{e}$ its extension in all of $\Omega$ by the invariance by rotations of angle $\frac{2\pi}{k}$. Then, by construction

[TABLE]

Observe that $Q_{u,\Omega}(\tilde{\xi}_{e},\varphi_{1})$ is well defined since $\varphi_{1}$ is supported in $B_{R_{0}}$ and $\xi_{e}$ and $\tilde{\xi}_{e}$ belong to $H^{1}_{0,loc}(S_{2k,e})$ and to $H^{1}_{0,k,loc}(\Omega)$ respectively.

Since $\varphi_{2}$ is supported in $B_{R}$ we can define the function $h(\psi)$ as in (4.5). We then obtain, as in Proposition 4.1, that there exists a direction $e^{\prime}\in\mathcal{S}$ such that $h(\psi_{e^{\prime}})=0$ showing that $\tilde{\xi}_{e^{\prime}}$ is orthogonal to $\varphi_{2}$ in $L^{2}(\Omega)$, since $h(\psi_{e^{\prime}})=\int_{\Omega}\tilde{\xi}_{e^{\prime}}\varphi_{2}=k\int_{S_{2k,e^{\prime}}}\xi_{e^{\prime}}\varphi_{2}=0$.

From the definition of $k$-Morse index of $u$ it is easy to deduce that $Q_{u,\Omega}(\rho,\rho)\geq 0$ for all $\rho\in C^{1}_{0}(\Omega)$ that satisfies (1.3) which is orthogonal in $L^{2}(\Omega)$ to $\varphi_{1}$ and $\varphi_{2}$. By density this holds also for functions in $H^{1}_{0,loc}(\Omega)$ vanishing outside a compact set, and so we denote by $\mathcal{H}$ the subspace of functions in $H^{1}_{0,loc}(\Omega)$ vanishing outside a compact set that satisfies (1.3) which are orthogonal in $L^{2}(\Omega)$ to $\varphi_{1}$ and $\varphi_{2}$. Hence $Q_{u,\Omega}$ defines a semidefinite scalar product in $\mathcal{H}$ with corresponding Cauchy Schwarz inequality

[TABLE]

We now fix $\phi\in C_{0}^{\infty}(\Omega)$ and put

[TABLE]

Then $\phi_{1}\in\mathcal{H}$ and, letting $\xi_{R}$ be a cut-off function supported in $B_{R}$, we have

[TABLE]

Now, by Lemma 2.3 (i) of [GPW] we have that

[TABLE]

for $j=1,2$ and, since $w_{e^{\prime}}^{j}(\sigma_{e^{\prime}}(x))=w_{e^{\prime}}^{j}(x)$ then by (4.1)

[TABLE]

for $j=1,2$, and also

[TABLE]

Combining this with (5.11) we obtain

[TABLE]

which, together with (5.10) gives

[TABLE]

Since $\phi\in C^{\infty}_{0}(\Omega)$ is arbitrary, then $\tilde{\xi}_{e^{\prime}}$ solves $L_{u}\tilde{\xi}_{e^{\prime}}=0$ and also $L_{u}\xi_{e^{\prime}}=0$ in $S_{2k,e^{\prime}}$. Then, defining the function $\widehat{\xi}_{e^{\prime}}$ as in (4.6) we obtain as in the proof of Proposition 4.1 that $\xi_{e^{\prime}}\equiv 0$, which is possibile, by (5.9), since $w_{e^{\prime}}^{1}$ and $w_{e^{\prime}}^{2}$ have disjoint supports only when $A_{e^{\prime}}=B_{e^{\prime}}=0$, namely

[TABLE]

The $k$-invariance of $\varphi_{1}$ and $\tilde{w}_{e^{\prime}}$ then also implies that

[TABLE]

But this is not possible by the same argument in the end of the proof of Theorem 1.3, Case 1, in [GPW] and the contradiction concludes the proof of Case 1.

Case 2: $S_{*}\neq\emptyset$

Let us assume that $e=e_{\psi}\in S_{*}$, so that $w_{\psi}\equiv 0$ in $S_{k,\psi}^{+}$. Since, by definition

[TABLE]

then, there exists $R_{1}>0$ such that

[TABLE]

For an $R>\max{R_{0},R_{1}}$ we define, as in proof of Proposition 4.1, the functions $g_{\psi}$, $\widehat{g}_{\psi}$, $\tilde{g}_{\psi}$. By symmetry reasons $\tilde{g}_{\psi}$ is orthogonal to $\varphi_{1}$ in $L^{2}(\Omega\cap B_{R})$ and satisfies

[TABLE]

To conclude we want to prove that either $w_{\psi+\frac{\pi}{2k}}$ has a sign in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ or $w_{\psi+\frac{\pi}{2k}}\equiv 0$ and

[TABLE]

since the thesis then follows by Propositions 5.1 and 5.2. i) First we show that whenever $w_{\psi+\frac{\pi}{2k}}\equiv 0$ in $S_{k,\psi+\frac{\pi}{2k}}^{+}$ then (5.12) holds. Suppose not, then for sufficiently large $R>\max{R_{0},R_{1}}$ we have

[TABLE]

By the same arguments of the proof of Proposition 4.1 $i)$ the function $\tilde{g}_{\psi+\frac{\pi}{2k}}$ is orthogonal in $L^{2}(\Omega)$ to $\varphi_{1}$ and $\tilde{g}_{\psi}$, and satisfies

[TABLE]

contradicting the bound $m_{k}(u)\leq 2$. This shows that, whenever $w_{\psi+\frac{\pi}{2k}}\equiv 0$ then (5.12) should hold.

ii) It last to prove that, whenever $w_{\psi+\frac{\pi}{2k}}\neq 0$ then it has a sign in $S_{k,\psi+\frac{\pi}{2k}}^{+}$. Assume by contradiction it changes sign and define the functions $w_{e}^{1}$ and $w_{e}^{2}$ as in (4.3) relative to the direction $e=e_{\psi+\frac{\pi}{2k}}$. Then, as in the proof of Proposition 4.1 $ii)$ the function $\tilde{\xi}_{e}$ is orthogonal in $L^{2}(\Omega)$ to $\varphi_{1}$ and to $\tilde{g}_{\psi}$. Then the proof follows as in the end of Case 1, where $\varphi_{2}$ is substituted by $\tilde{g}_{\psi}$ getting a contradiction and showing that, in this case, ${}_{\psi+\frac{\pi}{2k}}$ has a sign in $S_{k,\psi+\frac{\pi}{2k}}^{+}$. ∎

6. Applications

Let us consider the Lane-Emden problem

[TABLE]

where $p>1$ is a real parameter. In the paper [GGPS] the unique positive radial solution to (6.1) is studied when $\Omega$ is an annulus. In particular problem (6.1) admits a unique positive radial solution $u_{p}$ for every $p\in(1,\infty)$ and its Morse index converges to $+\infty$ as $p\to\infty$. This behavior produces infinitely many nonradial positive solutions that arise by bifurcation from the solution $u_{p}$. Moreover in [G2] it has been proved that, for every $k\in\mathbb{N}$, $k\geq 1$ there exists an exponent $p_{k}>1$ at which un unbounded continuum of $k$-invariant solutions to (6.1) bifurcates. This continuum exists for every $p>p_{k}$ and all the solutions in it are symmetric with respect to a direction $e\in\mathcal{S}$ and are monotone in the angular variable in the sectors $S_{k,e}^{\pm}$, by construction. Namely they present the same monotonicity properties that we proved under the $k$-Morse index bound.

It is also proved in [G2], by Morse index considerations, that, letting

[TABLE]

the least energy solution $u_{p}^{k}$ in $H^{1}_{0,k}$, that can be found minimizing the Energy functional associated with (6.1)

[TABLE]

on the $k$-invariant Nehari manifold

[TABLE]

is nonradial for every $p>p_{k}$, where $p_{k}$ is the same as before.

What it lasts to be proved is that these least energy solutions $u_{p}^{k}$ cannot have more invariances, namely they cannot belong to $H^{1}_{0,k^{\prime}}$ for $k^{\prime}>k$.

But this can now follows from our results. Indeed a least energy solution $u_{p}^{k}\in H^{1}_{0,k}$ satisfies

[TABLE]

so that, by Theorem 1.1, $u_{p}^{k}$ is symmetric with respect to a direction $e\in\mathcal{S}$ in $S_{2k,e}$ and it is strictly monotone in the angular variable in $S_{k,e}^{\pm}$ so that cannot belong to $H^{1}_{0,k^{\prime}}$ for $k^{\prime}>k$.

Summarizing we have:

Theorem 6.1**.**

For any $k\geq 1$ there exists an exponent $p_{k}$ for which $u_{p}^{k}$ is non radial for any $p>p_{k}$, it is symmetric with respect to a direction $e\in\mathcal{S}$ and it is strictly monotone in the angular variable in $S_{k,e}^{\pm}$. Moreover for $p>p_{k}$ $u_{p}^{k}\neq u_{p}^{h}$ with $h<k$ so that problem (6.1) admits at least $k+1$ positive different solutions.

The $k+1$ solutions in Theorem 6.1 are given by the radial solution $u_{p}$, the least-energy solution $u_{p}^{1}$ (corresponding to $k=1$) which is the least-energy solution in $H^{1}_{0}(\Omega)$, the least energy solution $u_{p}^{2}\in H^{1}_{0,2}$ and so on until the least energy solution $u_{p}^{k}\in H^{1}_{0,k}$.

This permits to say that the solutions found by bifurcation and the least energy solutions in the symmetric spaces $H^{1}_{0,k}$ possess the same symmetry and monotonicity properties in the sectors $S_{k,e}^{\pm}$ supporting the conjecture that indeed they are the same.

Finally we observe that solutions with a large number of peaks for large values of $p$ has been constructed in [EMP] in a more general domain than an annulus via the Lyapunov Schmidt reduction method. We believe that their solutions in the case of an annulus can coincide with ours $u_{p}^{k}$.

We are confident that a very similar result should hold also for the exponential nonlinearity, namely

[TABLE]

in an annulus, using the asymptotic behavior of the radial mountain pass solution $u_{\lambda}$ as ${\lambda}\to 0^{+}$, performed in [GG]. From this it should follow that the Morse index of $u_{\lambda}$ converges to $+\infty$ as ${\lambda}\to 0^{+}$, generating infinitely many nonradial solutions that arise by bifurcation as in the previous case. Applying the Mountain pass Theorem in the spaces $H^{1}_{0,k}$ then one ends with a $k$-Morse index one positive solution that cannot be radial by Morse index considerations. Then Theorem 1.1 applies and implies that solutions found in this way cannot coincide. Solutions with a large number of peaks for small values of ${\lambda}$ has been constructed in [EGP] and [DPKM] via the Lyapunov Schmidt reduction method.

Of course a result of this type deserves a deep study that we give to the interested reader.

Let us turn to the case of nodal solutions and consider now the Lane-Emden problem (6.1) in the unit ball $B$. In [GI] the radial nodal least-energy solution $u_{p}$ is studied. In particular it is shown that the Morse index of $u_{p}$ changes in its existence range $p\in(1,\infty)$, corresponding to some symmetric spaces $H^{1}_{0,k}$ when $k=3,4$ and $5$ giving rise to some bifurcating branches of solutions which arise from the nodal radial solution $u_{p}$. Along these branches solutions are symmetric with respect to a direction $e\in\mathcal{S}$ and are monotone in the angular variable in the sectors $S_{k,e}^{\pm}$, by construction. Namely they present the same monotonicity properties that we proved under the $k$-invariant Morse index bound.

It is also shown, using the $k$-Morse index, that, for $k=3,4$ and $k=5$, there exists an exponent $p_{k}$ such that least-energy nodal solutions in $H^{1}_{0,k}$, that we denote by $\widetilde{u}_{p}^{k}$, are nonradial when $p>p_{k}$, while it seems that they are radial when $p$ is near $p=1$.

Moreover it is known that $\tilde{u}_{p}^{1}$ is nonradial for every $p>1$, see [BWW] or [P], and it has also been noticed in [GI], by Morse index considerations, that $\widetilde{u}_{p}^{2}$ is nonradial when $p$ is near $1$ and when $p$ is large.

These solutions $\widetilde{u}_{p}^{k}$ are found minimizing the energy functional $\mathcal{E}(u)$ on the nodal $k$-invariant Nehari manifold

[TABLE]

($s^{+}$ ($s^{-}$) stands for the positive (negative) part of $s$) which has codimension $2$ by a result of [BWe], so that their $k$-Morse index is exactly two, namely

[TABLE]

Then Theorem 1.2 applies when $p\geq 2$, and hence a $k$-invariant nodal least energy solution $\widetilde{u}^{k}_{p}$ is symmetric with respect to a direction $e\in\mathcal{S}$ and strictly monotone in the angular variable in the sectors $S_{k,e}^{\pm}$, showing that $\widetilde{u}_{p}^{1}\neq\widetilde{u}_{p}^{2}\neq\widetilde{u}_{p}^{3}\neq\widetilde{u}_{p}^{4}\neq\widetilde{u}_{p}^{5}$ when they are nonradial and $p\geq 2$. For $k=1$, since $H^{1}_{0,1}$ coincides with $H^{1}_{0}(B)$ then $\widetilde{u}_{p}^{1}$ coincide with the least energy nodal solution and hence it is foliated Schwarz symmetric for every $p$, by [BWW]. Observe that the foliated Schwarz symmetry in the plane is nothing else that our $k$-invariance and $k$-monotonicity property for $k=1$.

Summarizing previous and new results we have:

Theorem 6.2**.**

The solution $\widetilde{u}_{p}^{1}$ is nonradial for every $p>1$. For $k=2,3,4,5$ there exists an exponent $p_{k}$ for which $\widetilde{u}_{p}^{k}$ is non radial for any $p>p_{k}$. When they are nonradial, the functions $\widetilde{u}_{p}^{k}$ are symmetric with respect to a direction $e\in\mathcal{S}$ and strictly monotone in the angular variable in $S_{k,e}^{\pm}$. Moreover $\widetilde{u}_{p}^{1}\neq\widetilde{u}_{p}^{k}$ for every $p>1$ and every $k$, and for $p>p_{k}$ $\widetilde{u}_{p}^{k}\neq\widetilde{u}_{p}^{h}$ with $h<k$ so that problem (6.1) admits for $p>p_{k}$ at least $k+1$ distinct nodal solutions.

The $k+1$ in Theorem 6.2 solutions are given by the radial solution $u_{p}$, the nodal least-energy solution $\widetilde{u}_{p}^{1}$ (corresponding to $k=1$) which is the nodal least-energy solution in $H^{1}_{0}(\Omega)$, the nodal least-energy solution $\widetilde{u}_{p}^{2}\in H^{1}_{0,2}$ and so on until the nodal least-energy solution $\widetilde{u}_{p}^{5}\in H^{1}_{0,5}$. Finally, Morse index considerations in [GI] suggest that $\widetilde{u}_{p}^{k}$ coincides with the radial nodal solution when $k\geq 6$.

As before we proved so far that the solutions found by bifurcation and the least energy nodal solutions in the symmetric spaces $H^{1}_{0,k}$ possess the same symmetry and monotonicity properties in the sectors $S_{k,e}^{\pm}$ supporting the conjecture that indeed they are the same.

Solutions with this type of symmetry have been construct by Lyapunov Schmidt reduction method in [EMP2].

Theorem 1.2 can be applied also considering the $\sinh$-Poisson problem

[TABLE]

where ${\varepsilon}>0$ is a small parameter when $\Omega$ is a ball or an annulus and the solutions change sign. In this case the known results, see [BPW], suggest that nonradial solutions can be found using our minimization procedure in the spaces $H^{1}_{0,k}$. We leave the interested reader to carry out this study.

The previous results can be applied also to positive and nodal solutions of the Hénon problem

[TABLE]

where $\Omega$ is a ball or an annulus, and can be used to distinguish solutions that belong to different spaces $H^{1}_{0,k}$. In the paper [AG], as an example, it has been used, when $\Omega$ is a ball, in order to obtain some multiplicity results, minimizing the energy functional associated with (6.5), namely

[TABLE]

on the $k$-invariant Nehari manifold

[TABLE]

or on the $k$-invariant nodal Nehari manifold

[TABLE]

Minimizing $\mathcal{E}(u)$ on $\mathcal{N}_{k}$, for any $k\in\mathbb{N}$, $k\geq 1$, produces positive solutions that we denote by $u^{k}$ and satisfies

[TABLE]

while minimizing $\mathcal{E}(u)$ on $\mathcal{N}_{k,nod}$, for any $k\in\mathbb{N}$, $k\geq 1$, produces nodal solutions that we denote by $\widetilde{u}^{k}$ and satisfies

[TABLE]

In [AG] it has been shown, by a careful study of radial solutions, that:

Theorem 6.3** (Theorem 4.2 in [AG]).**

Let ${\alpha}>0$ be fixed. Then, there exists an exponent $p^{\star}_{1}({\alpha})>1$ such that problem (6.5) admits at least $j_{1}=1+\lceil\frac{{\alpha}}{2}\rceil$ distinct positive solutions for every $p>p^{\star}_{1}({\alpha})$.

Here $\lceil t\rceil=\min\{k\in\mathbb{Z}:k\geq t\}$ denotes the ceiling function. One solution is radial while the others are not. Further

Theorem 6.4** (Theorem 4.4 in [AG]).**

Let ${\alpha}\geq 0$ be fixed. Then, there exists an exponent $p^{\star}_{2}({\alpha})>1$ such that problem (6.5) admits at least $j_{2}=1+\lceil\frac{2+{\alpha}}{2}\kappa\rceil$ distinct nodal least energy solutions for every $p>p^{\star}_{2}({\alpha})$.

Here $\kappa\approx 5.1869$. This number $\kappa$ has been found out in [GGP] in the study of nodal radial solutions to (6.1) in the unit ball and indeed it is responsible of the existence of the nonradial nodal solutions $\widetilde{u}_{p}^{k}$ for $k=2,3,4$ and $5$. As seen in Theorem 6.4 it plays a role also in the case of nodal solutions to (6.5), where its effects are combined with the ones of the parameter ${\alpha}$. When ${\alpha}=0$, $\lceil\frac{2+{\alpha}}{2}\kappa\rceil=5$ as said in Theorem 6.2.

Positive solutions with this type of symmetry have been construct by the finite dimensional reduction method in [EPW] for large values of $p$.

Nodal solutions with this type of symmetry have been construct by the finite dimensional reduction method in [ZY] for large values of $p$.

We believe that very similar results should hold also for positive solutions with exponential nonlinearities of the Hénon type

[TABLE]

and for positive and sign changing solution of the $\sinh$-Poisson problem of Hénon type

[TABLE]

where ${\lambda}$ and ${\varepsilon}$ are small parameters. Note that Theorem 1.1 can be applied in the first case, while Theorem 1.2 holds in the second example.

We quote the existence results in [GGNe], [D1], [D2]

We end observing that in the case of the unbounded domain $\mathbb{R}^{2}$, positive solutions to

[TABLE]

have been classified in the famous paper [PT] and they exhibit the same monotonicity properties that we have highlighted in our results.