Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

Abstract

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik–Schnirelman theory, and new estimates on the category of suitable configuration spaces.

1 Introduction

We consider the following system of singularly perturbed elliptic equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{2}\Delta u_{i}+u_{i}=\sum \limits _{j=1}^\ell \beta _{ij}|u_{j}|^{p}|u_{i}|^{p-2}u_{i},\\ u_{i} \in H^{1}_{0}(\Omega ), \quad u_i\ne 0, \qquad i=1,\ldots ,\ell , \end{array}\right. } \end{aligned}$$

(1.1)

where $\varepsilon >0$ is a small parameter, $\Omega$ is a bounded smooth domain in $\mathbb {R}^N$, $N\ge 2$, $\ell \ge 2$, $\beta _{ii}>0$, $\beta _{ij}=\beta _{ji}<0$ if $i\ne j$, and $2p\in (2,2^*)$, where $2^*$ is the critical Sobolev exponent, i.e., $2^*:=\infty$ if $N=2$ and $2^*:= \frac{2N}{N-2}$ if $N\ge 3$.

System (1.1) is used in physics to model a variety of phenomena; for instance, it is a model for a binary mixture of Bose–Einstein condensates in two different hyperfine states (then $\ell =2$) and in nonlinear optics. It also serves as a model in population dynamics. In this paper, we assume $\beta _{ii}>0$ and $\beta _{ij}<0$ for $i\ne j$ which means that the interaction between species (states) of the same type is attractive, while it is repulsive for species of different types. However, our point of view in this article is purely mathematical.

A solution $\textbf{u}=(u_1,\ldots ,u_\ell )$ to (1.1) is called fully nontrivial if $u_i\ne 0$ for every $i=1,\ldots ,\ell$, and we shall call it nonnegative if, in addition, $u_i\ge 0$ for all $i=1,\ldots ,\ell$.

Most of the research for (1.1) is focused on the case $\ell =p=2$ and $N=1,2,3$. In this setting, the asymptotic behavior (as $\varepsilon \rightarrow 0$) of nonnegative solutions of (1.1) was studied in detail in the seminal paper [17]. There, it is shown that the least energy solutions exhibit concentration. To be more precise, as $\varepsilon \rightarrow 0,$ the ith component is close to a rescaling and translation of the positive radially symmetric ground state solution of

$$\begin{aligned} -\Delta u+u=\beta _{ii}|u|^{2p-2}u,\qquad u \in H^1(\mathbb {R}^N). \end{aligned}$$

The concentration points approach a configuration that maximizes the distance between them and to $\partial \Omega$. The existence of two nonnegative solutions for $\ell =p=2$ and $N=1,2,3$ is shown in [24] using the Lusternik–Schnirelman theory.

This multiplicity result is interesting when compared to the case of a single equation. Consider for example

$$\begin{aligned} -\Delta u+u=|u|^{2p-2}u,\qquad u \in H_0^1(\Omega )\smallsetminus \{0\}. \end{aligned}$$

(1.2)

Uniqueness or multiplicity results for (1.2) rely on both the geometry and the topology of the domain $\Omega$ (see the introduction of [12] for an updated survey in this regard). In particular, if $\Omega$ is a ball, then (1.2) has a unique positive solution.

Therefore, a natural question is whether, for any domain, the number of nonnegative fully nontrivial solutions of (1.1) increases as the number of equations $\ell$ becomes larger. We give a positive answer to this question. Our main result is the following one.

Theorem 1.1

For $\varepsilon$ small enough, the system (1.1) has at least $\ell$ nonnegative solutions.

Note that this result concerns any dimension $N\ge 2$. If $p<2$, which is necessarily the case for $N\ge 4$, then neither the functional corresponding to (1.1) nor the Nehari-type manifold (defined later) is of class $\mathcal {C}^2$. It will therefore be convenient to employ the method of [10]. Then, we follow the ideas introduced in [4] which allow to estimate the number of critical points in the presence of concentration using the Lusternik–Schnirelman theory. For a multiplicity result in dimensions $N=2$ and $N=3$ with $p=2$, under symmetry assumptions on the domain and the coupling coefficients, we refer to [15].

Theorem 1.1 is a direct consequence of Theorems 1.2 and 1.3 below. To state these results, we introduce some notation.

Let X be a topological space. Recall that the Lusternik–Schnirelman category of a subset A of X in X, denoted $\textrm{cat}(A\hookrightarrow X)$ (or $\textrm{cat}_X(A)$), is the smallest number of subsets of A that cover A and each of them is open in A and contractible in X. If $A=X$, we write $\textrm{cat}(X)$ instead of $\textrm{cat}(X\hookrightarrow X)$.

Let $\Theta$ be a subset of $\mathbb {R}^N$. For $r\ge 0$ and $\ell \ge 2$, we consider the space

$$\begin{aligned} F_{\ell ,r}(\Theta ):=\{(\xi _1,\ldots ,\xi _\ell )\in \Theta ^\ell :\textrm{dist}(\xi _i,\mathbb {R}^N\smallsetminus \Theta )>r\text { and }|\xi _i-\xi _j|> 2r\text { if }i\ne j\}, \end{aligned}$$

endowed with the subspace topology of $\Theta ^\ell$. If $r=0$, we write $F_\ell (\Theta ):=F_{\ell ,0}(\Theta )$. This last space is called the ordered configuration space of $\ell$ points in $\Theta$. Finally, we set

$$\begin{aligned} \Theta ^+_r:=\{x\in \mathbb {R}^N:\textrm{dist}(x,\Theta )<r\}. \end{aligned}$$

The following theorem gives a lower bound on the number of nonnegative solutions of (1.1).

Theorem 1.2

Given $r>0$, there exists $\varepsilon _r>0$, such that the system (1.1) has at least

$$\begin{aligned} \textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)) \end{aligned}$$

(1.3)

nonnegative solutions for any $\varepsilon \in (0,\varepsilon _r)$.

The next result gives an estimate for (1.3).

Theorem 1.3

For $r>0$ sufficiently small, we have that

$$\begin{aligned} \textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+))\ge \ell . \end{aligned}$$

Furthermore, if $\Omega$ is convex and $r>0$ is sufficiently small, then $\textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega ))=\ell$.

Let us briefly discuss the strategy of the proofs. Existence of a nonnegative solution ${\textbf {u}}=(u_1,\ldots ,u_\ell )$ was established in [9] by minimization on a Nehari-type manifold following the approach in [10]. Denote such manifold corresponding to our system (1.1) by $\mathcal {N}_{\varepsilon }(\Omega )$. We show that all low-energy solutions, i.e., solutions $\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega )^{\le {\overline{d}}}$ for a suitable ${\overline{d}}$, do not change sign and exhibit concentration as $\varepsilon \rightarrow 0$ (see Sect. 2 for the definitions, and Theorem 3.2 and Proposition 3.3 for more rigorous statements). To count only the nonnegative solutions, we use a suitable quotient space. Let $\mathbb {Z}_2:=\{1,-1\}$ and consider the group $\mathcal {Z}:=(\mathbb {Z}_2)^\ell$ acting on $H_0^1(\Omega )^\ell$ by $\textbf{s}\textbf{u}:=(s_1u_1,\ldots ,s_\ell u_\ell )$ for $\textbf{s}=(s_1,\ldots ,s_\ell )\in \mathcal {Z}$ and $\textbf{u}=(u_1,\ldots ,u_\ell )\in H_0^1(\Omega )^\ell .$ Then, applying the method of [10] and some standard results in critical point theory with symmetries, we show that (1.1) has at least $\textrm{cat}(\mathcal {N}_{\varepsilon }(\Omega )^{\le {\overline{d}}}/\mathcal {Z})$ nonnegative solutions (see Theorem 2.1), where $\mathcal {N}_{\varepsilon }(\Omega )^{\le {\overline{d}}}/\mathcal {Z}$ is the space of $\mathcal {Z}$-orbits in $\mathcal {N}_{\varepsilon }(\Omega )^{\le {\overline{d}}}$. Now, the task is to explicitly estimate this number. Using the concentration behavior, we construct maps

$$\begin{aligned} F_{\ell ,r}(\Omega )\xrightarrow {\textbf{i}_\varepsilon }\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}\xrightarrow {q}\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}/\mathcal {Z}\xrightarrow {\widetilde{\textbf{b}}}F_\ell (\Omega _r^+), \end{aligned}$$

(1.4)

where $\textbf{i}_\varepsilon :F_{\ell ,r}(\Omega )\rightarrow \mathcal {N}_{\varepsilon }(\Omega )^{\le {\overline{d}}}$ (defined in (4.2)) places a copy of a suitable concentrated profile around each point of an $\ell$-tuple $(\xi _1,\ldots ,\xi _\ell )\in F_{\ell ,r}(\Omega )$, $q:\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}\rightarrow \mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}/\mathcal {Z}$ is the quotient map, and $\widetilde{\textbf{b}}$ is defined by $\textbf{b}=\widetilde{\textbf{b}}\circ q$, where $\textbf{b}:\mathcal {N}_{\varepsilon }(\Omega )^{\le d}\rightarrow F_\ell (\Omega _r^+)$ is a generalized barycenter map (see Proposition 4.3). The composition of these maps is the inclusion, whose category is a lower bound for $\textrm{cat}\left( \mathcal {N}_{\varepsilon }(\Omega )^{\le d}/\mathcal {Z}\right)$ (see Sect. 4).

Theorem 1.3 is proved by induction on $\ell$ using suitably constructed fibrations which allow to apply some basic tools from algebraic topology to obtain the estimate $\textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+))\ge \textrm{cat}(F_\ell (\mathbb {R}^N)) = \ell$ (see Sect. 5). The equality $\textrm{cat}(F_\ell (\mathbb {R}^N)) = \ell$ is shown in [21, Theorem 1.2].

Therefore, methodologically, our main contribution is to show how the category of $F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)$ can be estimated and used to prove multiplicity of nonnegative solutions of variational systems that exhibit concentration. We believe that this approach can be useful in other problems as well, for instance, to study systems of equations with suitable potentials in unbounded domains, and to obtain multiplicity results for low-energy nodal solutions of (1.1).

Our results hold true for systems with more general power nonlinearities, like those considered in [9,10,11]. We have considered the particular system (1.1) for the sake of simplicity.

System (1.1) has been extensively studied in recent years from several perspectives. Here, we just mention some of these results to provide an overview on different aspects of the problem. For a study of dependence of the concentration points on suitable potentials, see [20]. For results on sign-changing solutions, we refer to [9] and the references therein. For problems in $\mathbb {R}^N$, see [6, 22, 25], and for a recent study on the decay at infinity of solutions, we refer to [1]. For a relationship between nodal solutions and fully nontrivial solutions of (1.1) in the critical case, see [11].

We also mention that the use of configuration spaces, barycenter maps, and the Lusternik–Schnirelman theory is not completely new. It has been used, for instance, in [3] to study lower bounds for the number of nodal solutions to a singularly perturbed nonlinear Schrödinger equation.

The paper is organized as follows. In Sect. 2, we present the variational framework and give a lower bound for the number of solutions to (1.1) in terms of $\textrm{cat}\left( \mathcal {N}_{\varepsilon }(\Omega )^{\le d}/\mathcal {Z}\right)$. Sect. 3 is devoted to showing that low-energy solutions do not change sign. The maps (1.4) are constructed in Sect. 4, where we also give a proof of Theorem 1.2. Finally, in Sect. 5, we show the estimate in Theorem 1.3.

2 The variational problem

For $\varepsilon >0$ fixed, let

$$\begin{aligned} \Vert v\Vert _\varepsilon := \left( \frac{1}{\varepsilon ^N}\int _{\Omega }(\varepsilon ^2|\nabla v|^2+v^2)\right) ^{1/2} \end{aligned}$$

be the norm of v in $H^1_0(\Omega )$ (equivalent to the usual one). We write the elements in $H_0^1(\Omega )^\ell$ as $\textbf{u}=(u_1,\ldots ,u_\ell )$ and we introduce the norm $\Vert \textbf{u}\Vert _\varepsilon := (\Vert u_1\Vert _\varepsilon ^2 + \cdots +\Vert u_\ell \Vert _\varepsilon ^2)^{1/2}$. The solutions to the system (1.1) are the critical points of the functional $\mathcal {J}_\varepsilon :H_0^1(\Omega )^\ell \rightarrow \mathbb {R}$ given by

$$\begin{aligned} \mathcal {J}_\varepsilon (\textbf{u}):= \frac{1}{2\varepsilon ^N}\sum _{i=1}^\ell \int _{\Omega }\left( \varepsilon ^2|\nabla u_i|^2 + u_i^2\right) - \frac{1}{2p\varepsilon ^N}\sum _{i,j=1}^\ell \int _{\Omega }\beta _{ij}|u_j|^p|u_i|^p, \end{aligned}$$

which is of class $\mathcal {C}^1$. Its partial derivatives at $\textbf{u}$ are given by

$$\begin{aligned} \partial _i\mathcal {J}_\varepsilon (\textbf{u})v =\frac{1}{\varepsilon ^N}\left[ \int _{\Omega }(\varepsilon ^2\nabla u_i \cdot \nabla v + u_iv) -\sum _{j=1}^\ell \int _{\Omega }\beta _{ij}|u_j|^p|u_i|^{p-2}u_iv\right] , \end{aligned}$$

for $v \in H^1_0(\Omega )$ and $i=1,\ldots ,\ell$. The fully nontrivial solutions of (1.1) belong to the Nehari-type set

$$\begin{aligned} \mathcal {N}_{\varepsilon }(\Omega ):=\Big \{\textbf{u}\in H_0^1(\Omega )^\ell :u_i\ne 0, \ \partial _i\mathcal {J}_\varepsilon (\textbf{u})u_i=0, \ \forall \ i = 1,\ldots ,\ell \Big \}. \end{aligned}$$

(2.1)

Define

$$\begin{aligned} c_\varepsilon (\Omega ):= \inf _{\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega )}\mathcal {J}_\varepsilon (\textbf{u}). \end{aligned}$$

If $\textbf{u}\in \mathcal {N}_\varepsilon (\Omega )$ satisfies $\mathcal {J}_\varepsilon (\textbf{u})=c_\varepsilon (\Omega )$, then $\textbf{u}=(u_1,\ldots ,u_\ell )$ is a solution to the system (1.1); see [10, Theorem 3.4(a)]. It is called a least energy solution. Since $\Omega$ is bounded and 2p is subcritical, a standard argument yields a least energy solution for any $\varepsilon >0$; see [9, Theorem 2.6].

Let $\mathbb {Z}_2:=\{1,-1\}$. Recall that the group $\mathcal {Z}:=(\mathbb {Z}_2)^\ell$ acts on $H_0^1(\Omega )^\ell$ by

$$\begin{aligned} \textbf{s}\textbf{u}:=(s_1u_1,\ldots ,s_\ell u_\ell )\qquad \text {where }\textbf{s}=(s_1,\ldots ,s_\ell )\in \mathcal {Z}\text { and }\textbf{u}=(u_1,\ldots ,u_\ell )\in H_0^1(\Omega )^\ell . \end{aligned}$$

(2.2)

The $\mathcal {Z}$-orbit of ${\textbf{u}}$ is the set $\mathcal {Z}\textbf{u}:= \{\textbf{s}\textbf{u}: \textbf{s}\in \mathcal {Z}\}$. A subset $\mathcal {A}$ of $H_0^1(\Omega )^\ell$ is called $\mathcal {Z}$-invariant if $\mathcal {Z}\textbf{u}\subset \mathcal {A}$ whenever $\textbf{u}\in \mathcal {A}$, and a function $\Phi :\mathcal {A}\rightarrow \mathbb {R}$ is called $\mathcal {Z}$-invariant if $\Phi$ is constant on each $\mathcal {Z}$-orbit of $\mathcal {A}$. The $\mathcal {Z}$-orbit space of $\mathcal {A}$ is the set $\mathcal {A}/\mathcal {Z}:=\{\mathcal {Z}\textbf{u}:\textbf{u}\in \mathcal {A}\}$ endowed with the quotient space topology. Note that $\mathcal {N}_\varepsilon (\Omega )$ and $\mathcal {J}_\varepsilon$ are $\mathcal {Z}$-invariant, and so is the sublevel set

$$\begin{aligned} \mathcal {N}_{\varepsilon }(\Omega )^{\le d}:=\{\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega ):\mathcal {J}_\varepsilon (\textbf{u})\le d\},\qquad d\in \mathbb {R}. \end{aligned}$$

If $\textbf{u}$ is a critical point of $\mathcal {J}_\varepsilon$, then so is $\textbf{s}\textbf{u}$ for every $\textbf{s}\in \mathcal {Z}$, and $\mathcal {Z}\textbf{u}$ is called a critical $\mathcal {Z}$-orbit of $\mathcal {J}_\varepsilon$.

Theorem 2.1

Let $\varepsilon >0$ and $d\in \mathbb {R}$. Then, $\mathcal {J}_\varepsilon$ has at least

$$\begin{aligned} \textrm{cat}\left( \mathcal {N}_{\varepsilon }(\Omega )^{\le d}/\mathcal {Z}\right) \end{aligned}$$

critical $\mathcal {Z}$-orbits in $\mathcal {N}_{\varepsilon }(\Omega )^{\le d}$.

Proof

The proof follows by an adaptation of the argument in [10, Theorem 3.4] and well-known results in critical point theory with symmetries. For the reader’s convenience, we sketch it here.

Fix $\varepsilon >0$. Let $S:=\{v\in H^1_0(\Omega ): \Vert v\Vert _\varepsilon =1\}$, $\mathcal {T}:=S\times \cdots \times S$ ($\ell$ times) and

$$\begin{aligned} \mathcal {U}_\varepsilon := \{\textbf{u}\in \mathcal {T}: {\textbf{t}}\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega )\text { for some }{\textbf{t}}=(t_1,\ldots ,t_\ell )\in (0,\infty )^\ell \}. \end{aligned}$$

For $\textbf{u}\in \mathcal {U}_\varepsilon$, there exists a unique ${\textbf{t}}_\textbf{u}\in (0,\infty )^\ell$ such that ${\textbf{t}}_\textbf{u}\textbf{u}\in \mathcal {N}_\varepsilon (\Omega )$. Let $\textbf{m}_\varepsilon (\textbf{u}):= {\textbf{t}}_\textbf{u}\textbf{u}$. Then, $\textbf{m}_\varepsilon : \mathcal {U}_\varepsilon \rightarrow \mathcal {N}_\varepsilon (\Omega )$ is a homeomorphism. This and more can be seen from [10, Proposition 3.1]. Now, let

$$\begin{aligned} \Psi _\varepsilon (\textbf{u}):= \mathcal {J}_\varepsilon ({\textbf{t}}_\textbf{u}\textbf{u}),\qquad \textbf{u}\in \mathcal {U}_\varepsilon . \end{aligned}$$

According to [10, Theorem 3.3], $\Psi _\varepsilon \in \mathcal {C}^1(\mathcal {U}_\varepsilon ,\mathbb {R})$ and $\textbf{u}$ is a critical point of $\Psi _\varepsilon$ if and only if $\textbf{m}_\varepsilon (\textbf{u})$ is a fully nontrivial critical point of $\mathcal {J}_\varepsilon$. Now, a standard argument applies. One can construct a pseudogradient vector field for $\Psi _\varepsilon$ on $\mathcal {U}_\varepsilon$ and obtain a deformation lemma using its flow, as, e.g., in [26, Section 5.3]. By (iii) of [10, Theorem 3.3], the flow cannot reach the boundary of $\mathcal {U}_\varepsilon$.

Since

$$\begin{aligned} \mathcal {J}_\varepsilon (\textbf{u}) - \frac{1}{2p}\mathcal {J}_\varepsilon '(\textbf{u})\textbf{u} = \left( \frac{1}{2} - \frac{1}{2p}\right) \Vert \textbf{u}\Vert _\varepsilon ^2, \end{aligned}$$

it is easy to see using the compactness of the embedding $H^1_0(\Omega ) \hookrightarrow L^{2p}(\Omega )$ that $\mathcal {J}_\varepsilon$ satisfies the Palais–Smale condition on $\mathcal {N}_{\varepsilon }(\Omega )$ and is bounded from below there. Using [10, Theorem 3.3] again, it follows that the same is true of $\Psi _\varepsilon$ on $\mathcal {U}_\varepsilon$.

Note that $\mathcal {U}_\varepsilon$ and $\Psi _\varepsilon$ are $\mathcal {Z}$-invariant and the action of $\mathcal {Z}$ on $\mathcal {T}$ is free, that is, the $\mathcal {Z}$-orbit of every point $\textbf{u}\in \mathcal {T}$ is $\mathcal {Z}$-homeomorphic to $\mathcal {Z}$. Therefore, by [8, Theorem 1.1], $\Psi _\varepsilon$ has at least $\mathscr {G}$-$\textrm{cat}(\mathcal {U}_\varepsilon ^{\le d})$ critical $\mathcal {Z}$-orbits in

$$\begin{aligned} \mathcal {U}_\varepsilon ^{\le d}:=\{\textbf{u}\in \mathcal {U}_{\varepsilon }:\Psi _\varepsilon (\textbf{u})\le d\}, \end{aligned}$$

where $\mathscr {G}:=\{\mathcal {Z}\}$. Since $\mathcal {Z}$ acts freely on $\mathcal {U}_\varepsilon ^{\le d}$, one easily verifies that

$$\begin{aligned} \mathscr {G}\text {-}\textrm{cat}(\mathcal {U}_\varepsilon ^{\le d})=\textrm{cat}(\mathcal {U}_\varepsilon ^{\le d}/\mathcal {Z}), \end{aligned}$$

and, as ${\textbf{m}}_\varepsilon$ induces a homeomorphism of the $\mathcal {Z}$-orbit spaces, one has that $\textrm{cat}(\mathcal {U}_\varepsilon ^{\le d}/\mathcal {Z})=\textrm{cat}(\mathcal {N}_\varepsilon ^{\le d}/\mathcal {Z})$. It follows that $\mathcal {J}_\varepsilon$ has at least $\textrm{cat}\left( \mathcal {N}_{\varepsilon }(\Omega )^{\le d}/\mathcal {Z}\right)$ critical $\mathcal {Z}$-orbits in $\mathcal {N}_{\varepsilon }(\Omega )^{\le d}$, as claimed. $\square$

3 The sign of low-energy solutions

For each $i=1,\ldots ,\ell$, consider the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w+w=\beta _{ii}|w|^{2p-2}w,\\ w\in H^1(\mathbb {R}^N), \quad w\ne 0. \end{array}\right. } \end{aligned}$$

(3.1)

Its solutions are the critical points of the functional $J_{\infty ,i}:H^1(\mathbb {R}^N)\rightarrow \mathbb {R}$ given by

$$\begin{aligned} J_{\infty ,i}(w):=\frac{1}{2}\int _{\mathbb {R}^N}(|\nabla w|^2+w^2)-\frac{1}{2p}\int _{\mathbb {R}^N}\beta _{ii}|w|^{2p} \end{aligned}$$

on the Nehari manifold $\mathcal {M}_{\infty ,i}:=\{w\in H^1(\mathbb {R}^N):w\ne 0, \ J_{\infty ,i}'(w)w=0\}$. We set

$$\begin{aligned} \mathfrak {c}_{\infty ,i}:= \inf _{w\in \mathcal {M}_{\infty ,i}}J_{\infty ,i}(w). \end{aligned}$$

A solution $\omega _i$ to this problem that satisfies $J_{\infty ,i}(\omega _i)=\mathfrak {c}_{\infty ,i}$ is called a least energy solution. It is known to be radially symmetric up to translation; see [5, 14] or [26, Appendix C].

Recall that the energy functional for the system (1.1) in $\mathbb {R}^N$ with $\varepsilon =1$ is $\mathcal {J}_1:H^1(\mathbb {R}^N)^\ell \rightarrow \mathbb {R}$ given by

$$\begin{aligned} \mathcal {J}_1(\textbf{u}):= \frac{1}{2}\sum _{i=1}^\ell \int _{\mathbb {R}^N}\left( |\nabla u_i|^2 + u_i^2\right) - \frac{1}{2p}\sum _{i,j=1}^\ell \int _{\mathbb {R}^N}\beta _{ij}|u_j|^p|u_i|^p, \end{aligned}$$

the Nehari set is $\mathcal {N}_{1}({\mathbb {R}}^N):=\Big \{\textbf{u}\in H^1({\mathbb {R}}^N)^\ell :u_i\ne 0, \ \partial _i\mathcal {J}_1(\textbf{u})u_i=0, \ \forall \ i = 1,\ldots ,\ell \Big \}$, and

$$\begin{aligned} c_1(\mathbb {R}^N):= \inf _{\textbf{u}\in \mathcal {N}_1(\mathbb {R}^N)}\mathcal {J}_1(\textbf{u}). \end{aligned}$$

Lemma 3.1

$c_1(\mathbb {R}^N)=\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}=\lim _{\varepsilon \rightarrow 0}c_\varepsilon (\Omega )$, and $c_1(\mathbb {R}^N)$ is not attained if $\ell \ge 2$.

Proof

These are the statements of [9, Proposition 3.1 and Lemma 4.2] with G equal to the trivial group. Note that for such G, the set $\textrm{Fix}(G):= \{x\in \mathbb {R}^N: gx=x \text { for all }g\in G\}$ has positive dimension (in fact, $\textrm{Fix}(G) = \mathbb {R}^N$). This is needed to assure that $c_1(\mathbb {R}^N)$ equals the sum above. $\square$

The following result describes the behavior of minimizing sequences for the system (1.1) in $\mathbb {R}^N$ with $\varepsilon =1$. We write $B_1(x)$ for the unit ball in $\mathbb {R}^N$ centered at x.

Theorem 3.2

Let $\ell \ge 2$ and $\textbf{w}_k=(w_{1 k}, \ldots , w_{\ell k}) \in \mathcal {N}_{1}({\mathbb {R}}^N)$ be such that $\mathcal {J}_1({\textbf{w}}_k)\rightarrow c_1(\mathbb {R}^N)$. Then, there exist a positive number $\theta$ and, for each $i=1, \ldots , \ell$, a sequence $\left( \zeta _{i k}\right)$ in $\mathbb {R}^N$, and a least energy radial solution $\omega _i$ to the problem (3.1), such that after passing to a subsequence

• (i) $\int _{B_1(\zeta _{ik})}\beta _{ii}|w_{ik}|^{2p}\ge \theta >0,$

• (ii) $\lim _{k\rightarrow \infty }\left| \zeta _{i k}-\zeta _{j k}\right| =\infty$ if $i\ne j$,

• (iii) $\lim _{k \rightarrow \infty }\left\| w_{i k}-\omega _i\left( \cdot -\zeta _{i k}\right) \right\| _1=0$,

for $i,j=1,\ldots ,\ell$. Furthermore, if $w_{i k} \ge 0$ for all $k \in \mathbb {N}$, then $\omega _i \ge 0$.

Proof

This is the statement of [9, Theorem 3.3] when G is the trivial group. The inequality (i) is marked as (3.2) in the proof of that result. $\square$

Proposition 3.3

There exist $\varepsilon _0>0$ and $d>c_1(\mathbb {R}^N)$, such that, if $\varepsilon \in (0,\varepsilon _0)$ and $\textbf{u}=(u_1,\ldots ,u_\ell )$ is a critical point of $\mathcal {J}_\varepsilon$ with $\mathcal {J}_\varepsilon (\textbf{u})\le d$, then $u_i$ does not change sign for any $i=1,\ldots ,\ell$.

Proof

We argue by contradiction. Assume there are numbers $\varepsilon _k>0$ and $d_k>c_1(\mathbb {R}^N)$ and critical points $\textbf{u}_k=(u_{1 k}, \ldots , u_{\ell k})$ of $\mathcal {J}_{\varepsilon _k}$, such that $\varepsilon _k\rightarrow 0$, $d_k\rightarrow c_1(\mathbb {R}^N)$, $\textbf{u}_k\in \mathcal {N}_{\varepsilon _k}(\Omega )$, $\mathcal {J}_{\varepsilon _k}(\textbf{u}_k)\le d_k$, and $u_{ik}^+:=\max \{u_{ik},0\}\ne 0$ and $u_{ik}^-:=\min \{u_{ik},0\}\ne 0$ for some $i=1,\ldots ,\ell$. Without loss of generality, we assume that $i=1$.

Let ${\textbf{w}}_k=(w_{1 k}, \ldots , w_{\ell k})$ be given by $w_{ik}(z):=u_{ik}(\varepsilon _kz)$ if $z\in \Omega _k:=\{\varepsilon _kx:x\in \Omega \}$ and $w_{ik}(z):=0$ if $z\in \mathbb {R}^N\smallsetminus \Omega _k$. Then, ${\textbf{w}}_k\in \mathcal {N}_1(\mathbb {R}^N)$ and $\mathcal {J}_1({\textbf{w}}_k)=\mathcal {J}_{\varepsilon _k}(\textbf{u}_k)\rightarrow c_1(\mathbb {R}^N)$. By Lemma 3.1 and Theorem 3.2

$$\begin{aligned} J_{\infty ,i}(w_{ik})&= J_{\infty ,i}(w_{ik}(\cdot +\zeta _{ik})) \rightarrow J_{\infty ,i}(\omega _{i})=\mathfrak {c}_{\infty ,i}\qquad \text { and }\\ \mathcal {J}_1({\textbf{w}}_k)\rightarrow c_1(\mathbb {R}^N)&=\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}, \end{aligned}$$

where $\omega _i$ is a least energy solution to the problem (3.1). In particular

$$\begin{aligned} \lim _{k\rightarrow \infty }\sum \limits _{\begin{array}{c} i,j=1 \\ i\ne j \end{array}}^\ell \int _{\mathbb {R}^N}\beta _{ij}|w_{ik}|^p|w_{jk}|^p=2p \lim _{k\rightarrow \infty }\left( \sum _{i=1}^\ell J_{\infty ,i}(w_{ik})-\mathcal {J}_1({\textbf{w}}_k)\right) =0. \end{aligned}$$

Since $\textbf{u}_k$ is a critical point of $\mathcal {J}_{\varepsilon _k}$, one has that $\partial _1\mathcal {J}_1({\textbf{w}}_k)w_{1k}^+=\partial _1\mathcal {J}_{\varepsilon _k}(\textbf{u}_k)u_{1k}^+=0$. Hence

$$\begin{aligned} \int _{\mathbb {R}^N}(|\nabla w_{1k}^+|^2 + |w_{1k}^+|^2-\beta _{11}|w_{1k}^+|^{2p})=\sum _{j=2}^\ell \int _{\mathbb {R}^N}\beta _{1j}|w_{jk}|^p|w_{1k}^+|^{p}\rightarrow 0, \end{aligned}$$

(3.2)

and similarly for $w_{1k}^-$ [that the right-hand side above tends to 0 follows from (ii) and (iii) of Theorem 3.2]. Set

$$\begin{aligned} t_k:=\left( \frac{\int _{\mathbb {R}^N}(|\nabla w_{1k}^+|^2 + |w_{1k}^+|^2)}{\int _{\mathbb {R}^N}\beta _{11}|w_{1k}^+|^{2p}}\right) ^\frac{1}{2p-2}\text { and }s_k:=\left( \frac{\int _{\mathbb {R}^N}(|\nabla w_{1k}^-|^2 + |w_{1k}^-|^2)}{\int _{\mathbb {R}^N}\beta _{11}|w_{1k}^-|^{2p}}\right) ^\frac{1}{2p-2}. \end{aligned}$$

Then, $t_k w_{1k}^+,s_k w_{1k}^-\in \mathcal {M}_{\infty ,1}$, $t_k\rightarrow 1$ and $s_k\rightarrow 1$. Here, we have used (3.2) and the fact that $(w_{1k}^\pm ,w_{2k},\ldots ,w_{\ell k})\in \mathcal {N}_1(\mathbb {R}^N)$ which implies in particular that $w_{1k}^\pm$ are bounded away from 0 [10, Proposition 3.1(d)]. Thus, by the definition of $\mathfrak {c}_{\infty ,1}$

$$\begin{aligned} \mathcal {J}_1({\textbf{w}}_k)&=\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}+o(1)=\sum _{i=1}^\ell J_{\infty ,i}(w_{ik})+o(1)\\&=J_{\infty ,1}(w_{1k}^+)+J_{\infty ,1}(w_{1k}^-)+\sum _{i=2}^\ell J_{\infty ,i}(w_{ik})+o(1)\\&=J_{\infty ,1}(t_k w_{1k}^+)+J_{\infty ,1}(s_k w_{1k}^-)+\sum _{i=2}^\ell J_{\infty ,i}(w_{ik})+o(1)\ge 2\mathfrak {c}_{\infty ,1}+\sum _{i=2}^\ell \mathfrak {c}_{\infty ,i}+o(1)\\&=\mathfrak {c}_{\infty ,1}+\mathcal {J}_1({\textbf{w}}_k)+o(1), \end{aligned}$$

which is a contradiction. $\square$

4 The effect of the configuration space

Fix $r\in (0,\infty )$ and let $B_r$ be the ball of radius r in $\mathbb {R}^N$ centered at 0. For each $i=1,\ldots ,\ell$, consider the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta w+w=\beta _{ii}|w|^{2p-2}w,\\ w\in H^1_0(B_r), \quad w\ne 0. \end{array}\right. } \end{aligned}$$

(4.1)

Its solutions are the critical points of the functional $J_{\varepsilon ,r,i}:H^1_0(B_r)\rightarrow \mathbb {R}$ given by

$$\begin{aligned} J_{\varepsilon ,r,i}(w):=\frac{1}{2\varepsilon ^N}\int _{B_r}(\varepsilon ^2|\nabla w|^2+w^2)-\frac{1}{2p\varepsilon ^N}\int _{B_r}\beta _{ii}|w|^{2p} \end{aligned}$$

on the Nehari manifold $\mathcal {M}_{\varepsilon ,r,i}:=\{w\in H^1_0(B_r):w\ne 0, \ J_{\varepsilon ,r,i}'(w)w=0\}$. We set

$$\begin{aligned} \mathfrak {c}_{\varepsilon ,r,i}:= \inf _{w\in \mathcal {M}_{{\varepsilon ,}r,i}}J_{\varepsilon ,r,i}(w). \end{aligned}$$

A solution $w_{\varepsilon ,r,i}$ satisfying $J_{\varepsilon ,r,i}(w_{\varepsilon ,r,i})=\mathfrak {c}_{\varepsilon ,r,i}$ is called a least energy solution.

Lemma 4.1

$\lim _{\varepsilon \rightarrow 0}\mathfrak {c}_{\varepsilon ,r,i} =\mathfrak {c}_{\infty ,i}$ for each $i=1,\ldots ,\ell$.

Proof

This follows from Lemma 3.1, taking $\Omega =B_r$ and $\ell =1$. $\square$

For $r>0$, let

$$\begin{aligned} F_{\ell ,r}(\Omega ):=\{(\xi _1,\ldots ,\xi _\ell )\in \Omega ^\ell :\textrm{dist}(\xi _i,\mathbb {R}^N\smallsetminus \Omega )> r, \ |\xi _i-\xi _j|> 2r\text { if }i\ne j\}, \end{aligned}$$

endowed with the subspace topology of $\Omega ^\ell$. Note that $F_{\ell ,r}(\Omega )\ne \emptyset$ if r is sufficiently small. Define ${\textbf{i}}_\varepsilon :F_{\ell ,r}(\Omega )\rightarrow \mathcal {N}_{\varepsilon }(\Omega )$ by

$$\begin{aligned} {\textbf{i}}_\varepsilon (\xi _1,\ldots ,\xi _\ell ):=\left( \omega _{\varepsilon ,r,1}(\,\cdot \,-\xi _1),\ldots ,\omega _{\varepsilon ,r,\ell }(\,\cdot \,-\xi _\ell )\right) , \end{aligned}$$

(4.2)

where $\omega _{\varepsilon ,r,i}$ is the positive least energy solution to the problem (4.1) extended by 0 to $\mathbb {R}^N\smallsetminus \Omega$.

Lemma 4.2

Given $r>0$ and $d>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$, there exists $\widetilde{\varepsilon }_{r,d}>0$, such that

$$\begin{aligned} \mathcal {J}_\varepsilon ({\textbf{i}}_\varepsilon (\xi _1,\ldots ,\xi _\ell ))\le d\qquad \text {for every }(\xi _1,\ldots ,\xi _\ell )\in F_{\ell ,r}(\Omega )\text { and }\varepsilon \in (0,\widetilde{\varepsilon }_{r,d}). \end{aligned}$$

Proof

Let $d>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$ and $(\xi _1,\ldots ,\xi _\ell )\in F_{\ell ,r}(\Omega )$. For $\varepsilon >0$, set $u_i(x):=\omega _{\varepsilon ,r,i}(x-\xi _i)$. Then, since the components have disjoint supports

$$\begin{aligned} \mathcal {J}_\varepsilon ({\textbf{i}}_\varepsilon (\xi _1,\ldots ,\xi _\ell ))&= \frac{1}{2\varepsilon ^N}\sum _{i=1}^\ell \int _{\Omega }\left( \varepsilon ^2|\nabla u_i|^2 + u_i^2\right) - \frac{1}{2p\varepsilon ^N}\sum _{i,j=1}^\ell \int _{\Omega }\beta _{ij}|u_j|^p|u_i|^p\\&= \frac{1}{2\varepsilon ^N}\sum _{i=1}^\ell \int _{\Omega }\left( \varepsilon ^2|\nabla u_i|^2 + u_i^2\right) - \frac{1}{2p\varepsilon ^N}\beta _{ii}|u_i|^{2p}\\&=\sum _{i=1}^\ell J_{\varepsilon ,r,i}(\omega _{\varepsilon ,r,i}) = \sum _{i=1}^\ell \mathfrak {c}_{\varepsilon ,r,i} \rightarrow \sum _{i=1}^\ell \mathfrak {c}_{\infty ,i} \end{aligned}$$

as $\varepsilon \rightarrow 0$ by Lemma 4.1, and the claim follows. $\square$

Let $b:L^2(\mathbb {R}^N)\smallsetminus \{0\}\rightarrow \mathbb {R}^N$ be a generalized barycenter map as constructed in [2, Sec. 2]. It is shown in [3, (2.3)] that b can be taken to be equivariant with respect to scaling. In other words, b has the following properties: For every $u \in L^2(\mathbb {R}^N)\smallsetminus \{0\}$

$(B_1)$:

$b(u)=b(|u|)$,

$(B_2)$:

If $\xi \in \mathbb {R}^N$ and $u_{\xi }(x):=u(x-\xi )$, then $b(u_{\xi })=b(u)+\xi$,

$(B_3)$:

If u is radially symmetric with respect to $\xi \in \mathbb {R}^N$, then $b(u)=\xi$.

$(B_4)$:

$\varepsilon ^{-1}b(u)=b(u(\varepsilon \, \cdot \,))$ for every $\varepsilon >0$.

Let ${\textbf{b}}:(L^2(\mathbb {R}^N)\smallsetminus \{0\})^\ell \rightarrow (\mathbb {R}^N)^\ell$ be given by

$$\begin{aligned} {\textbf{b}}({{\textbf{u}}}):=(b(u_1),\ldots ,b(u_\ell )),\quad \text {where }{\textbf{u}}=(u_1,\ldots ,u_\ell )\in (L^2(\mathbb {R}^N)\smallsetminus \{0\})^\ell . \end{aligned}$$

For $r>0$ set $\Omega _r^+:=\{x\in \mathbb {R}^N:\textrm{dist}(x,\Omega )\le r\}$, and let

$$\begin{aligned} F_\ell (\Omega _r^+):=\{(\xi _1,\ldots ,\xi _\ell )\in (\Omega _r^+)^\ell :\xi _i\ne \xi _j\text { if }i\ne j\} \end{aligned}$$

be the ordered configuration space of $\ell$ points in $\Omega _r^+$.

Proposition 4.3

Given $r>0$, there exist $d_r>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$ and $\widehat{\varepsilon }_r>0$, such that $d_r>c_\varepsilon (\Omega )$ and

$$\begin{aligned} {\textbf{b}}(\textbf{u})\in F_\ell (\Omega _r^+)\qquad \text {for every }\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega )^{\le d_r}\text { with }\varepsilon \in (0,\widehat{\varepsilon }_r). \end{aligned}$$

Proof

We argue by contradiction. Fix $r>0$ and assume there are $d_k>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$, $\varepsilon _k>0$ and $\textbf{u}_k=(u_{1 k}, \ldots , u_{\ell k})\in \mathcal {N}_{\varepsilon _k}(\Omega )^{\le d_r}$, such that $d_k\rightarrow \sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$, and

$$\begin{aligned} {\textbf{b}}(\textbf{u}_k)\not \in F_\ell (\Omega _r^+). \end{aligned}$$

Then, passing to a subsequence, there are two possibilities:

(I):

either there exist $i\ne j$, such that $b(u_{ik})=b(u_{jk})$ for every k,

(II):

or there exists i, such that $\textrm{dist}(b(u_{ik}),\Omega )\ge r$ for all k.

Next, we prove that this leads to a contradiction.

Let ${\textbf{w}}_k=(w_{1 k}, \ldots , w_{\ell k})$ be given by $w_{ik}(z):=u_{ik}(\varepsilon _kz)$ if $z\in \Omega _k:=\{\varepsilon _kx:x\in \Omega \}$ and $w_{ik}(z):=0$ if $z\in \mathbb {R}^N\smallsetminus \Omega _k$. Then, ${\textbf{w}}_k\in \mathcal {N}_1(\mathbb {R}^N)$ and $\mathcal {J}_1({\textbf{w}}_k)=\mathcal {J}_{\varepsilon _k}(\textbf{u}_k)\rightarrow \sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}=c_1(\mathbb {R}^N)$. So for each $i=1, \ldots , \ell$, there exist a sequence $\left( \zeta _{i k}\right)$ in $\mathbb {R}^N$ and a least energy radial solution $\omega _i$ to the problem (3.1) satisfying statements (i), (ii), and (iii) of Theorem 3.2. Statement (iii) yields

$$\begin{aligned} w_{ik}(\,\cdot \, +\zeta _{ik})\rightarrow \omega _i\quad \text {in }H^1(\mathbb {R}^N). \end{aligned}$$

Set $\xi _{ik}:=\varepsilon _k\zeta _{ik}$. From properties $(B_2)$, $(B_3)$ and $(B_4)$ and the continuity of the barycenter map, we derive

$$\begin{aligned} \varepsilon _k^{-1}\left( b(u_{ik})-\xi _{ik}\right) =b(w_{ik})-\zeta _{ik}=b(w_{ik}(\,\cdot \, +\zeta _{ik}))\rightarrow b(\omega _i)=0. \end{aligned}$$

(4.3)

This implies that (I) cannot hold true; otherwise

$$\begin{aligned} |\zeta _{ik}-\zeta _{jk}|=\varepsilon _k^{-1}|\xi _{ik}-\xi _{jk}|\le \varepsilon _k^{-1}|\xi _{ik}-b(u_{ik})| + \varepsilon _k^{-1}|b(u_{jk})-\xi _{jk}|\rightarrow 0, \end{aligned}$$

contradicting statement (ii) of Theorem 3.2. Furthermore, statement (i) implies that $\textrm{dist}(\zeta _{ik},\Omega _k)\le 1$. Then, (4.3) yields that $\textrm{dist}(b(w_{ik}),\Omega _k)\le 2$ for large enough k and, using property $(B_4)$, we get that

$$\begin{aligned} \textrm{dist}(b(u_{ik}),\Omega )=\varepsilon _k\textrm{dist}(b(w_{ik}),\Omega _k)\le 2\varepsilon _k\rightarrow 0. \end{aligned}$$

This contradicts (II) and completes the proof. $\square$

Proof of Theorem 1.2

Let $\varepsilon _0>0$ and $d>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$ be as in Proposition 3.3 and, for the given r, let $d_r>\sum _{i=1}^\ell \mathfrak {c}_{\infty ,i}$ and $\widehat{\varepsilon }_r>0$ be as in Proposition 4.3. Set $\overline{d}:=\min \{d,d_r\}$ and let $\widetilde{\varepsilon }_{r,\overline{d}}$ be as in Lemma 4.2. Define $\varepsilon _r:=\min \{\varepsilon _0,\widetilde{\varepsilon }_{r,\overline{d}},\widehat{\varepsilon }_r\}$. Then, for every $\varepsilon \in (0,\varepsilon _r)$, the maps

$$\begin{aligned} F_{\ell ,r}(\Omega )\xrightarrow {{\textbf{i}}_\varepsilon } \mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}\xrightarrow {\textbf{b}}F_\ell (\Omega _r^+) \end{aligned}$$

are well defined and, by property $(B_3)$ of the barycenter map, their composition is the inclusion $F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)$.

Let $\mathcal {Z}:=(\mathbb {Z}_2)^\ell$ act on $H^1_0(\Omega )^\ell$ as stated in (2.2). Property $(B_1)$ yields ${\textbf{b}}(\textbf{s}\textbf{u})={\textbf{b}}(\textbf{u})$ for every $\textbf{s}\in \mathcal {Z}$, so ${\textbf{b}}$ can be written as ${\textbf{b}}=\widetilde{\textbf{b}}\circ q$, where $q:\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}\rightarrow \mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}/\mathcal {Z}$ is the map that associates to each $\textbf{u}\in \mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}$ its $\mathcal {Z}$-orbit $\mathcal {Z}\textbf{u}$, and $\widetilde{\textbf{b}}(\mathcal {Z}\textbf{u}):={\textbf{b}}(\textbf{u})$. Therefore, the composition

$$\begin{aligned} F_{\ell ,r}(\Omega )\xrightarrow {\textbf{i}_\varepsilon }\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}\xrightarrow {q}\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}/\mathcal {Z}\xrightarrow {\widetilde{\textbf{b}}}F_\ell (\Omega _r^+) \end{aligned}$$

equals $F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)$. From Theorem 2.1, Proposition 3.3 and property $(C_1)$ in Sect. 5, we derive that $\mathcal {J}_\varepsilon$ has at least

$$\begin{aligned} \textrm{cat}(\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}/\mathcal {Z})\ge \textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)) \end{aligned}$$

critical $\mathcal {Z}$-orbits in $\mathcal {N}_{\varepsilon }(\Omega )^{\le \overline{d}}$ whose components do not change sign for any $\varepsilon \in (0,\varepsilon _r)$.

Observe that, if the components of $\textbf{u}=(u_1,\ldots ,u_\ell )$ do not change sign, then there exists a unique $\textbf{s}\in \mathcal {Z}$, such that $\textbf{s}\textbf{u}$ is nonnegative. This completes the proof. $\square$

5 The category of configuration spaces

Let X and Y be topological spaces and $f:X\rightarrow Y$ be continuous. The category of f, denoted $\textrm{cat}(f)$, is the smallest number of open subsets of X that cover X and have the property that the image under f of each of them is contractible in Y. The category of X is the category of the identity map $\textrm{id}_X:X\rightarrow X$. If the map is an inclusion, then $\textrm{cat}(A\hookrightarrow X)$ is the category of A in X, as defined in the introduction.

The following properties are easily proved, cf. [7, Subsection 1.3].

$(C_1)$:

For any two continuous functions $f:X\rightarrow Y$ and $g:Y\rightarrow Z$

$$\begin{aligned} \textrm{cat}(g\circ f)\le \min \{\textrm{cat}(f),\textrm{cat}(g)\}. \end{aligned}$$

In particular, $\textrm{cat}(f)\le \min \{\textrm{cat}(X),\textrm{cat}(Y)\}$.

$(C_2)$:

If $f,g:X\rightarrow Y$ are homotopic, then $\textrm{cat}(f)=\textrm{cat}(g)$.

Observe that $(C_1)$ and $(C_2)$ imply

$(C_3)$:

If $h:X\rightarrow Y$ is a homotopy equivalence, then $\textrm{cat}(X)=\textrm{cat}(h)=\textrm{cat}(Y)$.

$(C_4)$:

If $h:X\rightarrow Y$ is a homotopy equivalence, $g:Y\rightarrow Z$ and $f: X\rightarrow Z$ is homotopic to $g\circ h$, then $\textrm{cat}(f)=\textrm{cat}(g)$.

Indeed, if $\widetilde{h}:Y\rightarrow X$ is a homotopy inverse of h (i.e., $\widetilde{h}\circ h$ is homotopic to $\textrm{id}_X$ and $h\circ \widetilde{h}$ is homotopic to $\textrm{id}_Y$), then $\textrm{cat}(X)=\textrm{cat}(\widetilde{h}\circ h)\le \textrm{cat}(h)\le \textrm{cat}(Y)$ and $\textrm{cat}(Y)=\textrm{cat}(h\circ \widetilde{h})\le \textrm{cat}(h)\le \textrm{cat}(X)$. This proves $(C_3)$. To prove $(C_4)$, note that $\textrm{cat}(f)=\textrm{cat}(g\circ h)\le \textrm{cat}(g)$ and, since $f\circ \widetilde{h}$ is homotopic to g, we also have $\textrm{cat}(g)=\textrm{cat}(f\circ \widetilde{h})\le \textrm{cat}(f)$.

Let $\Omega$ be a bounded smooth domain in $\mathbb {R}^N$, $N\ge 2$, $r\ge 0$ and $\ell \ge 2$. Recall that

$$\begin{aligned} F_{\ell ,r}(\Omega ):=\{(\xi _1,\ldots ,\xi _\ell )\in \Omega ^\ell :\textrm{dist}(\xi _i,\mathbb {R}^N\smallsetminus \Omega )>r\text { and }|\xi _i-\xi _j|> 2r\text { if }i\ne j\}, \end{aligned}$$

$F_\ell (\Omega ):=F_{\ell ,0}(\Omega )$, and $\Omega _r^+:=\{x\in \mathbb {R}^N:\textrm{dist}(x,\Omega )<r\}$. We set $B_r(0):=\{\xi \in \mathbb {R}^N:|\xi |<r\}$ and write $\overline{B}_r(0)$ for its closure. The following results will be used in the proof of Theorem 1.3.

Lemma 5.1

• (i) $\textrm{cat}(F_\ell (B_r(0))\hookrightarrow F_\ell (\mathbb {R}^N))=\textrm{cat}(F_\ell (B_r(0)))=\textrm{cat}(F_\ell (\mathbb {R}^N))=\ell$ for every $r>0$.

• (ii) If $\Omega$ is convex, then $\textrm{cat}(F_\ell (\Omega ))=\ell$.

Proof

• (i) Fix a (strictly increasing) homeomorphism $h:[0,r)\rightarrow [0,\infty )$. Then, the map $\widehat{h}:F_\ell (B_r(0))\rightarrow F_\ell (\mathbb {R}^N)$ given by

  $$\begin{aligned} \widehat{h}(\xi _1,\ldots ,\xi _\ell ):=(h(|\xi _1|)\xi _1,\ldots ,h(|\xi _\ell |)\xi _\ell ) \end{aligned}$$

  is a homeomorphism. The map $F_\ell (B_r(0))\times [0,1]\rightarrow F_\ell (\mathbb {R}^N)$ defined by

  $$\begin{aligned} (\xi _1,\ldots ,\xi _\ell ,t)\mapsto ((1-t)\xi _1+th(|\xi _1|)\xi _1,\ldots ,(1-t)\xi _\ell +th(|\xi _\ell |)\xi _\ell ) \end{aligned}$$

  is a homotopy between the inclusion $F_\ell (B_r(0))\hookrightarrow F_\ell (\mathbb {R}^N)$ and $\widehat{h}$. Therefore, from properties $(C_2)$ and $(C_3)$, we get

  $$\begin{aligned} \textrm{cat}(F_\ell (B_r(0))\hookrightarrow F_\ell (\mathbb {R}^N))=\textrm{cat}(\widehat{h})=\textrm{cat}(F_\ell (B_r(0)))=\textrm{cat}(F_\ell (\mathbb {R}^N)). \end{aligned}$$

  It is shown in [21, Theorem 1.2] that $\textrm{cat}(F_\ell (\mathbb {R}^N))=\ell$. (Note that the definition of category used in [21] equals our $\textrm{cat}(X)-1$.)

• (ii) As every nonempty open convex subset of $\mathbb {R}^N$ is homeomorphic to $\mathbb {R}^N$ [16, Chapter 3, Exercise J], this statement follows from [21, Theorem 1.2]. $\square$

Next, we define

$$\begin{aligned} E_{\ell ,r}(\Omega ):= & {} \{(\xi _1,\ldots ,\xi _\ell )\in \Omega ^\ell :\textrm{dist}(\xi _i,\mathbb {R}^N\smallsetminus \Omega )>2^{\ell -i+1}r\text { and } \\{} & {} \quad |\xi _i-\xi _j|> 2^{\ell -i+1}r\text { for all }j<i\}. \end{aligned}$$

Note that $E_{\ell ,0}(\Omega )=F_{\ell }(\Omega )$.

Proposition 5.2

If $\Omega$ is convex, then the inclusion $E_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega )$ is a homotopy equivalence for small enough r.

For the proof of this proposition, we need the following lemma which, for $r=0$, is a special case of [13, Theorem 3]. The definition of fiber bundle may be found, for instance, in [23, Chapter 2, Section 7].

Lemma 5.3

Let $\ell \ge 3$. Fix $\ell -1$ different points $q_1,\ldots ,q_{\ell -1}\in \Omega$. Then, for any $r\ge 0$ sufficiently small, the map

$$\begin{aligned} \psi :E_{\ell ,r}(\Omega )\rightarrow E_{\ell -1,2r}(\Omega ),\qquad \psi (\xi _1,\ldots ,\xi _\ell ):=(\xi _1,\ldots ,\xi _{\ell -1}), \end{aligned}$$

is a fiber bundle with fiber $\Omega \smallsetminus \{q_1,\ldots ,q_{\ell -1}\}$.

Proof

For $r\ge 0$ sufficiently small, we have that $E_{\ell -1,2r}(\Omega )\ne \emptyset$ and $\Omega _{2r}^-\smallsetminus \{q_1,\ldots ,q_{\ell -1}\}$ is homeomorphic to $\Omega \smallsetminus \{q_1,\ldots ,q_{\ell -1}\}$, where $\Omega _{2r}^-:=\{x\in \Omega :\textrm{dist}(x,\mathbb {R}^N\smallsetminus \Omega )>2r\}$. Take any such r, and let $(\xi _1,\ldots ,\xi _{\ell -1})\in E_{\ell -1,2r}(\Omega )$. Fix $\delta >0$ such that $2^{\ell -i+1}r+4\delta<\min _{j<i}|\xi _i-\xi _j|$ and $\textrm{dist}(\xi _i,\mathbb {R}^N\smallsetminus \Omega )>2^{\ell -i+1}r+2\delta$. Then, $U:=B_{\delta }(\xi _1)\times \cdots \times B_{\delta }(\xi _{\ell -1})\subset E_{\ell -1,2r}(\Omega )$ and $|\xi _i-\xi _j|>4r+4\delta$ if $i\ne j$ and $i,j\in \{1,\ldots ,\ell -1\}$. For each j, we fix a continuous function

$$\begin{aligned} h_j:U\times \overline{B}_{2r+2\delta }(\xi _j)\rightarrow \overline{B}_{2r+2\delta }(\xi _j), \end{aligned}$$

such that, for each $\overline{\zeta }=(\zeta _1,\ldots ,\zeta _{\ell -1})\in U$, the function $h_{j,\overline{\zeta }}(x):=h_j(\overline{\zeta },x)$ satisfies

• $h_{j,\overline{\zeta }}(x)=x$ if $x\in \partial \overline{B}_{2r+2\delta }(\xi _j)$,

• $h_{j,\overline{\zeta }}(x)=x-\zeta _j+\xi _j$ if $x\in \overline{B}_{2r}(\xi _j)$,

• $h_{j,\overline{\zeta }}(x):\overline{B}_{2r+2\delta }(\xi _j)\rightarrow \overline{B}_{2r+2\delta }(\xi _j)$ is a homeomorphism.

Note that $h_{j,\overline{\zeta }}$ maps $\overline{B}_{2r}(\zeta _j)$ onto $\overline{B}_{2r}(\xi _j)$. Now, define $h_{\overline{\zeta }}:\Omega \rightarrow \Omega$ by

$$\begin{aligned} h_{\overline{\zeta }}(x):= {\left\{ \begin{array}{ll} h_{j,\overline{\zeta }}(x) &{}\quad \text {if }x\in \overline{B}_{2r+2\delta }(\xi _j), \\ x &{}\quad \text {if }x\in \Omega \smallsetminus \bigcup \limits _{j=1}^{\ell -1}B_{2r+2\delta }(\xi _j). \end{array}\right. } \end{aligned}$$

Since $\overline{B}_{2r+2\delta }(\xi _j)\subset \Omega _{2r}^-$, we have that $h_{\overline{\zeta }}$ maps $\Omega _{2r}^-\smallsetminus \bigcup \nolimits _{j=1}^{\ell -1}\overline{B}_{2r}(\zeta _j)$ onto $\Omega _{2r}^-\smallsetminus \bigcup \nolimits _{j=1}^{\ell -1}\overline{B}_{2r}(\xi _j)$. Note that

$$\begin{aligned} \psi ^{-1}(U)=\Big \{(\zeta _1,\ldots ,\zeta _{\ell }):(\zeta _1,\ldots ,\zeta _{\ell -1})\in U \text { and } \zeta _\ell \in \Omega _{2r}^-\smallsetminus \bigcup \limits _{j=1}^{\ell -1}\overline{B}_{2r}(\zeta _j)\Big \}. \end{aligned}$$

Now, we fix homeomorphisms

$$\begin{aligned}{} & {} f:\,\Omega _{2r}^-\smallsetminus \bigcup \limits _{j=1}^{\ell -1}\overline{B}_{2r}(\xi _j)\cong \Omega _{2r}^-\smallsetminus \{\xi _1,\ldots ,\xi _{\ell -1}\}\cong \Omega _{2r}^-\smallsetminus \{q_1,\ldots ,q_{\ell -1}\}\\{} & {} \quad \cong \Omega \smallsetminus \{q_1,\ldots ,q_{\ell -1}\}. \end{aligned}$$

For the second one, see [19, Chapter 4, Homogeneity Lemma]. As indicated above, we denote the composition of these homeomorphisms by f. The map $\tau :\psi ^{-1}(U)\rightarrow U\times \left( \Omega \smallsetminus \{q_1,\ldots ,q_{\ell -1}\}\right)$ given by

$$\begin{aligned} \tau (\zeta _1,\ldots ,\zeta _{\ell }):=(\zeta _1,\ldots ,\zeta _{\ell -1},f(h_{(\zeta _1,\ldots ,\zeta _{\ell -1})}(\zeta _\ell ))) \end{aligned}$$

is a homeomorphism and satisfies $\textrm{proj}_U\circ \tau =\psi |_U$. This proves that $\psi$ is a fiber bundle. $\square$

Proof of Proposition 5.2

Without loss of generality, we assume that $0\in \Omega$. We prove the statement by induction on $\ell$.

Let $\ell =2$. We fix $s>0$, such that $\overline{B}_{s}(0)\subset \Omega _{4s}^-:=\{x\in \Omega :\textrm{dist}(x,\mathbb {R}^N\smallsetminus \Omega )>4s\}$. For each $r\in [0,s)$, we define maps

$$\begin{aligned}&\iota _r:\mathbb {S}^{N-1}\rightarrow E_{2,r}(\Omega ),\qquad \iota _r(x):=(sx,-sx), \\&\varrho _r:E_{2,r}\rightarrow \mathbb {S}^{N-1},\qquad \varrho _r(\xi _1,\xi _2)=\frac{\xi _1-\xi _2}{|\xi _1-\xi _2|}, \end{aligned}$$

where $\mathbb {S}^{N-1}$ is the unit sphere in $\mathbb {R}^N$. Then, $\varrho _r\circ \iota _r=\textrm{id}_{\mathbb {S}^{N-1}}$. Furthermore, since $\Omega$ is convex and $s>r$, the map $E_{2,r}(\Omega )\times [0,1]\rightarrow E_{2,r}(\Omega )$ given by

$$\begin{aligned} (\xi _1,\xi _2,t)\longmapsto \left( (1-t)\xi _1+ts\frac{\xi _1-\xi _2}{|\xi _1-\xi _2|},\,(1-t)\xi _2+ts\frac{\xi _2-\xi _1}{|\xi _1-\xi _2|}\right) \end{aligned}$$

is well defined and it is a homotopy between $\textrm{id}_{E_{2,r}(\Omega )}$ and $\iota _r\circ \varrho _r$. Since the composition

$$\begin{aligned} \mathbb {S}^{N-1}\xrightarrow {\iota _r} E_{2,r}(\Omega )\hookrightarrow F_2(\Omega )\xrightarrow {\varrho _0}\mathbb {S}^{N-1} \end{aligned}$$

is the identity map on $\mathbb {S}^{N-1}$ and $\iota _r$ and $\varrho _0$ are homotopy equivalences, we have that $E_{2,r}(\Omega )\hookrightarrow F_2(\Omega )$ is a homotopy equivalence.

Assume the statement holds true for $\ell -1$ with $\ell \ge 3$. For every $r\ge 0$ small enough, as in Lemma 5.3, the map $\psi :E_{\ell ,r}(\Omega )\rightarrow E_{\ell -1,2r}(\Omega )$ is a fiber bundle, and hence, it is a fibration [23, Corollary 2.7.14] with fiber $\Omega \smallsetminus Q_\ell$ where $Q_\ell :=\{q_1,\ldots ,q_{\ell -1}\}$. Note that $E_{\ell ,0}(\Omega )=F_{\ell }(\Omega )$. Note also that $E_{\ell ,r}(\Omega )$ and $F_{\ell }(\Omega )$ are path connected for every $\ell \ge 2$. This follows by induction using the fiber bundle structure, because they are homotopic to $\mathbb {S}^{N-1}$ if $\ell =2$ and $N\ge 2$. From the long exact homotopy sequence of a fibration [23, Theorem 7.2.10], we get a commutative diagram

$$\begin{aligned} \begin{matrix} \pi _{i+1}(F_{\ell -1}(\Omega )) &{} \rightarrow &{} \pi _i(\Omega \smallsetminus Q_\ell ) &{} \rightarrow &{} \pi _i(F_\ell (\Omega )) &{} \xrightarrow {\psi _*} &{} \pi _i(F_{\ell -1}(\Omega )) &{} \rightarrow &{} \pi _{i-1}(\Omega \smallsetminus Q_\ell )\\ \uparrow &{} &{} \uparrow &{} &{} \uparrow &{} &{} \uparrow &{} &{} \uparrow \\ \pi _{i+1}(E_{\ell -1,2r}(\Omega )) &{} \rightarrow &{} \pi _i(\Omega \smallsetminus Q_\ell ) &{} \rightarrow &{} \pi _i(E_{\ell ,r}(\Omega )) &{} \xrightarrow {\psi _*} &{} \pi _i(E_{\ell -1,2r}(\Omega )) &{} \rightarrow &{} \pi _{i-1}(\Omega \smallsetminus Q_\ell ) \end{matrix}, \end{aligned}$$

where all vertical arrows are induced by inclusions. The fact that the spaces above are path connected is needed to assert that the homotopy groups are independent of the choice of a base point. Using the induction hypothesis, we see that the two vertical leftmost arrows and the two vertical rightmost arrows in the diagram are isomorphisms for r sufficiently small. Therefore, by the five lemma (see, e.g., [23, Lemma 4.5.11]), the middle vertical arrow is an isomorphism too. This proves that $E_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega )$ is a weak homotopy equivalence. Since $E_{\ell ,r}(\Omega )$ is an open subset of $\mathbb {R}^{\ell N}$, it has the homotopy type of a CW-complex [18, Theorem 1 and Corollary 1]. Therefore, by [23, Corollary 7.6.24], $E_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega )$ is a homotopy equivalence. $\square$

Remark 5.4

Note that $\pi _1(\Omega \smallsetminus Q_\ell )$ may not be commutative (and it never is if $N=2$ and $\ell \ge 3$). The proof of the five lemma in [23] is for commutative groups. However, this lemma is known to hold also in the noncommutative case and it is easy to see that the proof in [23] still applies, with obvious changes.

Proof of Theorem 1.3

Without loss of generality, we may assume that $0\in \Omega$. Fix $s>0$, such that $B_s(0)\subset \Omega$. Applying $(C_1)$ to the composition of inclusions $E_{\ell ,r}(B_s(0))\hookrightarrow F_{\ell ,r}(B_s(0))\hookrightarrow F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)\hookrightarrow F_\ell (\mathbb {R}^N)$, we get that

$$\begin{aligned} \textrm{cat}(E_{\ell ,r}(B_s(0))\hookrightarrow F_\ell (\mathbb {R}^N))\le \textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+)). \end{aligned}$$

By Proposition 5.2, we have that $E_{\ell ,r}(B_s(0))\hookrightarrow F_{\ell }(B_s(0))$ is a homotopy equivalence for sufficiently small $r>0$. Therefore, from Lemma 5.1 and $(C_4)$, we get that

$$\begin{aligned} \ell =\textrm{cat}(F_{\ell }(B_s(0))\hookrightarrow F_\ell (\mathbb {R}^N))=\textrm{cat}(E_{\ell ,r}(B_s(0))\hookrightarrow F_\ell (\mathbb {R}^N)), \end{aligned}$$

and we derive

$$\begin{aligned} \textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+))\ge \ell . \end{aligned}$$

If $\Omega$ is convex, Proposition 5.2 states that $E_{\ell ,r}(\Omega )\hookrightarrow F_{\ell }(\Omega )$ is a homotopy equivalence for sufficiently small $r>0$. Note that $F_{\ell ,2^{\ell }r}(\Omega )\subset E_{\ell ,r}(\Omega )$. Therefore, from $(C_1)$, $(C_2)$ and Lemma 5.1, we obtain that

$$\begin{aligned} \textrm{cat}(F_{\ell ,2^{\ell }r}(\Omega )\hookrightarrow F_\ell (\Omega _{2^{\ell }r}^+))\le \textrm{cat}(E_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega ))=\textrm{cat}(F_\ell (\Omega ))=\ell . \end{aligned}$$

This shows that $\textrm{cat}(F_{\ell ,r}(\Omega )\hookrightarrow F_\ell (\Omega _r^+))=\ell$ for sufficiently small r if $\Omega$ is convex. $\square$