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

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 positive solutions increases with the number of equations. Our proofs use a combination of four key elements: a Nehari manifold method, the asymptotic behavior of solutions (concentration), the Lusternik Schnirelman theory, and new estimates on the category of suitable configuration spaces.

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 ℓ = 2) and in nonlinear optics.It also serves as a model in population dynamics.In this paper we assume β ii > 0 and β ij < 0 for i = 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.
Most of the research for (1.1) is focused on the case ℓ = p = 2 and N = 1, 2, 3.In this setting, the asymptotic behavior (as ε → 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 ε → 0, the i-th component is close to a rescaling and translation of the positive radially symmetric ground state solution of The concentration points approach a configuration that maximizes the distance between them and to ∂Ω.Furthermore, the existence of two nonnegative solutions for ℓ = 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, Uniqueness or multiplicity results for (1.2) rely on both the geometry and the topology of the domain Ω (see the introduction of [12] for an updated survey in this regard).In particular, if Ω 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 ℓ becomes larger.We give a positive answer to this question.Our main result is the following one.
Note that this result concerns any dimension N ≥ 2. If p < 2, which is necessarily the case for N ≥ 4, then neither the functional corresponding to (1.1) nor the Nehari-type manifold (defined later) is of class 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 cat(A ֒→ X) (or cat X (A)), is the 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 cat(X) instead of cat(X ֒→ X).
Let Θ be a subset of R N .For r ≥ 0 and ℓ ≥ 2, we consider the space endowed with the subspace topology of Θ ℓ .If r = 0 we write F ℓ (Θ) := F ℓ,0 (Θ).This last space is called the ordered configuration space of ℓ points in Θ.Finally, we set The following theorem gives a lower bound on the number of nonnegative solutions of (1.1).
The next result gives an estimate for (1.3).
Theorem 1.3 is proved by induction on ℓ using suitably constructed fibrations which allow to apply some basic tools from algebraic topology in order to obtain the estimate cat(F ℓ,r (Ω) Therefore, methodologically, our main contribution is to show how the category of F ℓ,r (Ω) ֒→ F ℓ (Ω + 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 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 Section 2 we present the variational framework and give a lower bound for the number of solutions to (1.1) in terms of cat N ε (Ω) ≤d /Z .Section 3 is devoted to showing that low-energy solutions do not change sign.The maps (1.4) are constructed in Section 4, where we also give a proof of Theorem 1.2.Finally, in Section 5, we show the estimate in Theorem 1.3.

The variational problem
For ε > 0 fixed, let be the norm of v in H 1 0 (Ω) (equivalent to the usual one).We write the elements in H 1 0 (Ω) ℓ as u = (u 1 , . . ., u ℓ ) and we introduce the norm . The solutions to the system (1.1) are the critical points of the functional J ε : which is of class C 1 .Its partial derivatives at u are given by for v ∈ H 1 0 (Ω) and i = 1, . . ., ℓ.The fully nontrivial solutions of (1.1) belong to the Nehari-type set The Z-orbit of u is the set The Z-orbit space of A is the set A/Z := {Zu : u ∈ A} endowed with the quotient space topology.Note that N ε (Ω) and J ε are Z-invariant, and so is the sublevel set If u is a critical point of J ε , then so is su for every s ∈ Z, and Zu is called a critical Z-orbit of J ε .
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 ε > 0. Let S := {v ∈ H 1 0 (Ω) : For u ∈ U ε there exists a unique This and more can be seen from [10, Proposition 3.1].Now, let fully nontrivial critical point of J ε .Now a standard argument applies.One can construct a pseudogradient vector field for Ψ ε on U ε 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 U ε .Since it is easy to see using the compactness of the embedding H 1 0 (Ω) ֒→ L 2p (Ω) that J ε satisfies the Palais-Smale condition on N ε (Ω) and is bounded from below there.Using [10, Theorem 3.3] again, it follows that the same is true of Ψ ε on U ε .
Note that U ε and Ψ ε are Z-invariant and the action of Z on T is free, that is, the Z-orbit of every point where G := {Z}.Since Z acts freely on U ≤d ε one easily verifies that and, as m ε induces a homeomorphism of the Z-orbit spaces, one has that cat(U ≤d ε /Z) = cat(N ≤d ε /Z).It follows that J ε has at least cat N ε (Ω) ≤d /Z critical Z-orbits in N ε (Ω) ≤d , as claimed.

The sign of low energy solutions
For each i = 1, . . ., ℓ, consider the problem (3.1) Its solutions are the critical points of the functional J ∞,i : A solution ω i to this problem that satisfies J ∞,i (ω i ) = c ∞,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 R N with ε = 1 is 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 Fix(G) := {x ∈ R N : gx = x for all g ∈ G} has positive dimension (in fact, Fix(G) = R N ).This is needed in order to assure that c 1 (R N ) equals the sum above.
The following result describes the behavior of minimizing sequences for the system (1.1) in R N with ε = 1.We write B 1 (x) for the unit ball in R N centered at x. Theorem 3.2.Let ℓ ≥ 2 and w k = (w 1k , . . ., w ℓk ) ∈ N 1 (R N ) be such that J 1 (w k ) → c 1 (R N ).Then, there exist a positive number θ and, for each i = 1, . . ., ℓ, a sequence (ζ ik ) in R N and a least energy radial solution ω i to the problem (3.1) such that, after passing to a subsequence, 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.Proposition 3.3.There exist ε 0 > 0 and d > c 1 (R N ) such that, if ε ∈ (0, ε 0 ) and u = (u 1 , . . ., u ℓ ) is a critical point of J ε with J ε (u) ≤ d, then u i does not change sign for any i = 1, . . ., ℓ.
Let w k = (w 1k , . . ., w ℓk ) be given by w ik (z where ω i is a least energy solution to the problem (3.1).In particular, and similarly for w − 1k (that the right-hand side above tends to 0 follows from (ii) and (iii) of Theorem 3.2).Set Here we have used (3.2) and the fact that (w ± 1k , w 2k , . . ., w ℓk ) ∈ N 1 (R N ) which implies in particular that w ± 1k are bounded away from 0 [10, Proposition 3.1(d)].Thus, by the definition of c ∞,1 , which is a contradiction.

The effect of the configuration space
Fix r ∈ (0, ∞) and let B r be the ball of radius r in R N centered at 0. For each i = 1, . . ., ℓ, consider the problem (4.1) Its solutions are the critical points of the functional J ε,r,i : A solution w ε,r,i satisfying J ε,r,i (w ε,r,i ) = c ε,r,i is called a least energy solution.
For r > 0 let endowed with the subspace topology of Ω ℓ .Note that F ℓ,r (Ω) = ∅ if r is sufficiently small.Define i ε : where ω ε,r,i is the positive least energy solution to the problem (4.1) extended by 0 to R N Ω.
Proof.We argue by contradiction.Fix r > 0 and assume there are Then, passing to a subsequence, there are two possibilities: (I) either there exist i = j such that b(u ik ) = b(u jk ) for every k, (II) or there exists i such that dist(b(u ik ), Ω) ≥ r for all k.
Next we prove that this leads to a contradiction.Let w k = (w 1k , . . ., w ℓk ) be given by w ik (z So, for each i = 1, . . ., ℓ, there exist a sequence (ζ ik ) in R N and a least energy radial solution ω i to the problem (3.1) satisfying statements (i), (ii) and (iii) of Theorem 3.2.Statement (iii) yields Set ξ ik := ε k ζ ik .From properties (B 2 ), (B 3 ) and (B 4 ) and the continuity of the barycenter map we derive This implies that (I) cannot hold true, otherwise Then, (4.3) yields that dist(b(w ik ), Ω k ) ≤ 2 for large enough k and, using property (B 4 ), we get that This contradicts (II) and completes the proof.
Observe that, if the components of u = (u 1 , . . ., u ℓ ) do not change sign, then there exists a unique s ∈ Z such that su is nonnegative.This completes the proof.

The category of configuration spaces
Let X and Y be topological spaces and f : X → Y be continuous.The category of f , denoted 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 id X : X → X.If the map is an inclusion, then cat(A ֒→ X) is the category of A in X, as defined in the introduction.
is a homotopy between the inclusion F ℓ (B r (0)) ֒→ F ℓ (R N ) and h.So from properties (C 2 ) and (C 3 ) we get 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].
Remark 5.4.Note that π 1 (Ω Q ℓ ) may not be commutative (and it never is if N = 2 and ℓ ≥ 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.