Intertwining semiclassical solutions to a Schr\"{o}dinger-Newton system

We study the problem (-\epsilon\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=\epsilon ^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}\epsilon\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magnetic potential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exterior electric potential, and $\epsilon$ is a small positive number. If A=0 and $\epsilon=\hbar$ is Planck's constant this problem is equivalent to the Schr\"odinger-Newton equations proposed by Penrose in \cite{pe2}\ to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that $A$ and $V$ are compatible with the action of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for any given homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential $V$ on the number of semiclassical solutions $u:\mathbb{R}% ^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=\tau(g)u(x)$ for all $g\in G$, $x\in\mathbb{R}^{3}$. We also study the concentration behavior of these solutions as $\epsilon\rightarrow0.\medskip$


Introduction
The Schrödinger-Newton equations were proposed by Penrose [23] to describe his view that quantum state reduction is a phenomenon that occurs because of some gravitational influence. They consist of a system of equations obtained by coupling together the linear Schrödinger equation of quantum mechanics with the Poisson equation from Newtonian mechanics. For a single particle of mass m this system has the form where ψ is the complex wave function, U is the gravitational potential energy, V is a given potential, is Planck's constant, and κ := Gm 2 , G being Newton's constant. According to Penrose, the solutions ψ of this system are the basic stationary states into which a superposition of such states is to decay within a certain timescale, cf. [22,23,18,19,24]. The second equation in (1.2) can be explicitly solved with respect toÛ , so this system is equivalent to the single nonlocal equation We shall consider a more general equation having a similar structure, namely where A : R 3 → R 3 is an exterior magnetic potential, i is the imaginary unit and * denotes the convolution operator. We are interested in semiclassical states, i.e. in solutions of this equation for ε → 0. The existence of one solution can be traced back to Lions' paper [15]. In the nonmagnetic case A = 0 equation (1.4) and related equations have been investigated by many authors, see e.g. [2,10,11,12,13,16,17,18,20,25,26,19] and the references therein. Recently, Wei and Winter [27] showed the existence of positive multibump solutions which concentrate at local minima, local maxima or nondegenerate critical points of the potential V as ε → 0. The magnetic case A = 0 was recently studied in [6] where it was shown that equation (1.4) has a family of solutions having multiple concentration regions located around the (possibly degenerate) minima of V .
In this paper we consider the situation where A and V are symmetric and we look for semiclassical solutions of equation (1.4) having specific symmetries. The absolute value of the solutions we obtain concentrates at points which need not be local extrema, nor nondegenerate critical points of V (in fact, we do not even assume that V is differentiable). We state our main results in the following section and give some explicit examples.

Statement of results
2.1. The results. Let G be a closed subgroup of the group O(3) of linear isometries of R 3 , A : R 3 → R 3 be a C 1 -function, and V : R 3 → R be a bounded continuous function with inf R 3 V > 0, which satisfy Given a continuous homomorphism of groups τ : G → S 1 into the group S 1 of unit complex numbers, we look for solutions to the problem This implies that the absolute value |u| of u is G-invariant, i.e.
whereas the phase of u(gx) is that of u(x) multiplied by τ (g). A concrete example is given in subsection 2.2 below. Note that if u satisfies (2.2) and (2.3) then e iθ u satisfies (2.2) and (2.3) for every θ ∈ R. We shall say that u and v are geometrically distinct if e iθ u = v for all θ ∈ R.
We introduce some notation. For x ∈ R 3 , we denote by Gx the G-orbit of x and by G x the G-isotropy subgroup of x, i.e.
of G-orbits of X with the quotient topology.
Let #Gx denote the cardinality of Gx, and define Observe that the points of M τ need not be neither local minima nor local maxima of V . Given ρ > 0 we set B ρ M τ := {x ∈ R 3 : dist(x, M τ ) ≤ ρ}, and write for the Lusternik-Schnirelmann category of M τ /G in B ρ M τ /G. Finally, we denote by E 1 the least energy of a nontrivial solution to problem We shall prove the following results.
Theorem 2.1. Assume there exists α > 0 such that the set is compact. Then, given ρ, δ > 0, there exists ε > 0 such that, for every ε ∈ (0, ε), problem (2.2) has at least cat BρMτ /G (M τ /G) geometrically distinct solutions u which satisfy (2.3) and The last inequality says that the energy of the solutions is arbitrarily close to ε 3 ℓ G,V E 1 for ε small enough. So considering different groups G and G ′ for which ℓ G,V = ℓ G ′ ,V will lead to solutions with energy in disjoint ranges.
For u ∈ H 1 (R 3 , R) set The following theorem describes the module of the solutions given by Theorem 2.1 as ε → 0.
Theorem 2.2. Let u n be a solution to problem (2.2) which satisfies (2.3) and (2.6) for ε = ε n > 0, δ = δ n > 0. Assume ε n → 0 and δ n → 0. Then, after passing to a subsequence, there exists a sequence (ξ n ) in R 3 such that ξ n → ξ ∈ M τ , G ξn = G ξ , and where ω ξ is the unique ground state of problem which is positive and radially symmetric with respect to the origin.
Next, we give an example which illustrates our results. u(e 2πi/m z, t) = e 2πij/m u(z, t) for all (z, t) ∈ C × R.
Solutions of this type arise in a natural way in some problems where the magnetic potential is singular and the topology of the domain produces an Aharonov-Bohm type effect, cf. [1,8]. Taking τ j (g) := g j we see that these are solutions of the type furnished by Theorem 2.1.
For each k dividing m the potentials A and V satify assumption (2.1) for G k and V satisfies (2.8) with k instead of m. Property (2.6) implies that the solutions obtained for G k are different from those for G m if k = m and ε is small enough. This paper is organized as follows. In section 3 we discuss the variational problem related to the existence of solutions to problem (2.2) satisfying (2.3). We also outline the strategy for proving Theorem 2.1. Sections 4 and 5 are devoted to the construction of an entrance map and a local baryorbit map which will help us estimate the Lusternik-Schnirelmann category of a suitable sublevel set of the variational functional for ε small enough. Finally, in section 6 we prove Theorems 2.1 and 2.2.

The variational problem
Set ∇ ε,A u := ε∇u + iAu and consider the real Hilbert space We write for the corresponding norm.
This is called the diamagnetic inequality [14]. Set for all f, h ∈ L 6/5 (R 3 ), where C is a positive constant independent of f and h. In particular, , is of class C 2 , and its derivative is given by Therefore, the solutions to problem (2.2) are the critical points of J ε,A,V . We write ∇ ε J ε,A,V (u) for the gradient of J ε,A,V at u with respect to the scalar product (3.1).
By the principle of symmetric criticality [21,28], the critical points of the restriction of J ε,A,V to the fixed point space of this G-action, denoted by are the solutions to problem (2.2) which satisfy (2.3). Those which are nontrivial lie on the Nehari manifold which is a C 2 -manifold radially diffeomorphic to the unit sphere in H 1 ε,A (R 3 , C) τ . The critical points of the restriction of J ε,A,V to N τ ε,A,V are precisely the nontrivial solutions to (2.2) which satisfy (2.3).
The radial projection π ε,A,V : Note that Recall that J ε,A,V : N τ ε,A,V → R is said to satisfy the Palais-Smale condition (P S) c at the level c if every sequence (u n ) such that The following holds.
Proof. This was proved in [5] for ε = 1. For ε > 0 the assertion follows after performing the change of variable u ε (x) := u(εx) since a straightforward computation shows that ε,A (R 3 , C) τ by scalar multiplication: (e iθ , u) → e iθ u. The Nehari manifold N τ ε,A,V and the functional J ε,A,V are invariant under this action. Two solutions of (2.2) are geometrically distinct iff they lie on different S 1 -orbits. Equivariant Lusternik-Schnirelmann theory yields the following result, see e.g. [7].
To obtain inequality (3.7) we shall construct maps The main ingredients for defining these maps are contained in the following two sections.

The entrance map
For any positive real number λ we consider the problem Its associated energy functional J λ : H 1 (R 3 , R) → R is given by Its Nehari manifold will be denoted by The critical points of J λ on M λ are the nontrivial solutions to (4.1). Note that u solves (2.4) if and only if u λ (x) := λu( √ λx) solves (4.1). Therefore, Minimizers of J λ on M λ are called ground states. Lieb established in [13] the existence and uniqueness of ground states up to sign and translations. Recently Ma and Zhao [17] showed that every positive solution to problem (4.1) is radially symmetric, and they concluded from this fact that the positive solution to this problem is unique up to translations. We denote by ω λ the positive solution to problem (4.1) which is radially symmetric with respect to the origin.
Fix a radial function ̺ ∈ C ∞ (R 3 , R) such that ̺(x) = 1 if |x| ≤ 1 2 and ̺(x) = 0 if |x| ≥ 1. For ε > 0 set ̺ ε (x) := ̺( √ εx), ω λ,ε := ̺ ε ω λ and Note that supp(υ λ,ε ) ⊂ B(0, 1/ √ ε) := {x ∈ R 3 : |x| ≤ 1/ √ ε} and υ λ,ε ∈ M λ . An easy computation shows that Observe that We assume from now on that there exists α > 0 such that the set is a compact G-invariant set and all G-orbits in M G,V are finite. We split M G,V according to the orbit type of its elements as follows: we choose subgroups G 1 , . . . , G m of G such that the isotropy subgroup G x of every point x ∈ M G,V is conjugate to precisely one of the G i 's, and we set M i := y ∈ M G,V : G y = gG i g −1 for some g ∈ G .
Since isotropy subgroups satisfy G gx = gG x g −1 , the sets M i are G-invariant and, since V is continuous, they are closed and pairwise disjoint, and We denote by V i the value of V on M i . Let υ i,ε := υ Vi,ε be defined as in (4.2) with λ : The proofs of the following two lemmas are similar to those of Lemmas 1 and 2 in [3], so we shall omit them.
It is well known that the map G/G ξ → Gξ given by gG ξ → gξ is a homeomorphism, see e.g. [9]. So, if G i ⊂ ker τ and ξ ∈ M i , then the map is well defined and continuous. Set Lemma 4.2. Assume that G i ⊂ ker τ . Then, the following hold: (a) For every ξ ∈ M i and ε > 0, one has that (b) For every ξ ∈ M i and ε > 0, one has that is well defined and continuous, and satisfies Moreover, given d > ℓ G E 1 , there exists ε d > 0 such that Proof. This follows immediately from Lemma 4.2.

A local baryorbit map
Let W : R 3 → R be a bounded, uniformly continuous function with inf R 3 W > 0 and such that W (gx) = W (x) for all g ∈ G, x ∈ R 3 . We assume that the set (5.1) y ∈ R 3 : (#Gy)W 3/2 (y) ≤ ℓ G,W + α is compact, where ℓ G,W := inf x∈R 3 (#Gx)W 3/2 (x), and consider the real-valued problem We write v, w ε,W := and set The nontrivial solutions of (5.2) are the critical points of the energy functional We wish to study the behavior of "minimizing sequences" for the family of problems (5.2), parametrized by ε, as ε → 0. This is described in Proposition 5.4 below. We start with some lemmas.
Lemma 5.2. Let ε n > 0 and ξ n ∈ R 3 such that ε n → 0 and (W (ξ n )) converges. Set W n (x) := W (ε n x + ξ n ) and W := lim n→∞ W (ξ n ). Then, for every sequence (u n ) in H 1 (R 3 , R) such that u n ⇀ u weakly in H 1 (R 3 , R) and every w ∈ H 1 (R 3 , R), the following hold: Proof. The argument is similar for both equalities. We prove the second one. Since .
Given ε > 0 we fix R > 0 such that Since W is uniformly continuous, there exists δ > 0 such that This concludes the proof.
Lemma 5.3. Let (z n ) be a sequence in R N . Then, after passing to a subsequence, there exist a closed subgroup Γ of G and a sequence (ζ n ) in R N such that (a) (dist(Gz n , ζ n )) is bounded, Proof. See Lemma 3.2 in [5].
Set M G,W := y ∈ R 3 : (#Gy)W 3/2 (y) = ℓ G,W Abusing notation we write again G i and M i for the groups and the sets defined as in Section 4 but now for W instead of V . So the value of W on M i is constant and we denote it by W i . We fix ρ > 0 such that For ρ ∈ (0, ρ), let M ρ i := {y ∈ R 3 : dist(y, M i ) ≤ ρ, G y = gG i g −1 for some g ∈ G}, and for each ξ ∈ M ρ i and ε > 0, define where ω i is unique positive ground state of problem (4.1) with λ := W i which is radially symmetric with respect to the origin. Set Θ ρ,ε := {θ ε,ξ : ξ ∈ M ρ 1 ∪ · · · ∪ M ρ m }. The following holds.
Proposition 5.4. Let ε n > 0 and v n ∈ H 1 (R 3 , R) G be such that ε,W and ∇ εn J εn,W is the gradient of J εn,W with respect to the scalar product ·, · εn,W . Then, passing to a subsequence, there exist an i ∈ {1, ..., m} and a sequence (ξ n ) in R 3 such that | v n | 2 .
Since c > 0, Lions' lemma [28,Lemma 1.21], together with inequality (3.4), yields that δ > 0. Choose z n ∈ R 3 such that and replace (z n ) by a sequence (ζ n ) having the properties stated in Lemma 5.3. Set v n (z) := v n (z + ζ n ). After passing to a subsequence, we may assume that v n ⇀ v weakly in H 1 (R 3 , R), v n (x) → v(x) a.e. on R 3 and v n → v in L 2 loc (R 3 , R). Choosing C ≥ dist(ζ n , Gz n ) for all n, we obtain Therefore, v = 0. Set ξ n := ε n ζ n and W n (x) := W (ε n x + ξ n ). Since W is bounded, a subsequence of W (ξ n ) converges. We set W := lim n→∞ W (ξ n ). The weak continuity of D ′ [2, Lemma 3.5], together with Lemma 5.2 and assumption (5.5) imply that v is a solution to problem (4.1) with λ := W .
Since v n and W are G-invariant we have that v n (g −1 x) = v n (ε n x+gξ n ), W n (g −1 x) = W (ε n x + gξ n ), and W := lim n→∞ W (gξ n ) for each g ∈ G. Fix g 1 , ..., g k ∈ G such that |g i ζ n − g j ζ n | → ∞ if i = j. Then, and performing the change of variable y = ε n x + g j ξ n we conclude that Iterating these equalities we conclude that This implies that 4 c ≥ k v This proves (iv) and gives also Moreover, (#Gξ n )W 3/2 (ξ n ) ≤ ℓ G,W +α for n large enough. Thus, assumption (2.5) implies, after passing to a subsequence, that ξ n → ξ. Hence, W (ξ) = W and We conclude that ξ ∈ M i for some i = 1, ..., m, as claimed in (ii). Then, W = W i , Γ = G ξ = gG i g −1 for some g ∈ G, and v is a ground state of problem (4.1) with λ = W i . Since the ground state is unique up to sign and translation we must have that v(z) = ±ω i (z − z 0 ) for some z 0 ∈ R 3 . Observe that v is Γ-invariant. So, if Γ is nontrivial, then z 0 = 0 and, since ω i is radial, equation (5.7) becomes (iii). If, on the other hand, Γ is the trivial group, we replace ξ n by ξ ′ n := ξ n + ε n z 0 . Since Gξ n ∼ = G and ε n → 0, ξ ′ n has the same properties as ξ n for n large enough. Moreover, since ω i is radially symmetric, and, again, equation (5.7) yields (iii). This completes the proof.
Proof. The proof is analogous to that of Proposition 5.3 in [4]. We omit the details.
Fix ρ ∈ (0, ρ) and ε ∈ (0, ε ρ ). Proposition 5.5 allows us to define a map Here, as usual, J c ε,W := {v ∈ H 1 (R 3 , R) : J ε,W (v) ≤ c}. The map β ρ,ε,0 is the Gequivariant analogon to the usual baricenter map. It is only defined for functions in M G ε,W with small enough energy. We call it the local baryorbit map. It reflects the fact that such functions concentrate at a unique G-orbit with minimal cardinality as ε → 0.
Observe that Let ι ε be the map defined in Proposition 4.3 and β ρ,ε,0 be as in (5.8). Then, for d ρ > ℓ G,V E 1 and ε ρ > 0 as in Proposition 5.5 the following holds.
Since v 2 ε ≤ C v 2 ε,W for some constant independent of ε, our claim follows.