SCHRÖDINGER-POISSON SYSTEMS IN THE 3-SPHERE

Transcription

1 SCHRÖDINGER-POISSON SYSTEMS IN THE 3-SPHERE EMMANUEL HEBEY AND JUNCHENG WEI Abstract. We investigate nonlinear Schrödinger-Poisson systems in the 3- sphere. We prove existence results for these systems and discuss the question of the stability of the systems with respect to their phases. While, in the subcritical case, we prove that all phases are stable, we prove in the critical case that there exists a sharp explicit threshold below which all phases are stable and above which resonant frequencies and multi-spikes blowing-up solutions can be constructed. Solutions of the Schrödinger-Poisson systems are standing waves solutions of the electrostatic Maxwell-Schrödinger system. Stable phases imply the existence of a priori bounds on the amplitudes of standing waves solutions. Unstable phases give rise to resonant states. We investigate in this paper nonlinear Schrödinger-Poisson systems in the 3- sphere. These are electrostatic versions of the Maxwell-Schrödinger system which describes the evolution of a charged nonrelativistic quantum mechanical particle interacting with the electromagnetic field it generates. We adopt here the Proca formalism. Then the particle interacts via the minimum coupling rule t t + i q ħ ϕ, i q ħ A with an external massive vector field ϕ, A) which is governed by the Maxwell- Proca Lagrangian. In particular, we recover as part of the full system the massive modified Maxwell equations in SI units, which are hereafter explicitly written down:.e = ρ/ε µ ϕ, ) E H = µ J + ε µ A, t.) E + H = and.h =. t These massive Maxwell equations, as modified to Proca form, appear to have been first written in modern format by Schrödinger [5]. The Proca formalism a priori breaks Gauge invariance. Gauge invariance can be restored by the Stueckelberg trick, as pointed out by Pauli [], and then by the Higgs mechanism. We refer to Goldhaber and Nieto [4, 5], Luo, Gillies and Tu [], and Ruegg and Ruiz-Altaba [4] for very complete references on the Proca approach. In the electrostatic case of the Maxwell-Schrödinger system, looking for standing waves solutions, we are led to the nonlinear Schrödinger-Poisson system we investigate in this paper. It is stated as follows: { ħ m g u + ω u + qvu = u p g v + m v = 4πqu,.) Date: September,. Revised January,.

2 EMMANUEL HEBEY AND JUNCHENG WEI where ω R, p 4, 6], g = div g is the Laplace-Beltrami operator, the constants ħ, m, m and q are positive, and u, v in S 3. Solutions of.) are standing wave solutions ψx, t) = ux)e iωt/ħ, with purely electrostatic field v, of the Maxwell-Schrödinger system in Proca form we mentioned above. The system.) is energy critical when p = 6. We refer to the temporal frequency ω as the phase and investigate both the question of the existence of one or more solutions to.), and the question of the stability of phases in.). Stability of a phase implies compactness of the set of associated solutions of.). We define the stability of a phase as follows. Definition.. Let S 3, g) be the unit 3-sphere, and p 4, 6]. A phase ω R is stable if for any sequence ψ α x, t) = u α x)e iω α t/ħ of standing waves, with purely electrostatic field v α, solutions of { ħ m g u α + ω αu α + qv α u α = u p α g v α + m v α = 4πqu α for all α N, the convergence ω α ω in R as α + implies that, up to a subsequence, the u α s and v α s converge in C S 3 ) to solutions u and v of.) as α +. In particular, if ω is stable then we get an upper bound on the L -norm of the amplitude of arbitray standing waves with phases close to ω. The first result we prove addresses the subcritical case p 4, 6) in.). The mountain pass solutions we obtain in our theorems are precisely defined in Section. These are variational solutions which inherit an additional ground state structure in the Nehari setting. Theorem. Subcritical case). Let S 3, g) be the unit 3-sphere, ħ, m, m >, and q >. Let p 4, 6). For any ω there exists a mountain pass solution of.). Moreover, all phases ω R are stable. As an interesting remark it can be noted that both the bounds 4 and 6 on the nonlinearity are sharp with respect to the stability issue in the theorem. Stability as in Theorem. is indeed false in general when p = 4 see Section 3). As shown in Theorem., it is also false when p = 6 and ω takes specific sufficiently) large values. When p = 6, a critical threshold for ω appears. In the case of S 3 that we consider in this paper this can be made very explicit. We let Λm ) be given by 3ħ Λm ) =..3) m The theorem we prove in the critical case answers positively the question of existence of special solutions and of stability of phases in the range Λm ), +Λm )), and asserts that resonant frequencies appear in the complementary range. Theorem. Critical case). Let S 3, g) be the unit 3-sphere, ħ, m, m >, and q >. Let p = 6. For any ω Λm ), +Λm )) there exists a mountain pass solution of.) and the solution is nonconstant when m q. Moreover: i) all phases ω Λm ), +Λm )) are stable, ii) there exists an increasing sequence ω k ) k of phases such that ω = Λm ), ω k + as k +, and both all the ω k s and ω k s are unstable. In particular, resonant frequencies appear outside Λm ), +Λm )), starting with ±Λm ), and the threshold Λm ) is critical.

3 SCHRÖDINGER-POISSON SYSTEMS 3 The mountain pass solution we obtain in Theorem. comes in addition to the constant solution when m q and we thus get two solutions in that case. As already mentioned, the stability of phases implies the existence of an upper iωt/ħ bound for the amplitude of standing waves ψx, t) = ux)e when ω is in compact subsets of Λm ), +Λm )). The resonant frequencies ω k break this upper bound. As we will see when proving the second part of Theorem., they come with blowing-up sequences of multi-spike solutions.. Coupling NLS with a massive field The nonlinear focusing Schrödinger equation NLS) is written as iħ ψ t = ħ m g ψ ψ p ψ. By coupling NLS) with a gauge vector field ϕ, A) governed by the Maxwell-Proca theory, the coupling being made via the minimum substitution rule, t t + i q ħ ϕ, i q ħ A, we get a system of particle-electromagnetic field describing the interactions of a matter scalar field ψ with its electromagnetic field ϕ, A). Here, ħ is the reduced Planck s constant, m > represents the mass of ψ, q its charge, and m > represents the mass of ϕ, A) in the Maxwell-Proca theory. To be more precise, let L NLS = iħ ψ t ψ qϕ ψ ħ ψ i q ) ħ Aψ + p ψ p, and L MP ϕ, A) = 8π A t + ϕ m 8π A + m 8π ϕ m 8π A, where is the curl operator, and define S = L NLS + L MP ) dv g dt to be the total action functional. Writing ψ = ue is ħ in polar form, u, and taking the variation of S with respect to u, S, ϕ, and A, we get that ) ħ S m g u + t + qϕ + S qa u = u p m u t +. S qa)u ) = m 4π. A t + ϕ) + m 4π ϕ =.) qu 4π A) + A t t + ϕ)) + m 4π A = q S qa) u. m Letting E = 4π A t + ϕ), H = 4π A, ρ = qu, and J = q S qa) u, m we recover the Maxwell-Proca equations.) with the two last equations in.), where µ = m /4π) and we normalize such that ε = and µ = the last two equations in.) are automaticaly satisfied due to the choice of E and H). The second equation in.) then reads as the charge continuity equation ρ t +.J =. We assume in what follows that A and ϕ depend on the sole spatial variables, thus we restrict our attention to the static case of.), and look for standing waves solutions of.), namely ψx, t) = ux)e iω t ħ. The fourth equation in.) then implies that A, while the second equation in.) is automatically satisfied since S = ω t. The first and third equations in.)

4 4 EMMANUEL HEBEY AND JUNCHENG WEI are rewritten as { ħ m g u + ω u + qvu = u p g ϕ + m ϕ = 4πqu..) Letting ϕ = v, the system.) is precisely the system.) we investigate in this paper. Solutions of.) are standing wave solutions of.) in the static or purely electrostatic) case.. Functional setting and existence of mountain pass solutions Let m, m >, ω R, and q >. We aim in getting mountain pass solutions for.). For this purpose we define an auxiliary functional Φ : H H by letting Φu) be the unique solution of g Φu) + m Φu) = 4πqu.) for u H. Then Φ is C and its differential Φ u at u, when computed over ϕ H, solves an equation like.) with a right hand side like 8πquϕ. In particular, Hu) = u Φu) is C with H u ϕ) = 4 uφu)ϕ for ϕ H. For p 4, 6], we define I p : H R by I p u) = ħ S3 4m u dv g + ω u dv g S 3 + q u Φu)dv g.) u + ) p dv g, 4 S p 3 S 3 where u + = maxu, ). If u is a critical point of I p, then u, Φu)) solves.). As is easily seen, Φtu) = t Φu) for all t and all u, and Φu) for all u. We prove the existence part of Theorem. in what follows. We say that u, Φu) ) is a mountain pass solution of.) when u is obtained from I p by the mountain pass lemma from to an endpoint u such that I p u ) <. Existence in the subcritical case somehow follows from a direct application of the mountain pass lemma. Proof of Theorem. - Existence part. Let p 4, 6) and u H be such that u +. There holds I p) =, and there exists T = T u ) such that I p T u ) <. For any < δ, there exists ε δ > such that Φu) ε δ for all u H satisfying that u H = and u L δ. It follows that there exists ε > such that S 3 u + Φu)u ) dv g ε for all u H satisfying that u H =. Since Φtu) = t Φu), we then get that there exist C, C > such that I p u) C u 4 H C u p H for all u such that u H. In particular the mountain pass lemma can be applied since p > 4. Let c p = inf P P max u P I pu),.3) where P is the set of continuous paths from to T u. Since u Φu) C u 4 H, mountain pass sequences associated to c p are bounded in H. Standard arguments then give the existence of u p such that I p u p ) = c p, and such that u p and v p = Φu p ) solve.). Then u p, v p are smooth and by the maximum principle u p, v p > in S 3. This ends the proof of the existence part in Theorem..

5 SCHRÖDINGER-POISSON SYSTEMS 5 From now on we assume p = 6. Let x S 3, β α ) α be a sequence such that β α > for all α and β α as α +, and define 3β ϕ α x) = α ) ) /4 βα cos r),.4) where r = d g x, x). The ϕ α s are -spike solutions of and they satisfy the energy estimates S 3 ϕ α dv g = K 3 3 g ϕ α ϕ α = ϕ 5 α.5) + o), S 3 ϕ 6 αdv g = K 3 3,.6) where K 3 = 3 S is the sharp constant for the Euclidean Sobolev inequality 3 /3 u L 6 K 3 u L. The proof of the existence part in Theorem. is as follows. Proof of Theorem. - Existence part. There holds ϕ α in L p for p < 6. Hence Φϕ α ) in H,p for p < 3, and we get that Φϕ α ) L as α +. By.6) there exists T such that I 6 T ϕ α ) < for all α. Let H ε u) = u dv g + 3 S 4 ε) u dv g m 3 S 3ħ u 6 dv g. 3 S 3 Since ω < Λm ) and Φϕ α ) L as α + we get that there exists ε > such that max I 6tϕ α ) ħ t T 4m max H ε tϕ α ) t T ħ 3 m 3 for all α. There also exist C, C > such that S 3 ϕ α dv g ε ) S 3 ϕ αdv g S 3 ϕ 6 αdv g ) /3 I 6 u) C u 4 H C u 6 H ) 3/ for all u such that u H. We let u = ϕ α for α sufficiently large, T = T, and we define c 6 = inf max I 6u),.7) P P u P where P is the set of continuous paths from to T u. According to the above and by.5), there exist δ > and ε > such that δ c 6 3K 3 3 ħ m ) 3 ε,.8) where c 6 is as in.7). Since I 6 ) =, the mountain pass lemma can be applied. We obtain the existence of a Palais-Smale sequence u α ) α such that I 6 u α ) c 6 and I 6u α ) as α +. Noting that u Φu) C u 4 L, the u 6 α s are bounded in H. In particular, there exists u H such that, up to a subsequence, u α u in H, u α u a.e., and u α u in L p for p < 6. Then Φu α ) Φu) in H,p for p < 3, and we get that Φu α ) Φu) in C,θ for some < θ. Mimicking the argument in Brézis and Nirenberg [4], it follows from.8) that u, that u α u in H, that u > in M, and that U = u, Φu)) solves.). In particular, I 6 u) = c 6.

6 6 EMMANUEL HEBEY AND JUNCHENG WEI The mountain pass solution u we obtain in the critical case is such that I 6 u) = c 6. As is easily checked,.) always possesses a constant solution U = u, v ) which, in the critical case, is given by v = 4πq m u and Then and we get that I 6 u ) = = u 4 = 4πq m u + ω..9) ω ) / + qv S 3 ω + q v ) 3 ω + qv ) ω ) / + qv S 3 3 ω + q ) 6 v I 6 u ) S3 qv ) 3/..) ) m. Let ε = 4πq Then εv = u 4 and by.9) we get that εv = q + q + 4εω. In particular, coming back to.), I 6 u ) 4π)3 S 3 and by.8) the mountain pass solution we obtain is nonconstant when m q. q Let N p be the Nehari manifold associated to.). By definition m N p = { u H, u, s.t. I pu).u = }..) Following an idea due to Rabinowitz, see Willem [8] for a presentation in book form, there holds that c p = inf I p u).) u N p for all p 4, 6], where N p is as in.), and c p is as in.3) and.7). In particular, the solutions we obtain are ground states in the sense of Willem [8]. We get.) by noting that for any u H, u +, there is one and only one t = t u), where t >, such that I ptu).tu) =. ) 6 3. Stability in the subcritical case Stability of the phases in the subcritical case follows from and can actually be reformulated into) the general theorem below, where we prove the existence of uniform bounds for arbitrary solutions of.). Let S p ω) be the set of all positive solutions U = u, v), u, v >, of.). Given θ, ) we define U C,θ = u C,θ + v C,θ for all U = u, v). The following theorem holds true. Theorem 3.. Let S 3, g) be the unit 3-sphere, ħ, m, m >, and q >. Let p 4, 6). For any θ, ), and any Λ >, there exists C > such that U C,θ C for all U S p ω) and all ω Λ. Let p 4, 6) and let ω α ) α be a sequence of phases such that ω α ω as α + for some ω R, and let U α = u α, v α ) be positive solutions of { ħ m g u α + ωαu α + qv α u α = u p α g v α + m v α = 4πqu α. 3.)

7 SCHRÖDINGER-POISSON SYSTEMS 7 Up to a subsequence we can assume that ω α ω as α + for some ω R. The proof of the existence of a priori bounds in Theorem 3. reduces to proving that the u α s and v α s are automatically bounded in C,θ S 3 ), < θ <. Proof of Theorem 3. - Existence of a priori bounds. We divide the first equation in 3.) by u α and integrate over S 3. Then S3 q v α dv g = ħ S3 u α m u dv g + uα p dv g ωα S 3 α S 3 3.) u p α dv g ωα S 3. S 3 Integrating the second equation in 3.) there also holds that m vα = 4πq u α. By 3.) and Hölder s inequality we then get that u p α dv g C + C S 3 u p α S 3 dv g ) /p ) for all α, where C, C < are independent of α. Then the u α s are bounded in L p S 3 ), and by the second equation in 3.), the v α s turn out to be bounded in H,p )/ S 3 ). By the Sobolev embedding s theorem we then get that the v α s are bounded in L q S 3 ) when p 4, 5), where q = 3p ) 5 p), and in C,θ S 3 ) for some θ, ) when p 5, 6). In particular, they are bounded in L 3 S 3 ). From now on we assume by contradiction that we can choose u α, v α ) such that max M u α + 3.3) as α +. Let x α M and µ α > be such that u α x α ) = u α L By 3.3), µ α as α +. Let ũ α be given by ũ α x) = µ p α u α expxα µ α x) ) = µ /p ) α. for x R 3. Let also g α x) = exp x α g ) µ α x) and ˆv α x) = v α expxα µ α x) ). There holds ħ m gα ũ α + ω αµ αũ α + qµ αˆv α ũ α = ũ p α 3.4) and there also holds that ũ α, ũ α ) =, and g α ξ in Cloc R3 ), where ξ is the Euclidean metric. Then there exists C > such that for any compact subset K R 3, µ αˆv α ũ α ) 3 dx C K for all α since the v α s are bounded in L 3. By elliptic theory it follows that ũ α ũ in C,θ loc R3 ) as α +, where ũ satisfies ũ and ũ) =. Moreover, by 3.4), we have that ħ ũ = ũ p, a contradiction with the Liouville m result of Gidas and Spruck [3]. Hence, 3.3) cannot happen, and for any ω α ) α such that ω α ω as α +, and any u α, v α ) solutions of 4.), there exists C > such that u α L C. By the second equation in 3.) it follows that u α L + v α L C for all α, and by standard elliptic theory, a C,θ -bound holds as well. This proves the existence of a priori bounds in Theorem..

8 8 EMMANUEL HEBEY AND JUNCHENG WEI As an interesting remark it is necessary in the above proof to assume that p > 4. Indeed, let p = 4, ω α = for all α, and 4πq = m. Then u α = α and v α = α /q solve 3.) and, obiously, u α L +, v α L + as α +. It is independently necessary to assume a bound on the ω α s since if not we get counter examples by the constant solutions which satisfy u α ωα /p ). As a remark, Theorem 3. is true on arbitrary compact Riemannian 3-manifolds. 4. Stability in the critical case Stability in the critical case is a consequence of, and is actually equivalent to, the following theorem where the existence of uniform bounds is obtained for phases in compact subsets of Λm ), +Λm )). Theorem 4.. Let S 3, g) be the unit 3-sphere, ħ, m, m >, and q >. Let p = 6. For any θ, ), and any ε >, there exists C > such that U C,θ C for all U S 6 ω) and all ω Λm ) ε. By the analysis in Druet and Hebey [9], we refer also to Druet and Laurain [] for a related reference, Theorem 4. can be extended to the case of arbitrary compact 3-dimensional manifolds. The result holds true as long as Λm ) min Λ, where Λ is such that g +Λ has a nonnegative mass. By the positive mass theorem, assuming the Yamabe invariant of g is positive, Λ 8 S g, where S g is the scalar curvature of g. In both cases we recover.3) when the manifold is the 3-sphere. The proof we present is a shortcut with respect to the analysis in Druet and Hebey [9]. We mix in our analysis ideas from Li and Zhang [9], Druet and Hebey [8], Hebey and Robert [6], and Hebey, Robert and Wen [7]. The proof extends almost as it is to compact conformally flat manifolds of positive scalar curvature. The 4- dimensional analogue of Theorem 4. for the Klein-Gordon equation is established in Hebey and Truong [8]. In what follows we let ω α ) α be a sequence of phases such that ω α α + for some ω R, and let U α = u α, v α ) be positive solutions of { ħ m g u α + ωαu α + qv α u α = u 5 α g v α + m v α = 4πqu α. ω as 4.) Dividing the first equation in 4.) by u α and integrating over S 3 we get as in Section 3 that S 3 u 4 αdv g C + C S 3 u 4 αdv g ) / for all α, where C, C < are independent of α. Then the u α s are bounded in L 4, and by the second equation in 4.), the v α s are in turn bounded in H. By the Sobolev embedding theorem we thus get that there exists v C,θ S 3 ), < θ <, such that, up to a subsequence, v α v in C,θ S 3 ) 4.) as α +. By standard elliptic theory, an L -bound on the u α s implies the C,θ -bound we are looking for in the theorem. We define h α = ω α + qv α, 4.3)

9 SCHRÖDINGER-POISSON SYSTEMS 9 and assume by contradiction that we can choose u α, v α ) such that max u α + 4.4) S 3 as α +. By 4.) the h α s converge in C,θ. The following lemma directly follows from the analysis in Li and Zhang [9]. Lemma 4. Li-Zhang [9]). Let û α > be a smooth positive solution of ħ m û α + ĥαû α = û 5 α 4.5) in R 3, where is the Euclidean Laplacian and ĥα) α is a converging sequence of functions in C loc R3 ). There exist C, δ > such that, up to a subsequence, sup B ε) û α inf B ûα C 4ε) ε 4.6) for all < ε < δ, and all α, where B ε) and B 4ε) are the Euclidean balls of center and radii ε and 4ε. Proof of Lemma 4.. We very briefly sketch the proof and refer to Li-Zhang [9] for more details. By contradiction we assume there exists ε α ) α and Λ α ) α, ε α > for all α, ε α and Λ α + as α +, such that max B ε α) û α min B 4ε α) û α Λ ε ε α 4.7) for all α. Let x α B ε α ) be a point where û α attains its maximum in B ε α ). There exist x α B xα ε α /) and σ α, ε α /4) such that û α x α ) û α x α ) for all α, û α x) Cû α x α ) for all α and all x B xα σ α ), and û α x α ) σ α + as α +. Let µ α = û α x α ), and define ˆv α by There holds σ α µ α ˆv α x) = µ / α û α x α + µ α x). 4.8) + by 4.7). By standard elliptic theory, ˆv α ˆv in C locr 3 ), 4.9) ħ where ˆv > satisfies ˆv = ˆv 5 and is given by the Caffarelli-Gidas-Spruck [5] m classification. Given λ > and x R 3, we let and Σ λ,x α = B x ε α µ α for C > let ˆv λ,x α y) = λ y x ˆv α x + λ y x) y x )\B x λ), where ˆv α is as in 4.8). Let w λ,x ) α 4.) = ˆv α ˆv α λ,x, and h λ,x α,c y) = Cλµ α y x λ). 4.) For any λ and any x, there exists C > such that wα λ,x + h λ,x α,c in Σ λ for all < λ λ and all α, where h λ,x α,c is as in 4.). Letting α + it follows that ˆv ˆv λ,x for all y x λ >, where ˆv is as in 4.9) and ˆv λ,x is built on ˆv as in 4.). This implies that ˆv is constant, and we get a contradiction with the equation for ˆv. This ends the proof of the lemma. Thanks to the estimates in Lemma 4., as noticed by Chen and Lin [6], the following holds true.

10 EMMANUEL HEBEY AND JUNCHENG WEI Lemma 4.. There exists C > such that u α H in 4.). C for all α, where u α is as Proof of Lemma 4.. Let x S 3 be any point in S 3. By the stereographic projection of pole x, there exists φ > smooth and positive such that ĝ = φ 4 g is flat in S 3 \{ x}, the set S 3 \{ x} can be assimilated with R 3, x with, and ĝ with the Euclidean metric, and such that ħ m ĝû α + ĥαû α = û 5 α, 4.) where û α = φ u α, φ 4ĥ α = m ħ h α 3 4, and h α = ωα + qv α. By 4.), ĥα ĥ in C,θ loc. Given δ >, let λ > be such that ĥ < λ in B R), R δ. Let Ĝ be the ħ Green s function of m ĝ + λ with zero Dirichlet boundary condition in B 5δ). Let also ˆv α solve { ħ ĝˆv m α + λˆv α = û 5 α in B 5δ) ˆv α = on B 5δ). By the maximum principle, ˆv α û α in B 5δ) for α. Let y α B 4δ) be such that û α y α ) = inf B4δ) û α. By standard estimates on G, see Robert [3], following Chen and Lin [6], we can write thanks to 4.3) and the estimates in Lemma 4. that u 6 αdv g C δ û 6 αy)dy B xδ) B δ) C δ δ sup û α) Gy α, y)û 5 αy)dy B δ) B δ) ) ħ C δ δ sup û α) Gy α, y) B δ) B 5δ) m ˆv α + λˆv α y)dy C δ δ sup û α B δ) inf B ûα C δ 4δ) for all α and δ > sufficiently small, where C δ > does not depend on α and change values from line to line in the above inequalities. In particular, since x is arbitrary, there exists C > such that u 6 S αdv 3 g C for all α. By 4.) this proves Lemma 4.. By Lemma 4. the u α s have bounded energy and Struwe s decomposition [6] can be applied. In particular, up to a subsequence, u α = u + where R α in H as α +, k N, u α u a.e., and k B i,α + R α, 4.3) i= ) m /4 ) / B i,α x) = µ i,α ħ µ i,α + 4.4) dgxi,α,x) 3

11 SCHRÖDINGER-POISSON SYSTEMS for some converging sequence x i,α ) α in S 3 and a sequence µ i,α ) α of positive real numbers such that µ i,α as α +. Moreover, there holds that ) ) µ m /4 i,α u α exp xi,α µ i,α x) ħ 4.5) + x 3 in C loc R3 \S i ) for all i, where S i consists of the limits of the µ i,α exp x i,α x j,α ) s as α + for j I i, and I i stands for the set of the j s which are such that d g x i,α, x j,α ) = Oµ i,α ) and µ j,α = oµ i,α ). Let D α, ˆD α : S 3 R + be defined by D α x) = ˆD α x) = min d gx i,α, x) and, i=,...,k min d gx i,α, x) + µ i,α ). i=,...,k 4.6) There holds that D α ˆD α and by the analysis in Druet and Hebey [8], since 4.) holds true, we can write that k ˆD α u α u B i,α in L S 3 ) 4.7) i= as α +. In particular, if S stands for the set consisting of the limits of the x i,α s as α +, then u α u in L loc S3 \S). Lemma 4.3. Let G α : S 3 S 3 \D R be the Green s function of ħ m g + h α, where h α is given by 4.3), and D is the diagonal in S 3 S 3. Suppose ω = and v =, where v is as in 4.). Then inf S 3 S 3 \DG α + as α +. Proof of Lemma 4.3. Let ε α = h α L and k α R be such that k α + and ε α k α as α +. Let Ĝα ħ be the Green s function of m g + ε α. By the maximum principle, G α Ĝα in S 3 S 3 \D and we can use the specific form of Ĝ α in S 3 or use the following more general S 3 -free argument. We let G be ħ a Green s function of m g. For any x S 3, if G x = Gx, ), there holds that ħ m g G x = δ x S 3. Let x S3 and V α solve ħ m g V α + ε α V α = ε α G x. There holds V α = G x so that, by Poincaré s inequality and standard estimates on G, V α is bounded in H uniformly with respect to x. By standard elliptic properties and standard estimates on G, it follows that V α L C for all α with a bound which is uniform with respect to x. Let Φ α = Ĝαx, ) G x k α + V α. Then ħ m g Φ α + ε α Φ α S 3 ε αk α for all α, and by the maximum principle and the above estimates it follows that Ĝ α x, ) k α C for all α and all x, where C is independent of α and x. This proves the lemma. The following key estimate is established in Druet and Hebey [8] see also Druet, Hebey and Robert []). A slight difference here is that we need to handle the noncoercive case where ω = and v =. We handle this case thanks to Lemma 4.3.

12 EMMANUEL HEBEY AND JUNCHENG WEI Lemma 4.4 Step 5. in Druet and Hebey [8]). There exists C > such that, up to a subsequence, ) u α C µ α Dα + u L 4.8) in S 3, for all α, where µ α = max i µ i,α. Proof of Lemma 4.4. We briefly sketch the proof and refer to Druet and Hebey [8] for more details. Given δ > we define η α δ) = max u α. 4.9) M\ k i= Bx i,α δ) ħ Let G be the Green s function of m g +. Given ε, ), we let Ψ α,ε be given by N k Ψ α,ε x) = µ ε) α G x i,α, x) ε + η α δ) G x i,α, x) ε, i= and let Ω α = M\ k i= B x i,α Rµ i,α ). We define y α Ω α be such that u i,α Ψ α,ε is maximum in Ω α at y α. Up to choosing δ > sufficiently small, and R sufficiently large, y α Ω α or D α y α ) > δ for α. By 4.5) and standard properties of the Green s function it follows that for any ε, ), there exist R ε, < δ ε, and C ε > such that, up to a subsequence, u α x) C ε µ ε) α D α x) n) ε) + η α δ ε )D α x) n)ε) 4.) for all α and all x M\ k i= B x i,α R ε µ i,α ). Now we claim that there exists δ > small such that for any sequence y α ) α of points in S 3, lim sup α + u α y α ) µ / α D α y α ) + η α δ) i= < +. 4.) By the definition of η α δ) and 4.7) we can assume that D α y α ) δ and that µ α D α y α ) + as α +. Let < λ be such that λ Sp ħ m g ), where Sp ħ ħ m g ) is the spectrum of m g, and let G be the Green s function of ħ m g λ. There exist C >, C, C 3 > such that C d g x, y) C Gx, y) C 3 d g x, y) and Gx, y) C 3 d g x, y) for all x, y S 3, x y. We choose δ > small such that d g x, y) 4δ for all x, y S, x y, and such that 4δC C. Let x S be such that d g x, y α ) δ + o). By the Green s representation formula and the above estimates on G, there exists C > such that ) ħ u α y α ) = G yα B x δ) m g u α λu α dv g + O η α δ)) C d g y α, x) u 5 αx)dv g x) + O η α δ)) S 3 for all α, since G yα in B yα δ) for α large by our choice of δ. By 4.), letting ε > be small, we get that d g y α, x) u 5 αx)dv g x) = O µ / α D α y α ) ) + O η α δ ε )). S 3

13 SCHRÖDINGER-POISSON SYSTEMS 3 Choosing δ, δ ε ), δ, we get that 4.) holds true. Now it remains to prove that if u, then η α δ) = Oµ / α ). As a consequence of 4.), assuming by contradiction that η α δ)µ α / + as α +, we get by standard elliptic theory that η α δ) u α H in Cloc S3 \S) as α +, where g H + hh = and H C in S 3 \S, H, and h = ω +qv. Then H is in the kernel of g +h and we get a contradiction if h >. In case h =, and thus in case ω = and v, we get thanks to Lemma 4., that we apply around a point where u α is maximum, that max M u α min M u α C for some C > and all α. Independently, by Lemma 4.3, ħ if x α is a point where u α is minimum, and G α is the Green function of m g +h α, then 3 max u α min u α max u α G α x α, )u 5 αdv g M M M S G α x α, )u 6 αdv g Λ α u 6 αdv g, S 3 S 3 where Λ α + as α +. Then u 6 S αdv 3 g as α + and we get a contradiction with 4.4) since if k =, then u α u uniformly in S 3. In other words, η α δ) = Oµ / α ) holds true, and by 4.), this ends the proof of the lemma. Up to now we did not use the assumption that ω < Λm ) neither ω α < Λm ). The conclusion of the proof does use this assumption. Proof of Theorem 4. - Existence of a priori bounds. We can assume that, up to a subsequence, µ α = µ,α for all α, where µ α is as in Lemma 4.4. In what follows we let x α = x,α for all α. First we claim that u. In order to prove this we proceed by contradiction and assume that u. Then v > in S 3, where v is as in 4.), since g v + m v = 4πqu 4.) in S 3. In particular, since h α = ωα + qv α by 4.3), there holds that h > in S 3, where h is the limit of the h α s. Let θ > be given, and let G θ be the Green s ħ function of m g + θh. By the maximum principle, G α G θ for α, where G α is as in Lemma 4.3. Let S = {x,..., x m } be the set consisting of the limits of the x i,α s and a S 3 \S. Let δ > be such that B a δ) M\S. Then, for any x S 3, u α x) = G α x, y)u 5 αy)dv g y) S 3 G θ x, y)u 5 αy)dv g y) B aδ) B aδ) G θ x, y)u 5 y)dv g y) + o) since u α u in L loc S \S). In particular, there exists ε > such that u α ε in S 3 for all α. Let y α S 3, given by 4.4), be such that u α y α ) + as α +. Up to a subsequence, y α y as α +. Coming back to the beginning of the proof of Lemma 4., by the stereographic projection of pole y, there exists φ >

14 4 EMMANUEL HEBEY AND JUNCHENG WEI smooth and positive such that ĝ = φ 4 g is flat in S 3 \{ y}, the set S 3 \{ y} can be assimilated with R 3, y with, and ĝ with the Euclidean metric, and such that ħ m ĝû α + ĥαû α = û 5 α 4.3) in R 3, where û α = φ u α, and φ 4ĥ α = m ħ h α 3 4. By construction we have that sup Bε) û α + for all ε > as α +. Since, according to the above, inf B4ε) û α ε > as long as u, we get a contradiction with Lemma 4.. This proves that u. In particular, by 4.), there holds that v. We assume in what follows that ω < Λ, where Λ is as in.3). By Lemma 4.4, for any K S 3 \S, there exists C K > such that µ / α u α C K in K for all α. There holds ħ m g µ / α u α ) + h α µ / α u α ) = µ αµ / α u α ) 5, where h α = ωα + qv α. By standard elliptic theory it follows that µ / α u α U in Cloc S3 \S) as α +. Splitting S 3 into the two subsets {D α Rµ α } and {D α Rµ α }, using 4.5) around x α, thanks to Lemma 4.4 and since u, there exists A > such that u 5 αdv g = A + o)) µ / α. 4.4) S 3 By the Green s representation formula we then get that ) u α x) inf G α u 5 αdv g, S 3 S 3 \D S 3 and since v, it follows from the bound µ / α u α C K, from 4.4), and from Lemma 4.3, that ω. In particular ħ m g + ω is coercive. Let G be its Green s function. Then, as in Hebey and Robert [6], we can write that Ux) = 3ħ ω m for all x S 3 \S, and U satisfies that ħ m g U + ω U = m i= µ / i Gx i, x) 3ħ ω m m i= µ / i δ xi in the sense of distributions, where µ i for all i, and µ > by 4.5). Sharper estimates would give that µ i µ α = + o)) µ i,α. There holds that Gx, y) = ηx, y) + Rx, y), ω d g x, y) where R : S 3 S 3 R is continuous, and η : S 3 S 3 R is smooth such that η, ηx, y) = if d g x, y) δ, and ηx, y) = if d g x, y) δ for δ > sufficiently small. Moreover, d g x, y) R x y) C for all y S 3 \{x}, where R x y) = Rx, y), and δ α max R xy) = o), 4.5) y B xδ α) where δ α as α +. At last, since ω < Λ, it follows from the maximum principle that Rx, x) > for all x S 3. Let x be the limit of the x α s. Let also

15 SCHRÖDINGER-POISSON SYSTEMS 5 φ > smooth be such that φ 4 ξ = g in a neighbourhood Ω of x, and φx ) =, where ξ is the Euclidean metric. Define û α = φu α. There holds d g, x) = x + φ), x) + o x ) ), where we assimilate x with and, ) stands for the Euclidean scalar product. Assume Ω S = {x }. Then µ / α û α H in Cloc Ω\{}) as α +, where ) 3ħ ω µ / Hx) = ω x + Kx), 4.6) m K C Ω), and K) = µ / Rx, x ) + j µ/ j Gx, x j ). Given δ > sufficiently small, let A δ = B δ)x k k H) ν Hdσ + ν) H B δ)x, dσ H ν Hdσ, B δ) where ν is the outward unit normal. By 4.5) and 4.6), we get that lim A δ = 3 ) ħ δ m ωµ / K). 4.7) By the Pohozaev identity applied to û α in B δ), there also holds that x k k û α ) + ) û α dx = A δ + o)) µ α. 4.8) ûα B δ) It remains to handle the left hand side in 4.8), the difficulty here being that we only have a C,θ -convergence for the h α s and not a C -convergence. We claim that there exists C > such that v α Cµ α D α 4.9) in S 3, for all α. In order to prove 4.9), let G be the Green s function of g + m. Then v α x) = 4πq Gx, y)u αy)dv g y) S 3 for all x S 3 and all α, and by standard properties of the Green s functions, see Druet, Hebey and Robert [], by Lemma 4.4, and since u, we get that v α x) Cµ α k i= S 3 dv g y) d g x, y) d g x i,α, y). In particular, 4.9) follows from Giraud s lemma, and this ends the proof of 4.9). There holds that for all α, where ħ m ĥ α = û α + ĥαû α = û 5 α m ħ h α 3 ) φ 4. 4

16 6 EMMANUEL HEBEY AND JUNCHENG WEI By 4.9), and since v α is bounded in H,3, we can write with Lemma 4.4 that v α u αdv g oµ α ) + v α u αdv g B δ) B δ)\ k i= Bx i,α µα) k oµ α ) + Cµ α i= B δ)\b xi,α µ α) dx i,α, x) 3 dx oµ α ) + Cµ α B xi,α δ)\b xi,α µ α) dx i,α, x) 3 dx i Ĩ = oµ α ) + O µ α ln ), µ α where Ĩ is the subset of {,..., k} consisting of the i s which are such that x i,α x as α +, d is the Euclidean metric, and δ > is sufficiently small. Integrating by parts the left hand side in 4.8), we then get that lim lim x k k û α ) + ) û α dx =. 4.3) δ α + ûα µ α B δ) Combining 4.7), 4.8), and 4.3), the contradiction follows since G and Rx, x ) > when ω < Λ. As a consequence, u α L + v α L C for some C >. By standard elliptic theory a C,θ -bound holds as well. Up to a subsequence we can pass to a C -limit. This proves the existence of a priori bounds in Theorem Unstable phases and resonant frequencies - Affine estimates The goal in this section and in the following one is to prove the second part of Theorem. by constructing multi-spikes solutions to.) when ω is close to resonant frequencies ω k. To each ω k is associated a sequence of n k -spikes solutions with n k + as k +. This can be considered as bifurcation from infinity see Bahri []). More precisely we use here the so-called localized energy method see Del Pino, Felmer and Musso [7], Rey and Wei [], and Wei [7]) which goes through the choice of suitable approximate solutions this is done in this section) and the use of finite-dimensional reduction carried over in the following section). Let P =,,, ) in S 3 and k N, k. We define the P i s, i =,..., k, by P i = e iθi, ) S 3 R R, where θ i = πi ) k. Let G k be the maximal isometry group of S 3, g) which leaves globally invariant the set {P,..., P k }. Let also Σ k S 3 be the slice { re Σ k = iθ, z ), r >, z C, r + z =, π k θ π }. 5.) k We consider the nonlinear critical equation ħ m g u + Λm ) u = u 5 in S 3, with u >. Its solutions are all known and given, see.4), by ħ 3β ) ) /4 U β,x = /4, m β cos r)

17 SCHRÖDINGER-POISSON SYSTEMS 7 where β > is arbitrary, r = d g x, ), and x S 3 is also arbitrary. These solutions can be rewritten as ) / U ε,x = 3/4 ħ ε 8 /4 m ε cos r + sin r, 5.) where ε, ). There holds that U ε,x = U βε,x by letting β ε = +ε ε. Also we do have an explicit expression for the Green s function G ω of ħ m g + ω. Namely, G ω x, y) = m sinh µ ω π r)) πħ sinhµ ω π) sin r 5.3) m ħ ω. When for all x, y S 3, x y, where r = d g x, y) and µ ω = ω = Λm ) we recover the Green s function of the conformal Laplacian. We write G instead of G ω when ω = Λm ) and there holds that G x, y) = m cos r πħ sin r 5.4) for all x, y S 3, x y, where r = d g x, y). At this point we let R ω be given by G ω = G + R ω, and we define η k ω) = R ω P, P ) + k G ω P, P i ), 5.5) where the second term in the right hand side of 5.5) is zero if k =. There holds i= R ω P, P ) = m πħ µ ω cothµ ω π) so that η ω) = if and only if ω = ±Λm ), while η Λm ) )) <. It is easily checked that η k ω) as ω ±, while η k ħ/ m > for k. There also holds that d dµ µ cothµπ) > while, by the maximum principle, G ω G ω if ω ω. Hence there exists a unique ω k Λm ) such that η k ω k ) =. We define ω k = inf {ω Λm ) s.t. η k ω) = }, = sup {ω Λm ) s.t. η k ω) = }, 5.6) where η k ω) is given by 5.5). Then η k ω) > if ω < ω k and η k ω) < if ω > ω k. When k =, ω = Λm ). Since sinhtx)/ sinx) > t for x π, π), there holds that ω k + as k +. Independently, we can check ħ m R ω,p = µ ω cothµ ω π) 4π + 8π m ħ ω 3 4 ) r + Or ), 5.7) where r = d g P, ) and R ω,p = G ω P, ) G P, ). The R ω,p s satisfy the equation ħ m g R ω,p + ω R ω,p = Λm ) ω ) G. 5.8) In what follows we define the projections U ε,pi, i =,..., k, by ħ m g U ε,pi + ω U ε,pi = U 5 ε,p i 5.9)

18 8 EMMANUEL HEBEY AND JUNCHENG WEI and we define ϕ ε,pi to be given by U ε,pi = U ε,pi + ϕ ε,pi, 5.) where U ε,pi is given by 5.). There holds that ħ m g ϕ ε,pi + ω ϕ ε,pi = Λm ) ω ) U ε,pi. 5.) Independently we let ψ Ḣ R 3 ) be the solution of ħ m ψ = 4 + x x 5.) in R 3. By the Green s representation formula we get that ψx) C ln+ x ) + x and ψx) C ln+ x ) + x ) as x +. The first lemma we prove is the following where we obtain the asymptotic expansion of the ϕ ε,pi s, and thus of the approximate solutions U ε,pi s. Lemma 5.. There holds that in Σ k, where A = 3 /4 π4 ϕ ε,p = A εr ω,p + B ω ε 3/ ψ ħ m ) 5/ r ) + o ε 3/) 5.3) ε and B ω = 3/4 ħ 8 /4 m Λm ) ω ), 5.4) Σ k is as in 5.), ϕ ε,p is as in 5.), R ω,p = G ω P, ) G P, ), ψ is as in 5.), and r = d g P, ). Proof of Lemma 5.. Thanks to the equations 5.8) and 5.) satisfied by R ω,p and ϕ ε,p, ħ m g ϕε,p A εr ω,p ) + ω ϕ ε,p A εr ω,p ) = CKε, where C = 3/4 ħ 8 /4 Λm m ) ω ) and K ε = v ε = ϕ ε,p A εr ω,p, ṽ ε, g ε be given by ṽ ε x) = ε 3/ v ε expp εx) ), ε ε cos r +sin r ) / ε sin. Let r and g ε x) = exp P g ) εx), where x R 3. There holds g ε ξ in C loc R3 ), where ξ is the Euclidean metric, and we have that where K ε = ε cos ε x ħ m + sin ε x gε ṽ ε + ε ω ṽ ε = Cε K ε, ) / sin ε x ). The functions ε K ε converge in C loc R3 ) to the right hand side of 5.), in the sense that the difference converges to in C loc, while by the Green s representation formula, ε 3/ v ε x ε ) if ε d gp, x ε ) +. In particular, we do have 5.3) and this proves Lemma 5..

19 SCHRÖDINGER-POISSON SYSTEMS 9 At this point we define W ε to be the sum of the U ε,pi s. Then W ε = k U ε,pi. 5.5) i= In particular W ε is G k -invariant. As a direct consequence of Lemma 5. we get that the following expansion of the W ε s holds true. Namely W ε = U ε,p + A ) k r ) ε R ω,p + G ω,pi + B ω ε 3/ ψ + o ε 3/) 5.6) ε i= in Σ k, where A, B ω are as in 5.4), Σ k is as in 5.), U ε,p is as in 5.), the G ω,pi = G ω P i, ) s are as in 5.3), R ω,p = G ω P, ) G P, ), ψ is as in 5.), and r = d g P, ). We define U : R 3 R to be given by ) / U x) =, 5.7) + x 4 and let K be the constant K = Also we let Φ k ω) : S 3 R be the solution of 3/4 ħ 8 /4 m. 5.8) g Φ k ω) + m Φ k ω) = K k i= G ω,pi ), 5.9) where G ω,p = G ω P, ), G ω is as in 5.3), and K = 8 ) 5. 3π 3 ħ m The right hand side in 5.9) is in L p for all p < 3. Thus, 5.9) makes sense. Now, thanks to 5.6), we get that the following asymptotics for the energy hold true. Lemma 5.. There holds that where A,k = kk6 3 A,k ω) = 6π I 6 W ε ) = A,k + A,k εη k ω) + A,k ω)q ε A 3,k ω) = 8π6/4 kk A 3,k ω)ε + O η k ω) ε ) + o ε ), U 6 dx, A,k = kk5 A U 5 dx, R 3 R 3 Φk ω) + m Φ k ω) ) dv g, S 3 m ) 3/ ω Λm ) ) + ħ dr 4 + r, 5.) 5.) I 6 is the functional given by.), A is as in 5.4), W ε is as in 5.5), U is as in 5.7), K is as in 5.8), and Φ is as in 5.9). Proof of Lemma 5.. There holds that ħ W ε dv g + ω Wε dv g = k Uε,P 5 W ε dv g + O ε 5/). S 3 S 3 Σ k m

20 EMMANUEL HEBEY AND JUNCHENG WEI Then, by 5.6), and thus by Lemma 5., ħ m W ε dv g + ω Wε dv g S 3 S 3 = k U ε,p + A r ) εψ + B ω ε 3/ ψ + o ε 3/)) dv g ε where Σ k U 5 ε,p + o ε ), Ψ = R ω,p + 5.) k G ω,pi. 5.3) Let δ ε > be such that δ ε as ε and δε 6 ε 3 = o ε ). There holds Uε,P 6 dv g = Uε,P 5 + o ε ). Σ k B P δ ε) We have ε ε cos r + sin r in B P δ ε ), while dv g = sin r r i= ) ε = + r ε + r + ε r 4 6ε + r 4 ) + or ) ) dx in geodesic normal coordinates. It follows that Σ k U 6 ε,p dv g = K 6 where K is as in 5.8). By 5.7), Ψ = η k ω) + C ω r + C i x i + Or ), R 3 U 6 dx + oε ), 5.4) where Ψ is as in 5.3), C ω is given by C ω = m4 πħ ω Λm 4 ) ), and the sum C i x i is given by C i x i = k 3 G ω,pj j= i= x i P )x i. Splitting Σ k into B P δ ε ) and Σ k \B P δ ε ), we obtain that Uε,P 5 Ψdv g = K 5 εηk ω) U 5 dx + KC 5 ω ε 3/ U 5 rdx + o ε 3/), 5.5) Σ k R 3 R 3 where r = x. In a similar way, thanks to the bounds at infinity we have on ψ, there holds that Σ k U 5 ε,p ψ r ε ) r ) dv g = Uε,P 5 ψ dv g + o ε 3/) B P δ ε) ε ε = K 5 Plugging 5.4) 5.6) into 5.) we get that ħ m W ε dv g + ω Wε dv g S 3 S 3 = kk 6 U 6 dx + kkaεη 5 k ω) U 5 dx R 3 R 3 + kkac 5 ω ε U 5 rdx + kkb 5 ω ε R 3 R 3 U 5 ψr)dx + o ε 3/). R 3 U 5 ψdx + o ε ). 5.6) 5.7)

21 SCHRÖDINGER-POISSON SYSTEMS Independently, still by 5.6), there holds that Wε 6 dv g = k Uε,P 6 dv g + 6k Uε,P 5 A r )) εψ + B ω ε 3/ ψ S 3 Σ k Σ k ε + 5A kε Uε,P 4 Ψ dv g + o ε ). Σ k Noting that Uε,P 4 Ψ dv g = Uε,P 4 Ψ dv g + oε) Σ k B P δ ε) = η k ω) Kε 4 U 4 dx + oε) R 3 we get as above that Wε 6 dv g = kk 6 U 6 dx + 6kKAεη 5 k ω) U 5 dx S 3 R 3 R 3 + 6kKAC 5 ω ε U 5 rdx + 6kKB 5 ω ε U 5 ψdx R 3 R 3 + O ε η k ω) ) + o ε ). 5.8) At last, by noting that ε ΦW ε) qφ k ω) in H, where Φ k ω) is as in 5.9), we get that q ΦW ε )Wε dv g = ΦWε ) + m S 4π ΦW ε ) ) dv g 3 S 3 = ε q Φk ω) + m 4π Φ k ω) ) 5.9) dv g + oε ) S 3 Combining 5.7), 5.8) and 5.9) we get 5.) where A 3,k ω) is given by ) A 3,k ω) = kk5 AC ω U 5 rdx + B ω U 5 ψdx R 3 R 3 = 6/4 kkm 5 3/ ω Λm ħ 3/ ) ) U 5 r ψ x)) dx, R 3 and ψ solves ψ = 4 + x x in R 3. Noting that U = 3 4 U 5, integrating by parts, we get the expression of A 3,k ω) stated in the lemma. This ends the proof of Lemma 5.. An additional result we prove is the following. Lemma 5.3. There holds that lim k + A 3,k ω k ) A,k ω k ) = +, 5.3) where the ω k s are as in 5.6), and A,k, A 3,k ω) are as in 5.).

22 EMMANUEL HEBEY AND JUNCHENG WEI Proof of Lemma 5.3. By the uniqueness of Φ k ω) in 5.9) it is G k -invariant. There holds that Φ k ω), and we can thus write by Hölder s inequalities that Φk ω) + m Φ k ω) ) dv g S 3 k C G ω,p i Φ k ω)dv g i= S 3 Ck G ω,p Φ k ω)dv g S 3 Ck G ω,p L /5 Φ k ω) L 6. By the maximum principle, G ω,p G ω,p for all ω ω. Since ω k +, it follows that A,k ω k ) Ck, where C > is independent of k. On the other hand, by the definition of ω k, µ k cothµ k π) CG ωk,p P ), where µ k Cω k, and we thus get that ω k Ck, where C > is independent of k. Then A 3,k ω k ) CkA,k ω k ), where C > is independent of k. This proves the lemma. 6. Equivariant finite-dimensional reduction We develop in this section the finite-dimensional reduction argument we need in order to prove the second part of Theorem., following in large parts previous arguments by Rey and Wei [] and Del Pino, Felmer and Musso [7], and we prove the second part of Theorem. for the finite dimensional reduction method in the subcritical case, we refer to the book by Ambrosetti and Malchiodi [] and the survey paper by Wei [7]). We let Θ k be given by Θ k = q + A 3,kω k ) A,k ω k ). 6.) Then we let ε = Λ ε, where C Λ C for C, and we define ε = η kω) for ω ω k δ, ω k ) with δ > small in case Θ k >, and ε = η k ω) for ω ω k, ω k + δ) with δ > small in case Θ k <. Since A 3,k ω ) = we have that Θ >. On the other hand, by Lemma 5.3, there holds that Θ k < for k. In the above constructions, ε > and ε as ω ω k. We let f ε : ε S3 S 3 be the map given by f ε x) = εx. If g ε is the standard metric on ε S3, induced from the Euclidean metric, then f ε g = ε g ε. Given u : S 3 R, we define the -procedure which, to u, associate ũ : ε S3 R, where ũ = εu f ε. We let Ỹ = W ε Λ, where W ε is obtained from W ε in 5.5) by the -procedure, and we define Z = ħ m g ε Ỹ + ε ω Ỹ. 6.) There holds that Ỹ, Z = γ + o), where γ > and, is the L -scalar product with respect to g ε. We say in what follows that a function ũ in ε S3 is

23 SCHRÖDINGER-POISSON SYSTEMS 3 G k -invariant if u is G k -invariant in S 3. In particular Ỹ and Z are G k -invariant. By the -procedure, the equation in S 3 is equivalent to ħ m ħ m g u + ω u + qφu)u = u 5 g ε ũ + ε ω ũ + q ε Φu)ũ = ũ 5 in ε S3, where Φu) = Φu) f ε. Now we define the norms,σ and,σ by u,σ = sup min + d g ε P ) ) σ i, x) ux), x i=,...,k ε S3 u,σ = sup x ε S3 min i=,...,k + d g ε P ) ) +σ i, x) ux) 6.3) for u L ε S3), where < σ < and f ε P i ) = P i, i =,..., k. Given a function h L ε S3) we consider the problem { ħ m g ε φ + ε ω φ 5 W ε 4 φ = h + c Z Zφdv ε S3 g ε =, 6.4) where c R, and Z is as in 6.). A key point in the equivariant finite-dimensional reduction argument we develop here is given by the following lemma. Lemma 6.. Let h ε ) ε be a family in L ε S3) of G k -invariant functions such that h ε,σ as ε, and φ ε ) ε be a family of G k -invariant solutions of 6.4) with h = h ε. There holds φ ε,σ as ε. Proof of lemma 6.. Let σ < σ. We prove by contradiction that φ ε,σ as ε. We can assume that φ ε,σ =. In what follows we let G ε be the Green s function of g ε + ε ω. Then G ε x, ỹ) εgx, y) C d g ε x, ỹ), 6.5) where f ε x) = x, f ε ỹ) = y, and G is the Green s function of g + ω. Thanks to the G k -symmetries, using the Green s representation formula and 6.5), we get that φ ε L C and that ) σ φ ε x) C min i d g ε P 6.6) i, x) for all x P i, i =,..., k. There also holds that c = O h ε,σ ) + o φ ε,σ ) = o). 6.7) Let ˆφ ε,i = φ ε exp Pi, i =,..., k. Then exp Pi g ε ξ in C loc R3 ) as ε, where ξ is the Euclidean metric, and by standard elliptic theory, ˆφ ε,i ˆφ in C loc R3 ) as ε, where ħ m ˆφ i = 5U 4 Λ, ˆφ i,

24 4 EMMANUEL HEBEY AND JUNCHENG WEI ˆφ i C, x σ ˆφ i C for all x, and U Λ, = K Λ / Λ + x /4 ) /. By Bianchi-Egnell [3] this implies that ˆφ U i = α Λ, i Λ since by the G k-invariance, ˆφ i is even. Still by the G k -invariance, α = = α k. Let α be the common value to the α i s. By 6.6) and since φ ε,σ =, there exist R > and δ > such that There holds Zφ ε dv ε S3 g ε = and we have that Zφ ε dv g ε kα ε S3 R 3 φ ε L B R) δ. 6.8) δα, U Λ, Λ ) UΛ, Λ dx where δ >. Hence α = and we get a contradiction with 6.8). This proves that φ ε,σ as ε. Noting that φ ε,σ C φ ε,σ + h ε,σ + c ), we then get with 6.7) that φ ε,σ as ε. This ends the proof of the lemma. At this point we define R, ε, R, ε, and R ε by R, ε = W 5 ε ħ m g ε Wε ω ε Wε, R, ε = q ε ΦW ε ) W ε, and R ε = R, ε + R, ε. 6.9) Thanks to the asymptotic expansion in Lemma 5., noting that ΦW ε ) = O ε σ ) for any < σ <, we get that R i, ε,σ C ε and D Λ R i, ε,σ C ε for all i =,. Following almost word by word the arguments in Rey and Wei [], see also Del Pino, Felmer and Musso [7], we get with Lemma 6. that there exist ε > and C > such that R) for any ε, ε ) and any G k -invariant function h L ε S3), 6.4) has a unique G k -invariant solution φ = L ε h) with φ,σ C h,σ. Moreover, the map L ε is C w.r.t. Λ and D Λ L ε h),σ C h,σ. R) for any ε, ε ), 6.) has a unique G k -invariant solution φ = φ ε with φ ε,σ C ε and D Λ φ ε,σ C ε, where 6.) is the problem ħ m g ε Ŵε + φ) + ε ω Ŵε + φ) +q ε ΦW ε + K ε + φ)ŵε + φ) = Ŵε + φ) 5 + c Z 6.) Z φdv ε S3 g ε =, Ŵ ε = W ε + L ε R ε ), c R, and K ε = L ε R ε ). ΦW ε + K ε + φ) = ΦW ε + K ε + φ) f ε, We get R) by an application of the Fredholm theorem, and R) by an application of the fixed point theorem and we assume σ is not to small). Now we let Û ε = W ε + L ε R ε ) + φ ε. 6.)

25 SCHRÖDINGER-POISSON SYSTEMS 5 There holds by R) that L ε R ε ),σ C ε. Thus Ûε >. We define ρ : R + R by ρλ) = ħ 4m Ûε dv g ε + ω ε Ûε dv g ε ε S3 ε S3 6.) + q ε 4 ΦU ε )Û ε dv g ε 6 ε S3 Ûε 6 dv g ε, ε S3 where U ε is such that Ũε = Ûε, namely such that Ûε is obtained from U ε by the -procedure. The following proposition holds true. Proposition 6.. The function Ûε > is a solution of ħ m in ε S3 if and only if Λ is a critical point of ρ. g ε Ũ + ε ω Ũ + q ε ΦU)Ũ = Ũ 5 6.3) Proof of Proposition 6.. We define I ε by ħ I ε Ũ) = 4m Ũ + ω ε ) Ũ dv ε S3 g ε + q ε ΦU)Ũ dv 4 g ε ε S3 Ũ + ) 6 dv 6 g ε. ε S3 Then I ε Ũ) = I 6U) and there holds that Ûε is a solution of 6.3) if and only if U ε = W ε + K ε + φ ε is a solution of ħ m g U + ω U + qφu)u = U 5. This is in turn equivalent to c =, where c is as in 6.), which is again equivalent to I ε Ûε).Ỹ ) = since I ε Ûε).Ỹ ) = c Ỹ, Z and Ỹ, Z = γ +o), where γ >. Independently, there holds that ρ Λ) = if and only if ) I ε Ûε. Ỹ + Ψ ) ε =, Λ where Ψ ε = K ε + φ ε, while if we let y = Ψε Λ, then y,σ Cε. We write that y = y + aỹ, where y, Ỹ ) ε = and, ) ε is the scalar product associated to ħ m g ε + ε ω. Then ρ Λ) = if and only if + a)i εûε).ỹ ) = since y, Z = y, Ỹ ) ε. There holds that y, Ỹ ) ε = o) and this implies that a = o). This ends the proof of the proposition. Now, thanks to Proposition 6., we are in position to prove point ii) in Theorem.. This is the subject of what follows.

26 6 EMMANUEL HEBEY AND JUNCHENG WEI Proof of the second part of Theorem.. Given σ, ) sufficiently close to, we compute ΦU ε )Ũ ε dv g ε = ΦW ε ) W ε dv g ε + o), ε S3 ε S3 Ũε 6 dv g ε = W6 ε dv g ε + 6 W5 ε ψ ε dv g ε ε S3 ε S3 ε S3 +5 W4 ε ψ ε dv g ε + o ε ), ε S3 Ũε 5 ψ ε dv g ε = W5 ε ψ ε dv g ε + 5 W4 ε ψ ε dv g ε + o ε ), ε S3 ε S3 ε S3 ΦU ε )Ũε ψ ε dv g ε = ΦW ε )Ũε ψ ε dv g ε + o), ε S3 ε S3 where U ε = W ε + ψ ε and ψ ε = K ε + φ ε. Then ρλ) = I 6 W ε ) R, ε ψ ε dv g ε + R, ε ψ ε dv g ε + o ε ), ε S3 ε S3 where R, ε and R, ε are as in 6.9). By our choices of ε, ε, and since C Λ C for C > fixed, we then get by direct computations that ρλ) = I 6 W ε ) + o ε ). Assume now that Θ k >, where Θ k is as in 6.). Then, by Lemma 5., ρλ) = A,k + A,k ε Λ + A,k ω)q ε Λ + A 3,k ω) ε Λ + o ε ) Λ = A,k + A,k ε Λ + A,k ω k )Θ k ε Λ + o ε ) Λ and since A,k < and Θ k >, ρ has an absolute minimum Λ ω in C, C) for C when ω ω k δ, ω k ) and < δ. Pick any sequence ω α ) α of phases in ω k δ, ω k ) such that ω α ω k as α +. By Proposition 6. we then get that there is an associated sequence U α, ΦU α )) of solutions of.) with ω = ω α, where U α = U εα and ε α = Λ ωα η k ω α ), such that U α ) α is a k-spikes type solution of the first equation in.). In particular, U α L + as α +. Similarly, if we assume that Θ k <, then by Lemma 5., ρλ) = A,k A,k ε Λ + A,k ω)q ε Λ + A 3,k ω) ε Λ + o ε ) Λ = A,k A,k ε Λ + A,k ω k )Θ k ε Λ + o ε ) Λ and ρ has an absolute maximum in C, C) for C when ω ω k, ω k + δ) and < δ. Pick any sequence ω α ) α of phases in ω k, ω k + δ) such that ω α ω k as α +. By Proposition 6. we then get that there is an associated sequence U α, ΦU α )) of solutions of.) with ω = ω α, where U α = U εα and ε α = Λ ωα η k ω α ), such that U α ) α is a k-spikes type solution of the first equation in.). In particular, U α L + as α +. We know that Θ k > for k = and, by Lemma 5.3, that Θ k < for k. This ends the proof of the second part of Theorem.. As a remark it can be noted that we obtain the existence of solutions to.) for ω sufficiently close to the ω k s with ω < ω k if Θ k > and ω > ω k if Θ k >. Acknowledgement: The first author was partially supported by the ANR grant ANR-8-BLAN The research of the second author is partially supported