Normalized Solutions to Strongly Indefinite
Semilinear Equations
Boris Buffoni
School of Mathematics, ´Ecole Polytechnique F´ed´erale-Lausanne, SB/IACS/ANA Station 8, 1015 Lausanne, Switzerland
e-mail: boris.buffoni@epfl.ch
Maria J. Esteban, Eric S´
er´
e
CEREMADE (UMR CNRS 7534) Universit´e de Paris-Dauphine, Pl. du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex 16, France
esteban, sere@ceremade.dauphine.fr Received 17 April 2006 Communicated by Shair Ahmad
Abstract
In this paper we discuss the existence of normalized solutions of nonlinear elliptic PDEs in gaps of the essential spectrum of the corresponding differential operator. This issue can be of interest, for instance, in nonlinear optics or in crystalline models with impurities.
2000 Mathematics Subject Classification.35J60, 35P30, 35J20, 35J10, 35Q55, 81Q05.
Key Words.Normalized solutions, nonlinear elliptic partial differential equations, Morse index, indefinite functionals, varia-tional methods, penalization.
1
Introduction
Finding normalized solutions of strongly indefinite semilinear elliptic equations in un-bounded domains has been a long standing question, the normalization being for instance in the space L2
(Rd). This kind of condition is natural for instance in quantum mechanics,
where one looks for particles which have unit charge. In this paper we give an answer to this question for some particular nonlinearities, local and nonlocal. We study the two
particular model equations
−∆u(x) + p(x)u(x) = a(x)|u(x)|γ−1u(x) + λu(x), u ∈ H1(Rd) , d ≥ 1 , (1.1) and −∆u(x)+p(x)u(x) − α |x|u(x) + β(u 2∗ 1 |x|)u = λu(x), u ∈ H 1 (Rd), d ≥ 3 , (1.2)
together with the normalization condition Z
Rd
u2(x) dx = 1 , (1.3)
where p ∈ L∞(Rd) is periodic (and real valued), α, β > 0, γ > 1, a ∈ L∞
(Rd), a ≥ 0,
a 6≡ 0 and either lim|x|→∞a(x) = 0 or a ∈ Lq(Rd) for some q ∈ [1, ∞).
The (unknown) spectral parameter λ is such that the unbounded self-adjoint linear op-erator in L2
(Rd) u → −∆u + pu − λu is invertible but not positive definite, that is, λ is in
a spectral gap of u → −∆u + pu, denoted by (λ0− m, λ0+ l) with m, l > 0 and λ0∈ R.
In the case of Equation (1.1), we prove that under the above conditions, there exist λ ∈ (λ0− m, λ0+ l) and u ∈ H1(Rd) such that
−∆u(x)+p(x)u(x) = a(x)|u(x)|γ−1u(x) + λu(x) in Rd, Z
Rd
u2(x) dx = 1, (1.4)
if 1 < γ < 1 +4d (d ≥ 3), 1 < γ < 3 (d = 1, 2), if the function a(x) goes slowly enough to 0 at x approaches ±∞ and if, in addition, there is no solution of
−∆u(x)+p(x)u(x) = a(x)|u(x)|γ−1u(x) + (λ
0− m)u(x) in Rd,
satisfying 0 <R
Rdu
2(x) dx ≤ 1.
This last condition is difficult to verify, but this can be done. For instance, when ||a||Lq(Rd) is small enough either for q = ∞ or for some appropriate q ∈ (1, ∞). For
precise statements about Equation (1.1), see Theorems 1, 2 and 3 below. Let us just men-tion a particular case of Theorem 2 where this last condimen-tion can also be verified: if d = 1, 1 < γ < 3, p is piecewise continuous and a ≥ 0 is continuous with non-empty compact support, then there exist λ ∈ (λ0− m, λ0+ l) and u ∈ H1(R) such that
−u00(x) + p(x)u(x) = a(x)|u(x)|γ−1
u(x) + λu(x) in R , Z
R
u2(x) dx = 1.
Note that the above results correspond to subcritical cases for Equation (1.1).
As far as Equation (1.2) is concerned, let us define H± ⊂ H1(Rd) as the
posi-tive/negative spectral space associated with −∆ + p − λ0. Then, we prove the existence of
inf v∈H+ ||v|| H1=1 Z Rd |∇v|2+ p − λ 0− α|x|−1v2dx > 0 , sup u∈H1 (Rd) ||u|| L2≤1 sup w∈H− ||w|| H1=1 Z Rd |∇w|2+ p − λ 0− α |x|w 2+ 3β(u2∗ 1 |x|)w 2dx < 0 . (1.5)
Note that the above conditions are satisfied if α, β are small enough while still satisfying 0 < β < α, thanks to the Hardy inequality : R
Rd(f 2/x2) dx ≤ 4 (d−2)2 R Rd|∇f | 2dx, for all f ∈ H1
(Rd) if d ≥ 3. From the point of view of physics, the relevant dimension for
Equation (1.2) is d = 3. The function p(x) can be interpreted as the Coulomb potential in a three-dimensional periodic crystal. Now, in this crystal, assume that the atomic nucleus located at the origin of coordinates has been replaced by a nucleus of higher atomic number (commonly called an “impurity”, see [8]). This impurity generates the additional Coulomb potential −α|x|−1. In some situations it can bind two electrons. To study this phenomenon, we make the so-called “restricted Hartree-Fock approximation” (see e.g. [15]), in which the two electrons have the same spatial wave function u, one electron has spin “up”, the other has spin “down”. This gives (1.2). In this interpretation, the nonlinear term β(u2∗ 1
|x|) is
the repulsive potential generated by an electron and felt by the other. The eigenvalue λ is the energy of each electron.
Our approach to prove the main results of this paper is the one developed in [7] for the Dirac-Fock equations and is closely related to the theory of bifurcation from the essential spectrum, particularly [14]. Note that of course much more general nonlinear elliptic equa-tions can be treated by the same arguments used to deal with (1.1) and (1.2). In Section 3 we give an abstract version of the theorems showing the requirements in the general case.
2
General setting
We are interested in problems of the form
−∆u(x)+p(x)u(x)+V (x)u(x) = ˜N (u)(x)+λu(x) in Rd,
Z
Rd
u2(x) dx = 1, (2.6) where λ is an unknown “spectral” parameter, p ∈ L∞(Rd) is periodic and ˜N (u) is a nonlinear function (or functional) of u, which is the derivative of a real-valued functional φ . Indeed, for Equation (1.1), ˜N (u) = a(x)|u|γ−1u and φ(u) = γ+11 RRda|u|
γ+1, while
for (1.2), ˜N (u) = |x|αu − β(|u|2∗ 1
|x|)u and φ(u) = α 2 R Rd u2 |x| − β 4 RR Rd×Rd u2(x)u2(y) |x−y| .
Being a solution of (2.6) is understood in the weak sense: Z
Rd
{∇u(x) · ∇v(x) + p(x)u(x)v(x) − λu(x)v(x)} dx = Z
Rd
˜
N (u)(x) v(x) dx for all v ∈ H1
(Rd). The unknown spectral parameter λ is such that the unbounded
self-adjoint linear operator in L2
that is, λ is in a spectral gap of u → −∆ + pu. That such a spectral gap may exist when p is not constant is a well-known phenomenon (see e.g. [13, 14, 16]). We denote it by (λ0− m, λ0+ l) with m, l > 0 and λ0∈ R:
λ0−m ∈ σ(−∆u+pu), λ0+l ∈ σ(−∆u+pu) and (λ0−m, λ0+l)∩σ(−∆u+pu) = ∅.
For u ∈ H1(Rd), the functional v → RRd{−∆u + pu − λu}v dx is well defined if
v ∈ H1
(Rd) and thus −∆u + pu − λu ∈ H−1
(Rd). If we now introduce in H1 (Rd) the inner product hu, viH1:= Z Rd ∇u · ∇v + (p − inf Rd p + 1) uv dx , (2.7) we can identify H1(Rd) and its dual H−1(Rd) and also −∆u + pu can be identified with
Lu = (−∆ + p − inf p + 1)−1(−∆u + pu) ∈ H1
(Rd). This defines a bounded operator
L on H1
(Rd). In other words,
Z
Rd
{−∆u + pu}v dx = hLu, viH1 for all v ∈ H1(Rd).
In the same way, there is a nonlinear operator N : H1
(Rd) → H1
(Rd) such that
Z
Rd
˜
N (u) v dx = hN (u), viH1for all v ∈ H1(Rd),
and a linear operator A : H1(Rd) → H1
(Rd) such that
Z
Rd
uv dx = hAu, viH1for all v ∈ H1(Rd).
With these notations, the problem is to find u ∈ H1
(Rd) such that
Lu = N (u) + λ Au , hAu, uiH1(Rd)= 1.
Note that L is bounded, self-adjoint and invertible, but neither positive nor negative defi-nite, whereas A is bounded, self-adjoint and positive defidefi-nite, but has no bounded inverse. Moreover LA = AL and N (u) is the gradient, for the H1-scalar product, of the C2
func-tional φ(u) defined by
φ(u) := 1 1 + γ
Z
Rd
a(x) |u|γ+1dx in the case of Equation (1.1) ,
φ(u) := α 2 Z Rd |u|2 |x| dx − β 4 Z Z Rd×Rd u2(x) u2(y)
|x − y| dx dy in the case of Equation (1.2) . The main advantages of these notations are that they are closely related to the variational structure of the problem and that all linear operators are bounded. They lead to a general and abstract framework, but first let us state our main results for (2.6), which ensures the existence of a normalized solution for at least one λ ∈ (λ0− m, λ0+ l).
Theorem 1. Let d ≥ 2, let p ∈ L∞(Rd) be periodic, let a ∈ L∞(Rd), a ≥ 0, a 6≡ 0, let γ satisfy 1 < γ < 1 +4d and define
τ (γ) = 2 −d
2(γ − 1) > 0 . Assume that
either a(x) → 0 as |x| → +∞ or a ∈ Lq
(Rd) for some q ∈ [1, +∞) (2.8)
and for all large |x|,
a(x) ≥ C|x|−τ for some τ < τ (γ) and some constant C > 0. (2.9) Then either there exist λ ∈ (λ0− m, λ0+ l) and u ∈ H1(Rd) such that
−∆u(x) + p(x)u(x) = a(x)|u(x)|γ−1u(x) + λu(x) in Rd, Z
Rd
u2(x) dx = 1 ,
or there is a solution of
−∆u(x) + p(x)u(x) = a(x)|u(x)|γ−1u(x) + (λ
0− m) u(x) in Rd,
satisfying 0 <R
Rdu
2(x) dx ≤ 1 ,
(or both hold simultaneously).
In the unidimensional case we can prove the impossibility of the second case in the above alternative (note that the hypothesis (2.9) on the decay of a(x) at ±∞ is no longer needed):
Theorem 2. Let p ∈ L∞(R) be periodic and piecewise continuous, let a ≥ 0, a 6≡ 0, be continuous and let γ satisfy 1 < γ < 3. Assume that a ∈ L∞(R) and that either a(x) → 0 as |x| → +∞ or a ∈ Lq(R) for some q ∈ [1, +∞). Assume furthermore thatR
R{|x|a(x)}
2/(3−γ)dx < ∞. Then there exist λ ∈ (λ
0− m, λ0+ l)
and u ∈ H1
(R) such that
−u00(x) + p(x)u(x) = a(x)|u(x)|γ−1u(x) + λu(x) in R , Z
R
u2(x) dx = 1.
In the higher dimensional case, a similar result can also be proved if the function a is “small enough”:
Theorem 3. Under the assumptions of Theorem 1, if ||a||Lq(Rd) is small enough,
either for q = ∞ or for some q > 2 1 +4
d − γ
, then there exist λ ∈ (λ0− m, λ0+ l)
and u ∈ H1(Rd) such that
−∆u(x) + p(x)u(x) = a(x)|u(x)|γ−1
u(x) + λu(x) in Rd, Z
Rd
In the case of Equation (1.2), when ˜N (u) = |x|αu − β(|u|2∗ 1
|x|)u , we can prove the
following.
Theorem 4. If p ∈ L∞(Rd) is periodic, d ≥ 3, 0 < β < α and if (1.5) holds, then there exist λ ∈ (λ0, λ0+ l) and u ∈ H1(Rd) such that
−∆u(x) + p(x)u(x) − α |x|u(x) + β(u 2∗ 1 |x|)u = λu(x) in R d, Z Rd u2(x) dx = 1 .
The main new feature of the above theorems is that these are not results “in the small” of the type “bifurcation from the essential spectrum”, although many technical steps are directly inspired by previous works in this field, particularly by [14] and [3, 10, 11, 12]. Our approach is the one developed in [7] for the Dirac-Fock equations based on an un-constrained penalization, a variational Lyapunov-Schmidt reduction [1, 5, 6, 4] and the mountain-pass theorem (see e.g. [9]). In the superlinear case (Equation (1.1)) some com-pactness is assumed in this paper (namely, we assume that at least in a weak sense a is small at infinity). In the case of Equation (1.2) the uniform spectral condition (1.5) is enough to ensure the necessary compactness.
3
Abstract results.
We consider instead of H1
(Rd) a general real Hilbert space H with inner-product h·, ·i and
norm k · k, and the equation
Lu = N (u) + λ Au in H ,
hAu, ui = 1, (3.10)
where L : H → H is a bounded linear self-adjoint operator with bounded inverse (this means that, in the examples, we assume λ0= 0, without loss of generality). The bounded
linear operator A : H → H is self-adjoint, positive definite, but 0 ∈ σ(A) is allowed. Moreover LA = AL.
Let H+ and H− be the eigenspaces corresponding to σ(L) ∩ R+ and σ(L) ∩ R− ,
supposed nontrivial; they satisfy
hLu, ui ≥ δkuk2∀u ∈ H+ and hLu, ui ≤ −δkuk2∀u ∈ H− (3.11)
for some δ > 0. We denote by P and I − P the orthogonal projections on H+ and H−.
Let
l = inf
u∈H+\{0}
hLu, ui
hAu, ui ≥u∈Hinf+\{0}
δkuk2 hAu, ui≥ δ kAk > 0 (3.12) and m = inf u∈H−\{0} −hLu, ui
hAu, ui ≥u∈Hinf−\{0}
δkuk2
hAu, ui≥ δ
kAk > 0. (3.13) Let φ ∈ C2(H, R) be such that
Together with A, we also consider Aµdefined by
Aµu = Au + µLP u − µL(I − P )u and µ > 0.
Our variational method is based on the functional Jr,µ(u) =
1
2hLu, ui − φ(u) − ψr,µ(u), u ∈ Uµ,
where Uµ= {u ∈ H : hAµu, ui < 1}, ψr,µ(u) is a penalization term defined by
ψr,µ(u) = fr(hAµu, ui) with fr(s) =
sr
1 − s (0 ≤ s < 1), and r > 2 is a parameter that will be chosen large (in fact we shall let r → ∞).
Here are some properties of the penalization ψr,µfor u ∈ Uµand v, v1, v2∈ H:
h∇ψr,µ(u), vi = 2fr0(hAµu, ui)hAµu, vi
with fr0(s) = rs r−1 1 − s + sr (1 − s)2 > r sfr(s) > 0 (0 < s < 1), and
hψ00r,µ(u)v1, v2i = 2fr0(hAµu, ui)hAµv1, v2i + 4fr00(hAµu, ui)hAµu, v1ihAµu, v2i
with fr00(s) = r(r − 1)s r−2 1 − s + 2 rsr−1 (1 − s)2 + 2 sr (1 − s)3 > 0 (0 < s < 1).
In particular ψr,µis convex and, for all u ∈ Uµ, h∇ψr,µ(u), ui ≥ r ψr,µ(u).
Next we assume
(C1) For every v ∈ H+such that hAv, vi < 1, the map
H− 3 w 7→
1
2hL(v + w), v + wi − φ(v + w)
has negative definite Hessian at every w ∈ H−such that hA(v + w), v + wi < 1.
Note that w 7→ −Jr,µ(v +w) is coercive on {w ∈ H−, v +w ∈ Uµ}. We now perform
a variational Lyapunov-Schmidt reduction in the spirit of the one in [1, 6, 5]. From (C1) and the convexity of the function ψr,µ, we see that for all v ∈ H+∩ Uµ, there exists a
unique w =: gr,µ(v) ∈ H−such that hAµ(v + gr,µ(v)), v + gr,µ(v)i < 1,
Jr,µ(v + gr,µ(v)) = max{Jr,µ(v + w) : w ∈ H−, hAµ(v + w), v + wi < 1}
and
which is equivalent to
0 = Lgr,µ(v) − (I − P )N (v + gr,µ(v))
− 2fr0 Aµ(v + gr,µ(v)), v + gr,µ(v) Agr,µ(v). (3.15)
By the Implicit Function Theorem, gr,µis of class C1and gr,µ(0) = 0, because the above
equality holds true when v = 0 and gr,µ(v) = 0.
For all r > 2 we can now define a reduced functional Fr,µ: H+∩ Uµ → R by
Fr,µ(v) = Jr,µ(v + gr,µ(v)) = 1 2hLv, vi + 1 2hLgr,µ(v), gr,µ(v)i − φ(v + gr,µ(v)) − fr Aµ(v + gr,µ(v)), v + gr,µ(v) ,
which is of class C1and satisfies ∇H+Fr,µ(v)
(3.14)
= P ∇HJr,µ(v + gr,µ(v)). (3.16)
This shows that Fr,µis in fact of class C2and
Fr,µ00 (v)(v1, v2) = Jr,µ00 (v + gr,µ(v))(v1, v2+ g0r,µ(v)v2) (3.14) = Jr,µ00 (v + gr,µ(v))(v1+ g0r,µ(v)v1, v2+ g0r,µ(v)v2), v1, v2∈ H+. (3.17) Also, by (3.15), L(gr,µ0 (0)v) = (I − P )N0(0)(v + g0r,µ(0)v) + 2fr0(0)Aµ(gr,µ0 (0)v) = (I − P )N0(0)(v + gr,µ0 (0)v), which shows that
gr,µ0 (0)v = {(I − P )(L − N0(0))}−1(I − P )N0(0)v does not depend on r, µ and A, and neither does
Fr,µ00 (0) = (P + gr,µ0 (0))T(L − φ00(0))(P + gr,µ0 (0)).
Indeed they only depend on L and N0(0). As a consequence, the fact that infnhF00
r,µ(0)v, vi | v ∈ H+, ||v|| = 1
o
> 0 (3.18) holds true does not depend on r and µ, and indeed depends only on L and N0(0). Hence (3.18) holds for L and φ if and only if it does for L and for the function H 3 u →
1 2φ
00(0)(u, u) (defined on H, without corresponding A
µ). Thus, we get that
and that (3.18) holds if infh(L − φ00(0))v, vi | v ∈ H+, ||v|| = 1 > 0. Hence, we further assume (C2) infnh(L − φ00(0))v, vi | v ∈ H+, ||v|| = 1 o > 0. Clearly, for all r > 2, µ > 0 and v ∈ H+, lim
t→1−Fr,µ(tv) = −∞ if hAµv, vi = 1.
As a consequence, Fr,µhas a mountain-pass structure:
cr,µ:= inf max t∈[0,1]Fr,µ(γ(t)) : γ ∈ C [0, 1], Uµ∩ H+, γ(0) = 0, Fr,µ(γ(1)) < 0 > 0.
This mountain-pass structure allows us to find a Palais-Smale sequence {vn}n
satisfy-ing
Fr,µ(vn) → cr,µ, ∇Fr,µ(vn) → 0 ,
which we call a mountain-pass critical sequence. If we can prove some compactness for the sequence {vn}n, we will be able to pass to the limit along some subsequence and find
v such that u = v + gr,µ(v) is a solution to
L(u) = N (u) + λr,µAµu ,
with λr,µ= 2 fr0(hAµu, ui).
The functional Fr,µis defined on the bounded set
{v ∈ H+ : hAv, vi + µhLv, vi < 1} (3.11)
⊂ {v ∈ H+ : kvk < (µδ)−1/2}. (3.19)
For an analogous reason, Jr,µtoo is defined on a bounded domain (namely Uµ). If r1≤ r2
and µ1≤ µ2, we get cr1,µ2 ≤ cr2,µ1 because Jr1,µ2 ≤ Jr2,µ1 on Uµ2.
Define c∞ := supr>2, µ>0cr,µ. The main abstract theorem in this paper states the
following.
Theorem 5. In addition to hypotheses (H1), (C1) and (C2), we assume that there is a constant l0< l such that :
(i) for r large enough and µ > 0 small enough, there exists a mountain-pass critical sequence (vn) of Fr,µ at level cr,µ satisfying
lim sup
n→+∞
fr0(hAµun, uni) ≤ l0/2 , where un= vn+ gr,µ(vn) , (3.20)
and such that the sequence (un) is relatively compact in H.
(ii) Any sequence (¯uk) of the form ¯uk = ¯vk+ grk,µk(¯vk) with ¯vk critical point of
Frk,µk at level crk,µk (rk→ ∞, µk → 0), satisfying lim sup k→+∞ fr0 k(hAµku¯k, ¯uki) ≤ l 0/2 , (3.21) is relatively compact in H.
1. either there exists ¯u ∈ H\{0} and ¯λ ∈ [0, l) such that L¯u − N (¯u) = ¯λ A¯u, hA¯u, ¯ui = 1 and 0 < 1
2hL¯u, ¯ui − φ(¯u) = c∞, 2. or there exists ¯u ∈ H\{0} such that
L¯u − N (¯u) = 0, hA¯u, ¯ui < 1 and 0 < 1
2hL¯u, ¯ui − φ(¯u) = c∞, (or both).
Proof of Theorem 5. Step 1. r < +∞ large, µ > 0 small.
We consider a sequence (vn) in the domain of Fr,µ, relatively compact in H, satisfying
(3.20) and such that ∇Fr,µ(vn) → 0 in H+ and limn→∞Fr,µ(vn) = cr,µ. Clearly this
sequence remains away from the boundary of the domain of Fr,µ. We shall use the notation
wn= gr,µ(vn), un= vn+ wn, and λn= 2fr0(hAµun, uni) ,
where hAµun, uni < 1. By assumption (3.20), if r is large enough and µ small enough,
we may assume λn→ λr,µ∈ [0, l0] ⊂ [0, l).
We can apply the relative compactness assumption to infer the existence of a subse-quence of (vn), denoted still by (vn), which converges towards some vr,µ ∈ H+. If we
define ur,µ= vr,µ+ gr,µ(vr,µ), by continuity we know that
∇Fr,µ(vr,µ) = 0 , Fr,µ(vr,µ) = cr,µ, hAµur,µ, ur,µi < 1 , (3.22) λr,µ= 2fr0(hAµur,µ, ur,µi) ∈ [0, l0] ⊂ [0, l) , (3.23) and 0 < 1 2hLur,µ, ur,µi − φ(ur,µ) = cr,µ+ ψr(ur,µ) < cr,µ+ (2r) −1λ r,µ. (3.24) Step 2. rn→ +∞, µn→ 0.
We can use Step 1 to obtain a sequence of critical points for Frn,µn satisfying (3.22),
(3.23), (3.24) and such that (rn) and (µn) are monotone sequences going to ∞ and 0.
We use (3.21) to extract a subsequence, still denoted by (¯urn,µn), converging to some
¯ u ∈ H, which satisfies, L¯u = N (¯u) + ¯λA ¯u , hA¯u, ¯ui ≤ 1 , 0 ≤ ¯λ ≤ l0< l , (3.25) 0 < cr1,µ1 ≤ 1 2hL¯u, ¯ui − φ(¯u) = c∞. (3.26) Now, either hA¯u, ¯ui = 1 or hA¯u, ¯ui < 1. When the latter happens,
0 ≤ ¯λ = lim
n→∞f 0
rn(hA¯urn, ¯urni) ≤ lim sup
n→∞ fr0n 1 + hA¯u, ¯ui 2 = 0 ,
Corollary 6. Assume that, for every s ∈ (−m, l), conditions (C1), (C2) and the assumptions of Theorem 5 remain true if L is replaced everywhere by L − sA and l by l − s, and Fr,µ, cr,µ and gr,µ modified accordingly. Then
1. either there exists u ∈ H\{0} and λ ∈ (−m, l) such that Lu − N (u) = λ Au, hAu, ui = 1, 2. or, for all λ ∈ (−m, l), there exists uλ∈ H\{0} such that
Luλ− N (uλ) = λAuλ, hAuλ, uλi < 1.
Proof. Apply Theorem 5 to the operator L − sA for any s ∈ (−m, l) instead of L.
4
Proofs of Theorems 1, 2 and 3.
In this section we consider a class of problems containing (1.1) as a particular case. We will deal with the general problem (3.10) and make the following hypotheses on N . First, as above, we assume that N = ∇φ for some φ of class C2.
(H2) There exists γ > 1 such that ∀u ∈ H , ∀t ∈ [0, 1] , φ(tu) ≥ tγ+1φ(u),
(H3) φ is convex and ∀u ∈ H , hN (u), ui ≥ 2φ(u),
(H4) ∃κ, ¯α, ¯β ≥ 0 such that ¯α < 2, ¯α + 2 ¯β > 2 and φ(u) ≤ κkukα¯hAu, uiβ¯ for all
u ∈ H,
(H5) N is a compact operator ,
(H6) there exist a sequence (vn) ⊂ H and a constant M > 0 such that
– ∀n ∈ N hAvn, vni = 1,
– (φ(vn)) is bounded,
– ∀n ∈ N k(L − lA)vnk2≤ M |h(L − lA)vn, vni|,
– ∀n ∈ N φ(vn) 6= 0 and lim n→∞
h(L − lA)vn, vni
φ(vn)
= 0.
Hypothesis (H6) was introduced by Heinz, K¨upper and Stuart [10, 11] in their theory of bifurcation from the essential spectrum.
We will then prove an abstract version of Theorem 1. But first consider the following (non penalized) functional:
J∞(u) =
1
2hLu, ui − φ(u)
for u ∈ H. We can, as for Jr,µ, make a variational reduction of J∞, leading to a reduction
map g∞: H+ → H−and to a reduced functional F∞: H+→ H−satisfying
Moreover, for all ˆv ∈ H+\{0},
c∞≤ sup 0≤t<hAˆv,ˆvi−1/2
F∞(tˆv). (4.28)
Theorem 7. Under hypotheses (H1) to (H6), the following alternative holds: 1. there exists u ∈ H\{0} and λ ∈ (−m, l) such that
Lu − N (u) = λAu, hAu, ui = 1, 2. or, for all λ ∈ (−m, l), there exists uλ∈ H\{0} such that
Luλ− N (uλ) = λAuλ, hAuλ, uλi < 1.
In the second case, lim supλ→−mφ(P uλ) > 0 and lim supλ→−mkuλk < ∞.
The second case cannot occur if, in addition, 2α+2 ¯¯ βκkAk1− ¯α/2 < δ, that is, if
the nonlinearity is “small” enough. Finally, if the additional assumption
(H7) If un* u∞ weakly in H, then N (un) * N (u∞) and φ(un) → φ(u∞)
holds and if there is no solution to
Lu − N (u) = λAu, hAu, ui = 1, for any λ ∈ (−m, l), then we can find um∈ H\{0} such that
Lum− N (um) = −mAum, hAum, umi ≤ 1.
Proof. Preliminary remarks.
If hypotheses (H1) to (H6) hold, they also hold when L is replaced by L−sA and l by l − s, for all s ∈ (−m, l). Hence it suffices to check the assumptions of Theorem 5 for s = 0 and then the statement of the alternative is a direct consequence of Corollary 6.
Hypothesis (H3) implies that φ ≥ 0 (because φ(0) = 0 and φ0(0) = 0) and (C1)
is then obvious. Hypothesis (H4) implies that φ00(0) = N0(0) = 0, and thus (C2) holds. Moreover g0r,µ(0) = 0. Indeed
L(g0r,µ(0)v) = (I − P )N0(0)(v + gr,µ0 (0)v) + 2fr0(0)Aµ(g0r,µ(0)v) = 0
and gr,µ0 (0)v = 0 for all v ∈ H+. Therefore Fr,µ00 = P L.
Step 1. Proof of the inequality c∞< l/2.
This result is an easy consequence of results originally due to Heinz, K¨upper and Stuart [10, 11]. Indeed, they proved that hypothesis (H6) implies the existence of a bounded sequence (zn) ⊂ H+ such that
∀n ∈ N hAzn, zni = 1, ∀n ∈ N φ(zn) 6= 0 and lim n→∞
h(L − lA)zn, zni
φ(zn)
See [3] for the proof rewritten with notations close to ours.
Then, we proceed as follows. We first assume that there exists a sequence (ˆvn) ⊂ H+\{0} such that
F∞(ˆvn) <
l
2hAˆvn, ˆvni ∀n ∈ N and n→∞lim kˆvnk = 0,
and prove
c∞<
l
2. (4.30)
Indeed, let us choose n so large that t → F∞(tˆvn) is increasing for 0 ≤ t ≤ 1
and hAˆvn, ˆvni < 1. Hypothesis (H3) implies that t → t−2φ(tx) is non-decreasing
for all x ∈ H, so that, for 1 ≤ t ≤ hAˆvn, ˆvni−1/2, we get φ(tˆvn + g∞(tˆvn)) ≥
t2φ(ˆvn+ t−1g∞(tˆvn)) and F∞(tˆvn) = J∞ tˆvn+ g∞(tˆvn) ≤ t2J∞ ˆvn+ t−1g∞(tˆvn) (4.27) ≤ t2F∞ ˆvn < t2 l 2hAˆvn, ˆvni ≤ l 2. Hence, by the above estimates and (4.28),
c∞≤ sup 0≤t<hAˆvn,ˆvni−1/2
F∞(tˆvn) <
l 2 for some fixed n chosen large enough, so that (4.30) holds true.
We now construct the sequence (ˆvn) from the sequence (zn) given by (4.29) as
follows. We set ˆvn = tnzn with limn→∞tn= 0+, and get for n large enough
F∞(ˆvn) = 1 2hL(ˆvn+ g∞(ˆvn)), ˆvn+ g∞(ˆvn)i − φ(ˆvn+ g∞(ˆvn)) (H3) ≤ 1 2hL(ˆvn+ g∞(ˆvn)), ˆvn+ g∞(ˆvn)i − 2φ(ˆvn/2) + φ(−g∞(ˆvn)) (H2) ≤ 1 2hL(ˆvn+ g∞(ˆvn)), ˆvn+ g∞(ˆvn)i − 2 −γφ(ˆv n) + φ(−g∞(ˆvn)) (3.11),(H4) ≤ 1 2hLˆvn, ˆvni − δ 2kg∞(ˆvn)k 2 − 2−γφ(ˆvn) + κkAk ¯ β kg∞(ˆvn)kα+2 ¯¯ β ¯ α+2 ¯β>2 ≤ 1 2hLˆvn, ˆvni − 2 −γφ(ˆv n) (H2) ≤ t2 n 1 2hLzn, zni − 2 −γt1+γ n φ(zn) see below < l 2t 2 nhAzn, zni = l 2hAˆvn, ˆvni if h(L − lA)zn, zni φ(zn) < 21−γtγ−1n ,
which holds for tn defined by
tn = 2 2γ−1h(L − lA)zn, zni φ(zn) 1/(γ−1) −→ n→+∞ 0.
Step 2. For r > 2 and µ > 0 fixed, existence of relatively compact mountain-pass critical sequences satisfying condition (3.20), as a conse-quence of the inequality c∞< l/2.
Let ξr, τr∈ (0, 1) be such that
c∞= fr0(ξr)ξr− fr(ξr) and 2fr0(τr) = l,
and set dr= fr0(τr)τr− fr(τr). By the properties of fr we have
ξr, τr r→+∞−→ 1 , dr r→+∞−→ l/2 ,
c∞ < dr for r large (because c∞ < l/2), ξr < τr (because s → fr0(s)s − fr(s) is
increasing) and, still for r large,
dr− c∞= fr0(τr)τr− fr(τr) − fr0(ξr)ξr+ fr(ξr) = Z τr ξr fr00(s)s ds ≤ Z τr ξr fr00(s) ds = (l/2) − fr0(ξr) . Hence, by Step 1, lim sup r→+∞ fr0(ξr) ≤ c∞< l/2 . (4.31)
Moreover for µ > 0 we also define σr,µ∈ (0, 1) by cr,µ= fr0(σr,µ)σr,µ− fr(σr,µ).
Let us now consider a (mountain-pass) critical sequence of Fr,µ at level cr,µ,
denoted by (vn). It is well known that such sequences exist at a mountain-pass level.
Defining un = vn+ gr,µ(vn), λn = 2 fr0(hAµun, uni) and adding h∇Fr,µ(vn), vni =
h∇Jr,µ(un), vni and h∇Jr,µ(un), wni = 0 (see (3.14) and (3.16)) , we get that
h∇Fr,µ(vn), vni = h∇Jr,µ(un), uni
= hLun, uni − hN (un), uni − λnhAµun, uni = nkvnk , (4.32)
for some sequence n → 0, because ∇Fr,µ(vn) → 0. By (H3),
λnhAµun, uni − 2fr Aµun, un ≤ λnhAµun, uni − 2fr Aµun, un + hN (un), uni − 2φ(un) (4.32) = hLun, uni − hN (un), uni − nkvnk − 2fr Aµun, un + hN (un), uni − 2φ(un) = 2Jr,µ(un) − nkvnk. It follows that lim n→∞{f 0 r Aµun, un hAµun, uni − fr Aµun, un } ≤ lim n→∞Jr,µ(un) = cr,µ= fr0(σr,µ)σµ− fr(σr,µ) ≤ c∞= fr0(ξr)ξr− fr(ξr).
Thus, by the strict monotonicity of s 7→ fr(s) and of s 7→ fr0(s)s − fr(s), we get
lim sup
n→∞
λn= 2 lim sup n→∞
and by (4.31), lim sup n→∞ λn = 2 lim sup n→∞ fr0(hAµun, uni) ≤ 2c∞< l ,
that is, (3.20) holds true.
The relative compactness of (un) follows from boundedness of (un) (see (3.19)),
the compactness of N (Hypothesis (H5)), the inequalities 0 ≤ lim supn→∞λn ≤
2c∞< l and from
Lun− N (un) − λnAµun→ 0 in H,
provided µ is small enough so that
lim sup n→∞ λn < lµ:= inf u∈H+\{0} hLu, ui hAµu, ui = l 1 + µl .
Step 3. Compactness holds for critical sequences satisfying (3.21). Let rn> 2, µn > 0 be such that (rn, µn) converges to (∞, 0) and let (vn) be a
sequence such that
Frn,µn(vn) = crn,µn, ∇Frn,µn(vn) = 0 (∀n ∈ N).
Note that the domain of Frn,µn depends on n.
Define now un= vn+ grn,µn(vn) and λn= 2f
0
rn(hAµnun, uni). We have
Lun= N (un) + λnAµnun, λn≤ l
0< l . (4.34)
Setting vn= P un and wn= (I − P )un, we obtain
δkunk2− l (3.11) ≤ hLvn, vni − hLwn, wni − λnhAµnvn, vni + λnhAµnwn, wni = h(L − λnAµn)(vn+ wn), vn− wni = hN (un), vn− wni = hN (un), 2vn− uni (H3) ≤ φ(2vn) − φ(un) (H4) ≤ 2α+2 ¯¯ βκku nkα¯
with 0 ≤ ¯α < 2. Hence, since ¯α < 2, we have the useful estimate lim sup
n→+∞
kunk ≤ C, where C depends only on l, δ, ¯α, ¯β, κ .
The relative compactness of (un) follows, as previously, from (4.34), the
com-pactness of N , the inequalities 0 ≤ lim supn→∞λn< l and µn → 0.
Step 4. End of the proof of Theorem 7.
The first part of the theorem is a direct consequence of Theorem 5, Corollary 6 and Steps 1 and 2.
Let us assume that the second case of the alternative occurs. Writing wλ =
(I − P )uλand vλ= P uλ, we get lim supλ→−mφ(2vλ) > 0 by the following standard
argument. If not, we would have for λ ∈ (−m, 0) δkvλk2+ (λ + m)hAwλ, wλi (3.11),(3.13) ≤ hLvλ, vλi + λhAwλ, wλi − hLwλ, wλi λ<0 ≤ hLvλ, vλi + λhAwλ, wλi − hLwλ, wλi − λhAvλ, vλi = hL(vλ+ wλ), vλ− wλi + λhA(wλ+ vλ), wλ− vλi = hN (uλ) + λAuλ, vλ− wλi + λhAuλ, wλ− vλi = hN (uλ), vλ− wλi = hN (uλ), 2vλ− uλi (H3) ≤ φ(2vλ) − φ(uλ) ≤ φ(2vλ) (H4) ≤ 2α+2 ¯¯ βκhAvλ, vλi ¯ βkv λkα¯ ≤ 2α+2 ¯¯ βκkAkβ¯kvλkα+2 ¯¯ β,
and limλ→−mkvλk = 0 (because limλ→−mφ(2vλ) = 0 by assumption). Hence
δkvλk2+ (λ + m)hAwλ, wλi ≤ 2α+2 ¯¯ βκkAk ¯ βkv λkα+2 ¯¯ β ≤ δ 2kvλk 2
for all λ near enough to −m, which leads to the contradiction vλ= wλ= uλ= 0.
Analogously, we get δkuλk2− m = δkvλk2+ δkwλk2− m ≤ hLvλ, vλi − hLwλ, wλi + λhAwλ, wλi ≤ hLvλ, vλi − hLwλ, wλi + λhAwλ, wλi − λhAvλ, vλi ≤ 2α+2 ¯¯ βκhAvλ, vλi ¯ β kvλkα¯ ≤ 2α+2 ¯¯ βκkuλkα¯
and therefore lim supλ→−mkuλk < ∞ (thanks to ¯α < 2).
We also obtain
δkvλk2+ (λ + m)hAwλ, wλi ≤ 2α+2 ¯¯ βκkAk1− ¯α/2kvλk2
and the same kind of argument shows that the second case of the alternative cannot occur if
2a+2 ¯βκkAk1− ¯α/2< δ.
Finally, to prove the last statement in the theorem, that is, the passage λ → −m in the second case of the above alternative, we first note that lim supλ→−mkuλk <
∞ and that there exists a sequence λn → −m and u ∈ H such that uλn * u,
Luλn * Lu, Auλn * Au. Hence from (H7), N (uλn) * N (u), φ(uλn) → φ(u),
Lu − N (u) = −mAu and hAu, ui ≤ 1. If u = 0, then P uλn* 0 and φ(2P uλn) → 0,
We will now deal with the particular equation (1.1) and prove what we stated in the introduction.
Proof of Theorem 1. Although Theorem 1 is stated for d ≥ 2, let us also mention the case d = 1. In (2.6), we suppose without loss of generality that the spectrum σ(−∆ + p) has a gap (−m, l) 3 0, that is, l, m > 0, σ(−∆ + p) does not intersect (−m, l) and {l, −m} ⊂ σ(−∆ + p). This can be achieved by replacing p by p − λ0
(see the Introduction for the meaning of λ0).
We set H = H1
(Rd) endowed with the inner-product defined in (2.7),
hLu, viH1:=
Z
Rd
(∇u · ∇v + puv) dx and hAu, viH1:=
Z
Rd
uv dx ∀u, v ∈ H,
which defines L : H → H and A : H → H uniquely. We also set
φ(u) = 1 1 + γ
Z
Rd
a(x)|u(x)|1+γdx ∀u ∈ H.
Since a ∈ L∞(Rd), hypothesis (H4) is satisfied provided that γ < 3, if d = 1, 2,
or γ < 1 + 4d, if d ≥ 3. Note that for d ≥ 2 and any q > 2
1 + 4d− γ , the condition a ∈ Lq
(Rd) implies (H4) with constants ¯α, ¯β, κ depending on q. (H5) holds because
of (2.8). (H2) and (H3) also hold true.
By results of Heinz, K¨upper and Stuart [10, 11], hypothesis (H6) holds when d ≥ 2 because, as assumed in the theorem, for |x| large, a(x) ≥ C|x|−τ, for some
τ < τ (γ), τ (γ) = 2 − d
2(γ − 1). Note that for d = 1 there is no assumption on
the speed of decay of a at infinity, but p is assumed piecewise continuous and a continuous (see condition (C4), Corollary 5.3 and Theorem 5.4 in [14]). Proof of Theorem 2. In the case d = 1, by a theorem of Rofe-Beketof [16, 13], the equation
−u00(x) + {p(x) + m}u(x) − a(x)|u(x)|γ−1
u(x) = 0 in R , 0 < Z
Rd
u2(x) dx ≤ 1 (4.35) seen as a linear equation for u in which a(x)|u|γ−1 is a coefficient, has no solution if
Z
R
|x| a(x) |u(x)|γ−1dx < ∞. (4.36)
Since |u|γ−1 ∈ L2/(γ−1)
(R), (4.36) holds if γ < 3 and | · |a(·) ∈ L2/(3−γ)
(R). See Theorem 3.2 in [14], where (4.36) is used in an analogous discussion. Proof of Theorem 3. This is a direct consequence of Theorem 7: when kakLq(Rd)
is small enough, then so is 2α+2 ¯¯ βκkAk1− ¯α/2 (see the remark made in the proof of
5
Proof of Theorem 4.
In order to prove this theorem, we notice that the assumptions (C1) and (C2) are implied by (1.5). Indeed remember that
φ(u) = α 2 Z Rd |u|2 |x| dx − β 4 Z Z Rd×Rd u2(x) u2(y) |x − y| dx dy, so that φ00(u)(z, z) = α Z Rd |z|2 |x| dx − β Z Z Rd×Rd u2(x) z2(y) |x − y| dx dy − 2β Z Z Rd×Rd u(x)z(x) u(y)z(y) |x − y| dx dy. Also note that
Z Z Rd×Rd u(x)z(x) u(y)z(y) |x − y| dx dy ≤ Z Z Rd×Rd u2(x) z2(y) |x − y| dx dy Z Z Rd×Rd z2(x) u2(y) |x − y| dx dy 1/2 = Z Z Rd×Rd u2(x) z2(y) |x − y| dx dy. Next, we will prove that the other assumptions of Theorem 5 hold true. For sim-plicity, we assume λ0= 0.
First Step : Condition (3.20) holds true.
Let (r, µ) ∈ (2, +∞) × [0, 1) and let (vn) be a sequence in H+ such that
∇Fr,µ(vn) → 0, Fr,µ(vn) → cr,µ.
Assume, moreover, that there is a sequence δn→ 0 such that the quadratic form
H+3 z → Fr,µ00 (vn) · z2+ δn||z||2H1(Rd)
has a negative space of dimension at most 1. It is known that such mountain-pass critical sequences with “Morse-type information” exist, by Theorems 4.1 and 6.2 in [9]. Let us set
un = vn+ gr,µ(vn), λn= 2fr0(hAµun, uni).
We may assume, after extraction, that (λn) has a limit λ∗. This limit is finite, since
for r, µ fixed the sequence (un) stays away from the boundary of Uµ. Since (un)
is bounded in H1 (see (3.19)), we may also assume, after a new extraction, that
this sequence has a weak limit u∗ in H1. Using the information ∇Fr,µ(vn) →n 0,
one easily shows that u∗ is in fact a limit for the Hloc1 topology. Then u∗ solves
Our goal now is to study the second derivative Fr,µ00 (vn).
Following [11], let us consider a function ψ ∈ H2
loc(Rd) ∩ C1(Rd) which is a
nontrivial solution of −∆u + pu = lu. We can choose it to be uniformly almost-periodic in the sense of Besicovich [2], and such that ψ2 is periodic. Then we consider f ∈ C0∞(Rd) such that ||f ||L2(Rd) = 1 and for every σ > 0 we define
kσ(x) := σ−N/2f (xσ) ψ(x) .
By the Riemann-Lebesgue lemma (see [11]), we have ((−∆ + p(x) − l)kσ, kσ)L2(Rd) = Z Rd {σ−2∆f (x)ψ(σx) + 2σ−1∇f (x) · ∇ψ(σx)}f (x)ψ(σx)dx = Z Rd σ−2f (x)∆f (x)ψ2(σx) + (1/2)σ−2∇(f2(x)) · ∇(ψ2(σx))dx = Z Rd σ−2f (x)∆f (x)ψ2(σx) − (1/2)σ−2∆(f2(x))ψ2(σx)dx = σ−2M (ψ2) Z Rd f (x)∆f (x) − (1/2)∆(f2(x))dx + o(σ−2) = O 1 σ2 ,
where M (ψ2) denotes the mean-value of ψ2. By the same computations,
||(−∆ + p(x) − l)kσ||L2(Rd)= O(σ−1) . (5.37)
So, setting P kσ:= k+σ and (I − P )kσ:= kσ−, we get
Const ||kσ−||2
H1(Rd)≤ ((−∆+p−l)kσ, kσ−2kσ−)L2(Rd)≤ O(σ−2)+O(σ−1)kk−σkH1(Rd),
and therefore
||k−σ||H1(Rd)= O(1/σ) . (5.38)
Now, for any z ∈ H1, we can estimate J00
r,µ(un) · (z)2 from above: Jr,µ00 (un) · (z)2≤ −∆ + p(x) − α |x| z, z + β Z Z Rd×Rd u2n(x) z(y)2 |x − y| dx dy +2β Z Z Rd×Rd un(x)z(x) un(y)z(y) |x − y| dx dy − λnhAµz, zi . The second member of this inequality is just (z, Lz − N0(un)z − λnAµz)H. If z
is fixed, then one easily proves that (z, Lz − N0(u∗)z − λ∗Aµz)H is the limit of
(z, Lz − N0(un)z − λnAµz)H as n goes to infinity.
Some tedious but straightforward computations show that as σ goes to +∞, Z Rd α |x||kσ| 2dx = α σM (ψ 2)Z Rd |f |2 |x| dx + o 1 σ ,
Z Z Rd×Rd u2 ∗(x) k2σ(y) |x − y| dx dy = Z Rd Z Rd k2 σ(y) |x − y|dy u2∗(x) dx = σ−1 Z Rd Z Rd f2(y)ψ2(σy) |x/σ − y| dy u2∗(x) dx ≤ 1 σM (ψ 2) Z Rd |f |2 |x| dx + o 1 σ ||u∗||2L2(Rd), and again by (5.37), ((−∆ + p(x) − l)kσ+, k+σ)L2(Rd)≤ ((−∆ + p − l)kσ, kσ− 2kσ−)L2(Rd)= O 1 σ2 .
Now, we want to compute Fr,µ00 (un) · (k+σ)
2. By (3.17),
Fr,µ00 (vn) · (kσ+) 2= J00
r,µ(un) · (k+σ + g0r,µ(vn)kσ+))
2. (5.39)
In order to estimate from above the r.h.s. of the above equality, let us show that ||g0
r,µ(vn)k+σ||H1(Rd) is also small when σ is large.
By (3.15), we have
L(gr,µ0 (vn)k+σ)−(I −P )N0(un)(k+σ+g0r,µ(vn)kσ+)−2fr0(hAµun, uni)Aµ(gr,µ0 (vn)k+σ)
− 4fr00(hAµun, uni)hAµun, k+σ + g 0
r,µ(vn)kσ+iAµgr,µ(vn) = 0.
Assuming that < Aµvn, kσ >= 0, the above equality together with (1.5) and the
properties of frgive Constkg0r,µ(vn)kσ+k 2 H1(Rd) ≤ −hL(gr,µ0 (vn)k+σ), g0r,µ(vn)kσ+i + hN0(un)g0r,µ(vn)kσ+, g0r,µ(vn)kσ+i = −hN0(un)k+σ, g 0 r,µ(vn)k+σi − 2f 0 r(hAµun, uni)hAµ(gr,µ0 (vn)k+σ), g 0 r,µ(vn)kσ+i
− 4fr00(hAµun, uni)hAµun, k+σ + g0r,µ(vn)kσ+ihAµgr,µ(vn), gr,µ0 (vn)k+σi
≤ −hN0(un)k+σ, g 0
r,µ(vn)k+σi.
Now, using the fact that since by (5.38), ||kσ− k+σ||H1(Rd) = O(σ−1) we easily
see that |hN0(un)k+σ, g 0 r,µ(vn)k+σi| ≤ O(σ−1)||gr,µ0 (vn)k+σ||L2(Rd)+ |hN0(un)|kσ|, |gr,µ0 (vn)k+σi| ≤ O(σ−1) +Const σ ||ψ||L∞(Rd)||∇f ||L2(Rd) ||g0r,µ(vn)kσ+||L2(Rd) = O(σ−1) ||gr,µ0 (vn)kσ+||L2(Rd). (5.40)
Hence, putting together the two above inequalities, we get
Finally, in order to estimate Jr,µ00 (un) · (kσ++ gr,µ0 (vn)kσ+))
2, we use Hardy
in-equality, (5.38) and (5.41) to obtain the following Z Rd α |x||k + σ + g 0 r,µ(vn)kσ+| 2dx Z Rd α |x||kσ− k − σ + g 0 r,µ(vn)k+σ| 2dx = Z Rd α |x||kσ| 2dx− 2Z Rd α |x|kσ(k − σ− gr,µ0 (vn)k+σ) dx + Z Rd α |x||k − σ− g0r,µ(vn)kσ+| 2dx = Z Rd α |x||kσ| 2dx + 2O s Z Rd α |x||kσ| 2dx Z Rd 2α |x|(|k − σ|2+ |g0r,µ(vn)kσ+|2) dx ! + Z Rd 2α |x|(|k − σ| 2+ |g0 r,µ(vn)k+σ| 2) dx = α σM (ψ 2) Z Rd |f |2 |x| dx + o 1 σ , Z Z Rd×Rd u2 n(x) kσ+(y)2 |x − y| dx dy ≤ 1 σM (ψ 2) Z Rd |f |2 |x| dx + o 1 σ ||u∗||2L2+ o(1)n→∞, Z Z Rd×Rd un(x)k+σ(x) un(y)k+σ(y) |x − y| dx dy = Z Z Rd×Rd u∗(x)k+σ(x) u∗(y)k+σ(y) |x − y| dx + o(1)n→∞, Z Z Rd×Rd u2n(x) k−σ(y)2 |x − y| dx dy = O(σ −2) , and Z Z Rd×Rd u2 n(x) (gr,µ0 (v)kσ+)(y) 2 |x − y| dx dy = O(σ −2) .
Moreover we may split the integral Z Z
Rd×Rd
u∗(x)kσ(x) u∗(y)kσ(y)
|x − y| dx dy as a sum of two integrals
I1 = Z Z Rd×Rd u∗(x)kσ(x)1|x|≤√σu∗(y)kσ(y) |x − y| dx dy ≤ ( Z Z Rd×Rd k2 σ(x)1|x|≤√σu2∗(y) |x − y| dx dy Z Z Rd×Rd k2 σ(y) u2∗(x) |x − y| dx dy )1/2 ≤ Const σ n k∇f k2 L2(Rd)kkσ1|x|≤√σkL2(Rd)kkσkL2(Rd) o1/2 ku∗k2L2(Rd)
and I2 = Z Z Rd×Rd u∗(x)kσ(x)1|x|>√σu∗(y)kσ(y) |x − y| dx dy ≤ ( Z Z Rd×Rd k2 σ(x) u2∗(y) |x − y| dx dy Z Z Rd×Rd k2 σ(y)1|x|>√σu2∗(x) |x − y| dx dy )1/2 ≤ Const σ−1k∇f kL2(Rd)kkσkL2(Rd) Z Rd u2∗(y)dy Z Rd 1|x|>√σu 2 ∗(x)dx 1/2
it is easy to see that each of these integrals is o(σ−1). As a consequence, Z Z Rd×Rd un(x)kσ+(x) un(y)kσ+(y) |x − y| dx dy = o(σ −1) + o(1) n→∞ .
The above estimates show that, for large enough σ, lim sup n→+∞ Jr,µ00 (un) · (kσ++ g 0 r,µ(vn)k+σ) 2 ≤ l||k+ σ|| 2 L2(Rd)− (α − βR Rd|u∗| 2dx) σ M (ψ 2) Z Rd |f |2 |x| dx − λ∗hAµk + σ, k + σi + o 1 σ ≤ l − (α − β R Rd|u∗| 2dx) σ M (ψ 2)Z Rd |f |2 |x| dx + o 1 σ − λ∗ 1 + O(σ−2) .
The r.h.s. of the above inequality is negative if σ is large enough and
λ∗> l −
¯
C(α, β, ψ, f ) σ , with ¯C(α, β, ψ, f ) > 0.
We finally extend this method to a 3-dimensional space X by considering ad-ditional functions ˜f , ¯f such that f, ˜f , ¯f generate a 3-dimensional space. Choosing σ large enough, we find a constant l0 < l, which depends only on ¯σ, α, β, f, ˜f , ¯f and ψ, such that the inequality λ∗ > l0 would imply that the negative space of
Fr,µ00 (vn) + δn has dimension at least 2 for µ small and r and n large, which
contra-dicts our assumptions (the dimension being reduced from 3 to 2 because we assumed < Aµvn, kσ>= 0). So we have proved (3.20) by contradiction.
Note that this kind of argument, relating the Morse index and the location of a nonlinear eigenvalue in a Hartree-Fock equation, was first used by P.L. Lions [15].
Second Step : Compactness holds for critical sequences satisfying (3.20) or (3.21).
Let (rn, µn) → (r, µ) ∈ (2, ∞] × [0, 1] and let (vn) be a sequence satisfying
Fr0n,µn(vn) → 0 and
lim sup
n→∞
with un = vn+grn,µn(vn). Define also wn = grn,µn(vn) and λn= 2f
0
rn(hAµnun, uni).
Thus, (un) ⊂ H = H1(Rd) is a sequence such that
R Rd|un| 2dx ≤ 1, and −∆ + p(x) − α |x|+ β u2n∗ 1 |x| un= λnun+εn, ||εn||H−1(Rd)→n 0 . (5.43)
To see that (un) is bounded in H, multiply (5.43) by un and integrate, to get
Z Rd |∇un|2dx ≤ (kpkL∞(Rd)+ λn) Z Rd u2ndx + α Z Rd |x|−1u2ndx + ||εn||H−1(Rd)||un||H1(Rd)
≤ Const ||un||2L2(Rd)+ Const kukH1(Rd)||un||L2(Rd)+ ||εn||H−1(Rd)||un||H1(Rd).
As (un) is bounded in L2(Rd), it is therefore also bounded in H1(Rd).
It is important to note that the operator u → |x|−1u is compact from H1
(Rd) to
its dual H−1(Rd). Moreover u → (u2∗ |x|−1)u sends weakly convergent sequences
in H1
(Rd) to weakly convergent sequences in H−1
(Rd) with corresponding limits
(this is a consequence of the Hardy inequality and the presence of |x|−1).
Extracting a subsequence if needed, we can assume that un * ¯u weakly in
H1(Rd) for some ¯u ∈ H1(Rd) and λn → ¯λ for some ¯λ ∈ [0, l). By the previous
remarks, we get −∆ + p(x) − α |x|+ β ¯ u2∗ 1 |x| ¯ u = ¯λ ¯u, Z Rd ¯ u2dx ≤ 1. Hence, setting ¯un = un− ¯u, −∆ + p(x) − α |x|− λn un− −∆ + p(x) − α |x|− ¯λ ¯ u = εn− β u2n∗ 1 |x| un+ β ¯ u2∗ 1 |x| ¯ u and −∆ + p(x) − α |x|+ β ¯u 2 n∗ |x| −1 − ¯λu¯ n= (λn− ¯λ)un+ εn − β ¯ u2∗ 1 |x| ¯ un− β 2(¯unu) ∗¯ 1 |x| ¯ un− β ¯ u2n∗ 1 |x| ¯ u − β 2(¯unu) ∗¯ 1 |x| ¯ u.
Note that the right-hand member converges to 0 strongly in H−1(Rd), so that −∆ + p(x) − α |x|+ β ¯u 2 n∗ |x| −1 − ¯λu¯ n→ 0 (5.44) strongly in H−1(Rd).
Define the operator Ln: H1(Rd) → H−1(Rd) by Ln= −∆ + p(x) − α |x|+ β ¯u 2 n∗ |x|−1 − ¯λ .
Equation (5.44) can be written Lnu¯n→ 0 (strongly in H−1(Rd)).
Now, by (1.5), the map
w ∈ H−7→ Gn(w) := (¯un+ w, Ln(¯un+ w))L2
is strictly concave. So, denoting ¯vn= P ¯un, ¯wn= (I − P )¯un,
Gn(− ¯wn) ≤ Gn(0) − G0n(0) · ¯wn≤ 3 ||¯un||H1(Rd)||Lnu¯n||H−1(Rd).
Hence, lim supnGn(− ¯wn) ≤ 0. As Gn(− ¯wn) = (¯vn, Lnv¯n)L2 and |x|−1¯vnconverges
strongly to 0 in H−1(Rd), we have lim sup n ¯ vn, (Ln+ α|x|−1)¯vn L2 → 0. As, by (1.5), ¯ vn, (Ln+ α|x|−1)¯vn L2≥ h0||¯vn|| 2 H1(Rd)
for some h0> 0, we deduce ¯vn → 0 strongly in H1(Rd) and ( ¯wn, Lnw¯n)L2 → 0.
But again by (1.5),
( ¯wn, Lnw¯n)L2 ≤ −h0|| ¯wn||2H1(Rd)
for some h0> 0. Hence, k ¯wnkH1(Rd)→ 0 and
||¯un||H1(Rd)= ||un− ¯u||H1(Rd) −→ n→+∞ 0 .
Last Step.
Let us now check that the second case of the alternative in Theorem 5 does not occur and that, in Theorem 4, λ > λ0 with λ0 = 0 (without loss of generality).
Define the operator ˜Lu: H1(Rd) → H−1(Rd) by
˜ Lu= −∆ + p(x) − α |x|+ β u 2∗ |x|−1 for u ∈ H1 (Rd) such that 0 <R Rdu 2dx ≤ 1, assume that ˜L
uu = 0 and let us look
for a contradiction. By (1.5), the map w ∈ H− 7→ Gu(w) :=
u + w, ˜Lu(u + w)
L2
is strictly concave. Letting v = P u, w = (I − P )u, we get
Gu(−w) ≤ Gu(0) − G0u(0) · w ≤ 3 ||u||H1(Rd)|| ˜Luu||H−1(Rd)= 0
and Gu(−w) ≤ 0. As Gu(−w) = (v, ˜Luv)L2, we get v = 0, thanks to (1.5). Hence
w, ˜Luw
L2 = 0 and w = 0 (by (1.5) again), which is in contradiction with u 6= 0.
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