Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part II A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I A BSTRACT. Author: Please supply.
1. I NTRODUCTION AND S TATEMENT OF THE M AIN R ESULTS The main goal of this paper is to complete the study, announced in [5] and initiated in [6], on positive radial solutions concentrating on spheres to singularly perturbed elliptic problems. While nonlinear Schr¨odinger (NLS) equations in Rn was the theme of Part I, [6], here we shall focus on Neumann and Dirichlet boundary-value problems on bounded domains. More precisely, letting Ω ⊆ Rn , n ≥ 2, denote either B1 = {x ∈ Rn : |x| < 1}
or A = {x ∈ Rn : a < |x| < 1},
where a ∈ (0, 1), we consider the following problems 2 p −ε ∆u + V (|x|)u = u
(N)
∂u = 0 on ∂Ω, ∂ν −ε2 ∆u + V (|x|)u = up
(D)
u = 0 P
on ∂Ω,
in Ω, u > 0 in Ω;
in Ω, u > 0 in Ω,
where ∆ = ni=1 ∂ 2 /∂xi2 is the usual Laplace operator, V : Rn → R is a radial potential, p > 1, and ν is the unit outward normal to Ω.
2 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I When V ≡ 1, problem (N) is equivalent to the following system:
(S)
Up d1 ∆U − U + q = 0 ξ Z Ur = ξ s+1 |Ω| Ω ∂U =0 ∂ν
in Ω, in (0, +∞), on ∂Ω × (0, +∞),
where |Ω| is the measure of Ω, and p , q, r , s are non-negative satisfying 0
0. As in Part I, it turns out that the auxiliary potential M(r ) = r n−1 V θ (r ),
where θ =
p+1 1 − , p−1 2
plays a crucial role in the concentration phenomena. Let us recall the main result of Part I. Theorem 1.1. Let (V1)-(V2) hold, let p > 1, and suppose that M has a point of local strict maximum or minimum at r = r¯ . Then, for ε > 0 small enough, the equation (1.1)
−ε2 ∆u + V (|x|)u = up ,
u > 0, u ∈ H 1 (Rn ),
has a radial solution which concentrates near the sphere |x| = r¯ , in the sense that ∀ δ > 0, ∃ ε0 > 0 and R > 0 such that (1.2)
uε (r ) ≤ δ,
for ε ≤ ε0 , and |r − r¯ | ≥ εR.
Conversely, suppose that, for all ε > 0 small, (1.1) has a radial solution uε concentrating near the sphere |x| = r¯ . Then uε has a unique maximum at r = rε , rε → r¯ and M 0 (r¯ ) = 0.
4 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I The goal of this Part II is to highlight that for problems like (N) and (D) new phenomena, which are created by the imposed boundary conditions, will take place. Below, we use the notation a ∼ b if the quantities a and b are of the same order when ε → 0. Our main result on (N) is the following theorem. Theorem 1.2. Let (V1)-(V2) hold, p > 1, and let Ω ⊆ Rn be the ball B1 (resp. the annulus A). Suppose that the function M satisfies the condition (1.3)
M 0 (1) > 0
(resp. M 0 (a) < 0).
Then there exists a family of radial solutions uε of (N) concentrating on |x| = rε , where rε is a local maximum for uε such that 1 − rε ∼ ε| log ε| (resp. rε − a ∼ ε| log ε|). Note that in Theorems 1.1, 1.2 there is no restriction on p other than p > 1, see Remark 1.10. A relevant particular case of Theorem 1.2 is the following. Corollary 1.3. Let p > 1, and consider the problem ˜ (N)
−ε2 ∆u + u = up , ∂u = 0 on ∂B1 , ∂ν
x ∈ B1 , u > 0.
˜ concentrating on |x| = rε . Then there exists a family of radial solutions uε of (N) where rε is a local maximum point of uε for which 1 − rε ∼ ε| log ε|. If one is willing to sacrifice the information concerning the location of the concentration set |x| = rε in Theorem 1.2 above, a more general existence result is in order. Theorem 1.4. Suppose that Ω = B1 (resp. Ω = A), p > 1, and that V : B1 → R (resp. V : A → R) satisfies the assumptions (V1)-(V2). Then problem (N) admits a family of solutions concentrating on a sphere in the interior of B1 . The radii of concentration in Theorem 1.4 may in general vary with ε. Requiring concentration at a fixed set, one should pass to a subsequence. The counterparts of Theorems 1.2 and 1.4 for the Dirichlet case (D) may be stated as follows. Theorem 1.5. Let Ω ⊆ Rn be the ball B1 (resp. the annulus A). Suppose that the function M satisfies the condition (1.4)
M 0 (1) < 0
(resp. M 0 (a) > 0).
Then there exists a family of radial solutions uε of (D) concentrating on |x| = rε , where rε is a local maximum for uε , such that 1 − rε ∼ ε| log ε| (resp. rε − a ∼ ε| log e|).
Singularly Perturbed Elliptic Equations with Symmetry II
5
Theorem 1.6. Suppose that Ω = A, p > 1, and that V : A → R satisfies the assumptions (V1)-(V2). Then problem (D) admits a family of solutions concentrating on a sphere. Remark 1.7. (i) It seems interesting to note that inequalities concerning
M 0 (1) and M 0 (a) in Theorems 1.2 and 1.5 are reversed. See Remark 5.1.
(ii) The solution obtained in Theorem 1.2 is different from the one obtained in [20] as the latter concentrates precisely on the boundary ∂Ω. Anyway, our method can be easily adapted to achieve also this kind of solutions, see Remark 3.5. (iii) The counterpart of Theorem 1.4 for problem (D) holds only for annulus. Indeed, in the case of problem (D) in the unit ball with V ≡ 1, the only solution is the spike at the origin (for p subcritical) by the results in [12] and [18]. For V ≡ 1, and for more general nonlinearities, Theorem 1.6 has been proven in [9] using ODE techniques. In some sense, the solutions obtained in Theorems 1.2 and 1.5 are created by the boundary effect. It is possible to obtain solutions concentrating on spheres near |x| = r¯ , where r¯ is a local extreme of M , in the same spirit of Theorem 1.1, as our next result shows. Theorem 1.8. Let (V1)-(V2) hold, p > 1, and suppose that M has a point of strict local maximum or minimum at r = r¯ . Then, for ε > 0 small enough, (N) has a radial solution which concentrates near the sphere |x| = r¯ . The same statement is true for problem (D). Remark 1.9. The hypothesis of the strictness above can be relaxed as follows. Assume the existence of an interval Λ for which sup M > sup M Λ
∂Λ
(resp. inf M < inf M). Λ
∂Λ
Then, passing to a subsequence, one would obtain concentration at a radius r¯ ∈ Λ. Using this fact and Theorems 1.2, 1.5, under the additional assumptions M 0 ≠ 0 at ∂Ω, one could prove the existence of solutions concentrating on a sphere bypassing Theorems 1.4 and 1.6. Remark 1.10. In all the above results, we do not impose any upper bound on the exponent p . We note that, in the case of spike-layers, the sub-criticality condition p < (n + 2)/(n − 2) is necessary (at least to get uniformly bounded solutions, which is stronger than (1.2)), see [8] for the case of NLS. For the critical case, a blow-up phenomenon may take place, see e.g. [1], [2], [26] and references therein. We do not discuss this phenomenon here, which might occur in higher codimension, see the comments below. The above results could be extended to the case of concentration at spheres of higher codimension. For example, one has the following result.
6 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I Theorem 1.11. Let (V1)-(V2) hold, p ∈ (1, (n − k + 2)/(n − k − 2)) (p > 1 if n − k ≤ 2), and let Ω = B1 ⊆ Rn . For k integer, 1 ≤ k ≤ n − 1, let x = (x 0 , x 00 ), x ∈ Rn , x 0 ∈ Rk+1 , x 00 ∈ Rn−k−1 , and define the function Mk (r ) = r k V θk (r ),
θk =
p+1 1 − (n − k). p−1 2
Suppose Mk satisfies the condition Mk0 (1) > 0.
(1.5)
Then there exists a family of solutions uε of (N), radial in the variable x 0 , concentrating near at the set {|x 0 | = rε < 1, x 00 = 0}, with 1 − rε ∼ ε| log ε|. The upper bound on the exponent p in the above theorem means that p is sub-critical with respect to Rn−k , and provides existence of solutions for the limit problem p −∆Uλ,k + λ2 Uλ,k = Uλ,k
(1.6)
in Rn−k ,
Uλ,k > 0, Uλ,k ∈ H 1 (Rn−k ).
In particular, for k = n − 1, n − 2, no upper bound on p is assumed. Our approach mainly relies upon a finite dimensional reduction which is related to the procedure introduced in [3, 4] and used also in [6] for concentration on spheres for NLS in Rn . In particular, here we take the advantage of many technical details in [6]. Let us give, for example, an outline of the arguments in the case of equation (N) (for Ω = B1 ) whose solutions, after a rescaling, are critical points of the functional Iε (u) =
1 2
Z 1/ε 0
r n−1 [(u0 )2 + V (εr )u2 ] dr −
1
Z 1/ε
p+1 0
r n−1 |u|p+1 dr , u ∈ Hr1 .
Here Hr1 denotes the space of radial H 1 functions. More precisely, following the same arguments carried out in [6], we look for critical points of the form zρ + w , where zρ is in a suitable family Z N of approximate solutions of (N). See the precise definition in Subsection 3.1. The general abstract procedure establishing the equation Iε0 (zρ + w) = 0 is equivalent to (i) finding w = wε,ρ ∈ (Tz Z N )⊥ such that P Iε0 (zρ + w) = 0, and (ii) finding a stationary point of Φε (ρ) := Iε (zρ + wε,ρ ), where P denotes the orthogonal projection onto W = (Tz Z N )⊥ . As for the former step (i), the invertibility of P Iε00 (zρ ) allows us to find the solution w = wε,ρ as a
Singularly Perturbed Elliptic Equations with Symmetry II
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fixed point of a map Sε . An important fact is that Sε turns out to be a contraction in a suitable set Cε (see (2.12)) such that any w ∈ Cε is L∞ uniformly bounded. It is this fact that enables us to handle critical or supercritical exponents p , via a truncation argument. In order to accomplish the procedure by using the latter step (ii) we need to study Φε . Roughly, to find stationary points of Φε we have to balance three terms. The first two are, like in the problems discussed in Part I, a volume energy and an energy due to the potential V . The third one is the boundary effect, created by the boundary conditions. Quantitatively, these three parts are measured by the following expansion εn−1 Φε (ρ) = M(ερ)[α − βe−2λ(1/ε−ρ) ] + h.o.t.,
λ2 = V (ερ),
for some positive constants α, β. With respect to NLS equations discussed in Part I, the new feature here is the additional term −βe−2λ(1/ε−ρ) M(ερ) which has a striking effect near the boundary r = 1. Actually, if M has a maximum or a minimum far from the boundary r = 1, then this effect is negligible, and the dominating term in the expansion is M(ερ) (hence Theorem 1.8 follows). On the other hand, if M 0 (1) > 0, then Φε has a maximum near r = 1 and this yields Theorem 1.2. Finally, in the case of Dirichlet boundary conditions, let us consider the corresponding Euler functional Jε , see Section 4. The reduced functional Ψε (ρ) := Jε (zρ + wε,ρ ) is given by εn−1 Ψε (ρ) = M(ερ)[α + βe−2λ(1/ε−ρ) ] + h.o.t.,
λ2 = V (ερ),
and similar arguments allows us to prove the results concerning problem (D), like Theorem 1.5. See also Remark 5.1. Remark 1.12. We point out that all the above results, except Corollary 1.3, remain valid also for n = 1, where M coincides with V . In this case, there is no volume energy in the expansions. Remark 1.13. For the construction of the approximate solutions in Z N (or in Z D for the Dirichlet case, see Sections 3, 4) we follow mainly the approach of [16]. This consists in adding (or subtracting) a virtual spike outside ∂Ω, in order to match the boundary conditions, at least to some order of precision. Notation. – Hr1 denotes the set of the functions u ∈ H 1 (Ωε ) which are radially symmetric and will be endowed with the norm
2
u 1 = Hr
Z 1/ε a/ε
r n−1 (|u0 (r )|2 + V (εr )u2 (r )) dr ,
where we take a = 0 for Ω = B1 . The corresponding scalar product will be denoted by (·, ·).
8 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I – We will often use the notation C to denote generic positive constants. The value of C is allowed to vary from place to place. – We use the notation ∼ to denote quantities which, in the limit ε → 0, are of the same order. Similarly, the notation a Ü b means that the order of a not exceed that of b when ε converges to zero. – The symbol o(f ) denotes a quantity that tends to 0 faster than f as s → 0. Similarly, O(f ) stands for |O(f )| Ü f as s → 0. – Throughout the paper we will omit the coefficient ωn−1 (the measure of S n−1 ) when we write integrals in polar coordinates, consistently with [6]. 2. P RELIMINARIES In this section we collect some preliminary facts that will be used frequently in the following. Consider the problem (
(2.1)
−∆u + u = up u>0
in Rn , in Rn ,
with n ≥ 1, p > 1 and u ∈ H 1 (Rn ). We recall that in the sequel if n ≥ 3 we will first assume that p ∈ (1, (n + 2)/(n − 2)], so the variational approach is consistent, see Remark 2.2 below. It is known (see e.g. [7], [12]) that problem (2.1) possesses a ground state solution U¯ which satisfies the properties
(2.2)
¯ ¯ U(x) = U(|x|), ¯ U(r ) > 0, lim er r (n−1)/2 U(r ¯ ) = αn,p > 0, r →∞
for all x ∈ Rn , for all r > 0, lim U¯ 0 (r )/U¯ (r ) = −1, r →∞
where αn,p is a constant depending only on the dimension n and the exponent p . As a consequence, When n = 1, there exist two real functions g and g˜ , defined in a neighborhood of +∞ with the following properties (
(2.3)
¯ ) − α1,p e−r = g(r )e−r , U(r ˜ )e−r , U¯ 0 (r ) + α1,p e−r = g(r
where g(r ) → 0 as r → +∞, ˜ ) → 0 as r → +∞. where g(r
¯ Hence all the solutions of (2.1) belonging to H 1 (R) are of the form U(x−x 0 ), with x0 ∈ R. For all λ > 0, let Uλ denote the solution of the problem (2.4)
( −U 00 + λ2 U = U p 0
U (0) = 0, U > 0
in R, U ∈ H 1 (R).
Singularly Perturbed Elliptic Equations with Symmetry II
9
¯ It is immediate to check that Uλ (s) = λ2/(p−1) U(λs) . The Uλ ’s are critical points of the functional I¯λ (u) =
1 2
Z
R
(|∇u|2 + λ2 u2 ) −
Z
1 p+1
R
|u|p+1 ,
u ∈ H 1 (R),
hence by translation invariance one has I¯λ00 (Uλ )
(2.5)
∂Uλ ∂x
= 0.
We point out that, by (2.3), the Uλ ’s satisfy the following decay properties (2.6)
Uλ (s), Uλ0 (s), Uλ00 (s) ≤ Ce−λs ,
|s| > 1.
We recall the following non-degeneracy property of the functions Uλ ’s, see [25]. Proposition 2.1. For λ > 0, define Zλ = {Uλ (x − x0 ) : x0 ∈ R} ⊆ H 1 (R).
Then Zλ is non-degenerate for the functional I¯λ . Precisely, there exists a positive constant C such that
2 I¯λ00 (Uλ )[Uλ , Uλ ] ≤ −C −1 Uλ λ ,
2 I¯λ00 (Uλ )[v, v] ≥ C −1 v λ ,
for all v ∈ H 1 (R), v ⊥ Uλ , v ⊥ TUλ Zλ , Z
Z
where we are using the scalar product hu, viλ = ∇u∇v the induced norm k · kλ .
+ λ2
uv on H 1 (R) and
In the following, we will always use the notation λ = λ(ερ),
(2.7)
where λ(ερ)2 = V (ερ).
From a straightforward calculation, based on rescaling arguments, one finds Z
(2.8)
p+1
R
Uλ
Z = Vθ
R
U¯ p+1 ,
where V = λ2 .
In the sequel we will work in the space Hr1 (Ω) = Hr1 of the functions u ∈ H 1 (Ω) ˜ ) which are radially symmetric. For each u ∈ Hr1 there is a real function u(r ˜ such that u(x) = u(|x|) . With an abuse of notation, we will simply write u(r ) ˜ ). For future reference, let us recall that (see [27]) if u ∈ Hr1 one instead of u(r has (2.9)
|u(r )| ≤ Cr (1−n)/2 kukHr1 ,
r ≥ 1.
10 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I The constant C in (2.9) depends only on n (and on a if Ω = A). Performing the change of variable x , εx , the equation in (N) becomes −∆u + V (ε|x|)u = up ,
(2.10)
1 in Ωε = Ω, u > 0. ε
If u˜ ε is a solution of (2.10) with Neumann boundary conditions, then uε (x) = ˜ ε (x/ε) is a solution of (N). u Radial solutions of (2.10) are found as critical points of the C 2 functional Iε : Hr1 → R, Z
Z
1 1 [|∇u|2 + V (ε|x|)u2 ] dx − |u|p+1 dx 2 Ωε p + 1 Ωε Z Z 1/ε 1 1/ε n−1 1 0 2 2 = r [(u ) + V (εr )u ] dr − r n−1 |u|p+1 dr . 2 a/ε p + 1 a/ε
Iε (u) =
When Ω = B1 , we take a = 0 in the last formula. Similarly, when dealing with problem (D), we are led to studying equation (2.10) with zero boundary data at ∂Ωε . Letting Hr1,0 denote the space of radial functions vanishing on ∂Ωε , we study critical points of the functional Z
Z
1 1 [|∇u|2 + V (ε|x|)u2 ] dx − |u|p+1 dx 2 Ωε p + 1 Ωε Z Z 1/ε 1 1/ε n−1 1 = r [(u0 )2 + V (εr )u2 ] dr − r n−1 |u|p+1 dr , 2 a/ε p + 1 a/ε
Jε (u) =
u ∈ Hr1,0 .
We remark that replacing u with its positive part u+ , one finds that any critical point uε of Iε (resp. Jε ) satisfies uε ≥ 0 and hence, by the maximum principle, u+ is positive. We will look for critical points of Iε (resp. Jε ) near suitable approximate solutions of (2.10). Precisely, for ρ > 0, let Uλ = Uλ(ερ) be the solution of (2.4) with λ2 = V (ερ): −U 00 + V (ερ)U = U p , U 0 (0) = 0, U ∈ H 1 (R+ ), U > 0.
(2.11)
For γ > 0 and λ1 ∈ (0, λ0 ), let us introduce the following set
(2.12) Cε = w ∈ Hr1 : kwkHr1 ≤ γεkzkHr1 , |w(r )| ≤ γ e−λ1 (ρ−r ) for r ∈ [0, ρ] ,
Singularly Perturbed Elliptic Equations with Symmetry II
11
Remark 2.2. As we will see in Proposition 3.2-(iv) and Proposition 4.1-(iv), functions w in Cε are uniformly bounded in the L∞ norm. As in [6], this allows us to carry over all the proofs in the following way: (i) we first assume that the exponent p ∈ (1, (n + 2)/(n − 2)]; (ii) we use the L∞ a-priori bounds and a suitable truncation argument to handle the general case p > 1. This second step can be performed in a way quite similar to the one used in [6]. So, for the sake of brevity, throughout all the proofs in this paper we will limit ourselves to indicate in details step (i), only. 3. T HE N EUMANN P ROBLEM In this section we prove Theorem 1.2. We only discuss the case of Ω = B1 . The case of the annulus is easier since the L∞ estimates are immediate (see formula (2.9)), and there is no need to use the set Cε . It is convenient to scale as before the problem to the set Ωε = (1/ε)B1 , and to use the functional Iε introduced above Iε (u) =
1 2
Z Ωε
(|∇u|2 + V (ε|x|)u2 ) −
1 p+1
Z Ωε
|u|p+1 ,
u ∈ Hr1 .
3.1. Abstract setting and preliminary estimates. For any r0 < 12 , let ϕε (r ) be a smooth cutoff function such that
(3.1)
ϕε (r ) = 0, ϕε (r ) = 1, |ϕε0 (r )| ≤ Cε, |ϕ00 (r )| ≤ Cε2 , ε
for r ∈ [0, r0 /(8ε)]; for r ∈ [r0 /(4ε), 1/ε]; for r ∈ [r0 /(8ε), r0 /(4ε)]; for r ∈ [r0 /(8ε), r0 /(4ε)].
We define Z N to be the following manifold (3.2)
Z N = {ϕε · (zρ + α1,p λ2/(p−1) e−λ(1/ε−ρ) e−λ(1/ε−·) )}ρ = {zρN = ϕε · (zρ + vρ )}ρ ;
ρ≥
3 . 4ε
Here α1,p is the constant in (2.2) for n = 1, zρ (r ) = Uλ (r − ρ), and λ is given by (2.7). The range of ρ will be chosen appropriately later. We need first the following lemma, which can be proved by reasoning as in [6, Section 3]. Lemma 3.1. Let Z N be as above, and let w ∈ Cε . Then the following properties hold true (E1) kIε00 (zρN + sw)k ≤ C , (0 ≤ s ≤ 1);
12 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I (p−1) (E2) kIε00 (zρN + sw) − Iε (zρN )k ≤ C max{kwk∞ , kwk∞ }, (0 ≤ s ≤ 1).
The same arguments of [6] Section 4, with some minor modifications, yield the following result. Proposition 3.2. There exists a positive constant µ with the following property. For ε sufficiently small, and for all ρ ∈ [r0 /ε, 1/ε − µ] there exist a function w(zρN ) = wρN satisfying (i) (ii) (iii) (iv)
Iε0 (zρN + w(zρN )) = αρN (∂/∂ρ)zρN ; w(zρN ) ⊥ TzρN Z N ;
kw(zρN )k ≤ CkIε0 (zρN )k; kw(zρN )k∞ ≤ C(ε + ε(n−1)/2 kIε0 (zρN )k),
where αρN ∈ R and C is a positive constant depending only on n, p and µ . Moreover, if for some ε 1, ρε is a stationary point of Φε (ρ) = Iε (zρN + wρN ), then u˜ ε = zρNε + wρNε is a critical point of Iε . Proposition 3.2 is proved using the Contraction Mapping Theorem in the set
Cε . The condition ρ ∈ [r0 /ε, 1/ε−µ], for µ sufficiently large and for ε sufficiently small, makes the functional Iε00 (zρN ) qualitatively similar to Iλ00 (Uλ ) (see Section 2
and in particular Proposition 2.1) and allows to conclude Iε00 (zρN )[v, v] ≥ δkvk2 ;
for all v ⊥ zρN ⊕ TzρN Z N ,
for some δ > 0, as in [6], Lemma 4.1. This is a key inequality for performing the finite-dimensional reduction. We note that in the case Ω = A the functions in Hr1 belong to L∞ : the argument carries over directly in Hr1 and there is no need to introduce the set Cε . For proving Theorem 1.2 we need to be careful in the expansion of Φε , since we want to consider values of ρ which are close to the exterior boundary of Ωε . In the next lemma we estimate the quantity kIε0 (zρN )k, which by (iii) is an upper bound for kw(zρN )k. Lemma 3.3. Let Z N be defined in (3.2). Then there holds kIε0 (zρN )k ≤ Cε(1−n)/2 (ε + o(e−λ(1/ε−ρ) )),
for every zρN ∈ Z N .
for some constant C depending only and a, n and p . Proof. In the proof we will often omit the index ρ in zρ and vρ . Since p zρ = Uλ (· − ρ) and vρ satisfy respectively the equations −zρ00 + V (ερ)zρ = zρ
Singularly Perturbed Elliptic Equations with Symmetry II
13
and −vρ00 + V (ερ)vρ = 0, we have Iε0 (zN )[u]
Z =
N
N
N p
n−1
1
N 0
1
1
u (−∆z + V (εr )z − (z ) )u + (z ) ε ε ε Z 1 1 = ε1−n (zN )0 u + (V (εr ) − V (ερ))zN u ε ε Z Z 1 N 0 − (n − 1) (z ) u − (2ϕε0 (zρ + vρ )0 + ϕε00 (zρ + vρ ))u r Z p N p − ((z ) − ϕε zρ ).
From formulas (2.3) and (2.9) we find (3.3)
N 0 1 1 1 ˜ u − ρ e−λ(1/ε−ρ) ε(1−n)/2 kuk ε1−n ) g λ (z ≤ C ε ε ε = ε(1−n)/2 o(e−λ(1/ε−ρ) )kuk.
From the H¨older inequality and the smoothness of V , we deduce the inequality (3.4)
Z Z (V (εr ) − V (ερ))zN u ≤ Cε |r − ρ| |zN | |u| ≤ Cεε(1−n)/2 kuk.
On the other hand, since the function Z N is supported in {r ≥ r0 /(8ε)}, one has also (3.5)
Z 1 N 0 ≤ Cε(3−n)/2 kuk. ) u (z r
From the decay of zρ (see (2.6)) and vρ , from the fact that ϕε0 , ϕε00 have support in [r0 /(8ε), r0 /(4ε)] and from the fact that ρ ≥ 3/(4ε), it is easy to deduce the estimates (3.6a) (3.6b)
Z ϕ0 (zρ + vρ )0 u ≤ Cεε(1−n)/2 e−λr0 /(4ε) kuk; ε Z ϕ00 (zρ + vρ )u ≤ Cε2 ε(1−n)/2 e−λr0 /(4ε) kuk. ε Z
Let us consider now the term ((zN )p − ϕε zρp )u. We can write p
p
p
p
p
(zN )p − ϕε zρ = ϕε ((z + v)p − zp ) + ϕε (ϕε zp − ϕε zρ )
Since z is uniformly bounded, we have |(z + v)p − zp − pzp−1 v| ≤ C|v|2∧p .
14 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I It follows that Z Z Z [(z + v)p − zp ]u ≤ p zp−1 v|u| + C |v|2∧p |u| .
Again from the H¨older inequality we obtain Z Z |v|2∧p |u| ≤ e−λ(2∧p)(1/ε−ρ) e−(1/ε−r ) |u| ≤ Ce−(2∧p)λ(1/ε−ρ) ε(1−n)/2 kuk. R
R
We have also | zp−1 v|u|| ≤ ( z2(p−1) v 2 )1/2 kuk. We divide the last integral in the two regions r ≤ (ρ + ε−1 )/2 and r ≥ (ρ + ε−1 )/2. When r ≤ (ρ + ε−1 )/2, v satifies |v| ≤ e−(3/2)λ(1/ε−ρ) and hence Z r ≤(ρ+ε)/2
z
2(p−1)
v
2
1/2
≤ Ce
−(3/2)λ(1/ε−ρ)
Z r ≤(ρ+ε)/2
z
2(p−1)
1/2
≤ Ce−(3/2)λ(1/ε−ρ) ε(1−n)/2 .
On the other hand, when r ≥ (ρ + ε−1 )/2, z satisfies |z(r )| ≤ e−(1/2)λ(1/ε−ρ) so we have Z r ≥(ρ+ε)/2
z
2(p−1) 2
1/2
v
≤ Ce
−((p−1)/2)λ(1/ε−ρ)
Z |v|
2
1/2
≤ Ce−(1+(p−1)/2)λ(1/ε−ρ) ε(1−n)/2 .
We have also the estimate Z Z 1/2 p 2 2p (ϕεp zρp − ϕε zρp )u ≤ C (ϕ − ϕ ) U kuk ε ε λ ≤ Ce−λpr0 /(2ε) ε(1−n)/2 kuk.
The above estimates yield (3.7)
Z ((zN )p − ϕε U p )u λ ≤ Cε(1−n)/2 (e−(3∧(p+1)/2)λ(1/ε−ρ) + e−λpr0 /(4ε) )kuk.
Hence estimates (3.3)-(3.7) imply
This concludes the proof of the lemma.
❐
kIε0 (zρN )k ≤ Cε(1−n)/2 (ε + o(e−λ(1/ε−ρ) ) + e−λr0 /(4ε) ).
Singularly Perturbed Elliptic Equations with Symmetry II
15
3.2. Expansion of Iε on Z N . In this subsection we expand Iε (zρN ) as a function of ρ and ε. Integrating by parts and using the equations satisfied by z and v (see the proof of Lemma 3.3), we find Z
Z
1 1 (|∇zN |2 + V (εr )(zN )2 ) − |zN |p+1 2 p+1 Z 1 = (−∆zN + V (εr )zN )zN 2 Z 1 1 n−1 N 1 1 1 + z |zN |p+1 (zN )0 − 2 ε ε ε p+1 1 1 n−1 N 1 1 (zN )0 = z 2 ε Z ε ε Z 1 1 p N ϕε z z − |zN |p+1 + 2 p+1 Z Z n − 1 (zN )0 zN − ϕε0 zN (z + v)0 − 2 r Z Z 1 1 00 N − ϕε z (z + v) + (V (εr ) − V (ερ))(zN )2 . 2 2
Iε (zN ) =
(3.8)
Let us estimate each of the seven terms in the last expression. From equation (2.3) we find (3.9)
ε
1 1 1 (zN )0 ≤ Cε1−n e−2λ(1/ε−ρ) g˜ λ −ρ z ε ε ε = ε1−n o(e−2λ(1/ε−ρ) ).
1−n N
To estimate the second and the third term, we can write Z
Z
1 1 ϕε zp zN − |zN |p+1 2 p+1 !Z Z 1 1 1 p+1 p+1 p+1 = − + (ϕε2 − ϕε )zp (z + v) ϕε z 2 p+1 2 Z Z 1 1 p+1 p p+1 − ϕε z v − ϕε (|z + v|p+1 − zp+1 − (p + 1)zp v). 2 p+1
(3.10)
We have Z Z Z p+1 p+1 n−1 ϕεp+1 zp+1 − ρ n−1 Uλ Uλ (r − ρ) ≤ρ R r ≥1/ε Z Z p+1 p+1 n−1 n−1 + + (r − ρ )U (r − ρ) r n−1 (1 − ϕε )zp+1 . λ [0,1/ε]
[0,1/ε]
16 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I Using a Taylor expansion for the function r n−1 − ρ n−1 and the fact that r ≤ C(r0 )ρ (since ρ ≥ r0 /ε), we obtain Z
(r
n−1
[0,1/ε]
−ρ
n−1
p+1 )Uλ (r
Z
≤ C(n, r0 )ρ n−2
− ρ)
p+1
[0,1/ε]
|r − ρ|Uλ
(r − ρ) ≤ Cρ n−2 .
On the other hand, from (2.6) we get Z
ρ
p+1
n−1 r ≥1/ε
Z
Uλ
(r − ρ) ≤ Cε1−n (e−(p+1)λ(1/ε−ρ) + e−λ(p+1)r0 /(4ε) ), p+1
[0,1/ε]
r n−1 (1 − ϕε
p+1
)Uλ
≤ Cε1−n e−λ(p+1)r0 /(4ε) .
Hence from the last three equations we deduce
Z Z p+1 ϕεp+1 zp+1 − ρ n−1 ≤ Cε1−n (e−(p+1)λ(1/ε−ρ) + ε). U λ
(3.11)
R
R
The term ϕεp+1 (|z + v|p+1 − zp+1 − (p + 1)zp v) in (3.10) can be estimated as follows. From the inequality |z + v|p+1 − zp+1 − (p + 1)zp v − p(p + 1)zp−1 v 2 ≤ Cv 3∧(p+1) ,
one finds Z
[0,1/ε]
|z+v|p+1 −zp+1 −(p+1)zp v ≤ C
Z
Z [0,1/ε]
zp−1 v 2 +C
v 3∧(p+1) . [0,1/ε]
The first integral in the last expression can be estimated dividing the domain in the two regions r ≤ (ρ + ε−1 )/2 and r ≥ (ρ + ε−1 )/2, as before, while for the second it is sufficient to use the explicit expression of v . In this way we find (3.12)
Z
[0,1/ε]
p+1 ϕε (|z
+ v|
p+1
−z
p+1
− (p + 1)z v) p
≤ Cε1−n (e−3λ(1/ε−ρ) + e−((p+3)λ/2)(1/ε−ρ) + e−λ(3∧(p+1))(1/ε−ρ)) . R
The term Aε ϕεp+1 zp v in (3.10) turns out to be of order ε1−n e−2λ(1/ε−ρ) . We need to have a rather precise expansion of this term, so we treat it in some detail. There holds Z
p+1 p
[0,1/ε]
ϕε
z v = α1,p ρ n−1 λ2/(p−1) e−2λ(1/ε−ρ)
Z
p
Uλ eλr Z p − α1,p ρ n−1 λ2/(p−1) e−2λ(1/ε−ρ) Uλ (r − ρ)eλ(r −ρ) r ≥1/ε Z Z p+1 + (r n−1 − ρ n−1 )zp v + (ϕε − 1)zp v. [0,1/ε]
R
[0,1/ε]
Singularly Perturbed Elliptic Equations with Symmetry II
17
Reasoning as above, we obtain ρ n−1 e−2λ(1/ε−ρ)
Z
p
r ≥1/ε
Uλ (r − ρ)eλ(r −ρ) ≤ Cε1−n e−(p+1)λ(1/ε−ρ) ;
Z n−1 n−1 p (r −ρ )z v ≤ Cε2−n e−2λ(1/ε−ρ) ; [0,1/ε] Z p+1 (1 − ϕε )zp v ≤ Cε1−n e−((p+1)r0 λ)/(4ε) . [0,1/ε]
Hence the last three equations and the expression of Uλ (see Section 2) imply Z
(3.13)
p+1 p
ϕε
z v = α1,p ρ n−1 λ2θ e−2λ(1/ε−ρ)
Z R
U¯ p er
+ ε1−n e−2λ(1/ε−ρ) O(ε + e−(p−1)(1/ε−ρ) ) Z = α1,p ε1−n M(ερ)e−2λ(1/ε−ρ) U¯ p er R
+ ε1−n e−2λ(1/ε−ρ) O(ε + e−(p−1)(1/ε−ρ) ),
for ε small. The fourth term in (3.8) can be estimated as for (3.5), and gives Z (zN )0 zN ≤ Cε2−n . r
(3.14)
The fifth and the sixth terms in (3.8) can be estimated in the following way Z ϕ0 zN (z + v)0 ≤ Cε2−n e−λr0 /(2ε) ; ε Z ϕ00 zN (z + v) ≤ Cε3−n e−λr0 /(2ε) . ε
(3.15a) (3.15b)
The seventh term can be estimated in the following way Z Z (V (εr ) − V (ερ))(zN )2 ≤ Cε |r − ρ|(zN )2 ≤ Cε2−n .
(3.16)
From (3.9)-(3.16) we deduce the following result. Lemma 3.4. Let zN be defined in (3.2). Then one has 1 Iε (zρN ) = ε1−n M(ερ) −
2
−
1 p+1 Z
1 α e−2λ(ερ)(1/ε−ρ) 2 1,p
for all ρ ∈ [3/(4ε), 1/ε].
Z R
R
U¯ p+1
U¯ p er
+ O(ε2−n ) + ε1−n o(e−2λ(ερ)(1/ε−ρ) )
18 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I 3.3. Proof of Theorem 1.2. For s ∈ [0, 1], using (E1) and (E2) in Lemma 3.1, we have kIε0 (zN + sw N ) − Iε0 (zN )k
Z1
00 N N 00 N
(I (z + ζsw ) − I (z ))[w] dζ ≤ kIε00 (zN )[sw N ]k +
ε ε 0
= O(kw k) + O(max{kw N k2 , kw N kp }). N
Hence using property (iii) in Proposition 3.2 and the smallness of kIε0 (zN )k, we deduce Iε (zN + w(zN )) = Iε (zN ) + Iε0 (zN )[w(zN )] +
Z1 0
(Iε0 (zN + sw N ) − Iε0 (zN ))[w] ds
= Iε (zN ) + O(kIε0 (zN )k2 ).
Hence from Lemmas 3.3 and 3.4 it turns out that (3.17)
Iε (zρN + wρN ) = ε1−n M(ερ)[α − βe−2λ(1/ε−ρ) ] + O(ε2−n ) + ε1−n o(e−2λ(1/ε−ρ) ),
where (3.18)
1 1 α= − 2 p+1
!Z R
U¯ p+1 ;
1 β = α1,p 2
Z R
U¯ p er .
We are going to show that the function ρ , Iε (zρN + wρN ) possesses a critical point ρε with |1/ε − ρε | ∼ | log ε|. We give first an heuristic argument, which justifies the assumption M 0 (1) > 0 and the choice of the numbers ρ0,ε , ρ1,ε and ρ2,ε below. The main term in (3.17) is M(ερ)[α − βe−2λ(1/ε−ρ) ], apart from the factor ε1−n . Recalling that λ(ερ)2 = V (ερ), and differentiating the main term with respect to ρ we obtain εM 0 (ερ)[α − βe−2λ(1/ε−ρ) ] 1 + 2βM(ερ) ελ0 (ερ) − ρ − λ(ερ) e−2λ(1/ε−ρ) . ε
Since |1/ε − ρε | ∼ | log ε|, the term e−2λ(1/ε−ρ) converges to 0 as ε goes to 0, hence to get a critical point we must require, roughly εαM 0 (ερ) = 2βλ(ερ)M(ερ)e−2λ(1/ε−ρ) .
Singularly Perturbed Elliptic Equations with Symmetry II
19
Since the right-hand side is positive, this equation is solvable if M 0 (ερ) ∼ M 0 (1) is positive. Taking the logarithm, and using the fact that all the terms except ε and e−2λ(1/ε−ρ) are uniformly bounded from above and from below by positive constants, we obtain the condition | log ε| ∼ 2λ(1)
(3.19)
1 ε
−ρ
⇐⇒
| log ε| −ρ ∼ . ε 2λ(1)
1
We now begin our justification of the above arguments. Given C0 > 0 (to be fixed later sufficiently large), consider the three numbers (3.20) ρ0,ε =
1 ε
−
1 2λ(1)
| log ε|; ρ1,ε =
1 ε
−
1 C0
| log ε|; ρ2,ε =
1 ε
− C0 | log ε|.
By condition (3.19) we expect ρ0,ε to be almost critical for the function ρ , Φε (ρ) = Iε (zρN + wρN ). Using Lemma 3.4 and some elementary computations, one finds Φε (ρ0,ε ) = =ε
1−n
! ε| log ε| 0 M (1) + o(ε| log ε|) [α − βελ(1−ε| log ε|/(2λ(1)))/λ(1) ] M(1) − 2λ(1) + O(ε2−n ) + ε1−n o(ελ(1−ε| log ε|/(2λ(1)))/λ(1) ).
From the differentiability of λ we obtain ελ(1−ε| log ε|/(2λ(1)))/λ(1) = ε1+O(ε| log ε|) = O(ε) ε| log ε|,
and hence Φε (ρ0,ε ) = ε
1−n
! ε| log ε| 0 α M(1) − M (1) + o(ε| log ε|) . 2λ(1)
On the other hand, there holds Φε (ρ1,ε ) = =ε
1−n
! ε| log ε| 0 M(1) − M (1) + o(ε| log ε|) [α − βε2λ(1−ε| log ε|/C0 )/C0 ] C0 + O(ε2−n ) + ε1−n o(ε2λ(1−ε| log ε|/C0 )/C0 ).
If C0 > 2λ(1), we use the estimate ε2λ(1−ε| log ε|/C0 )/C0 = ε2λ(1)/C0 +O(ε| log ε|) = ε2λ(1)/C0 (1 + o(1)) ε| log ε|,
20 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I to obtain Φε (ρ1,ε ) = ε1−n M(1)[α − βε2λ(1)/C0 + o(ε2λ(1)/C0 )].
For the third term, we can write Φε (ρ2,ε ) = = ε1−n (M(1) − C0 ε| log ε|M 0 (1) + o(ε| log ε|))[α − βε2C0 λ(1−C0 ε| log ε|) ] + O(ε2−n ) + ε1−n o(ε2C0 λ(1−C0 ε| log ε|) ).
If C0 > 1/(2λ(1)), we obtain ε2C0 λ(1−C0 ε| log ε|) = ε2C0 λ(1)+O(ε| log ε|) = O(ε2C0 λ(1)+O(ε| log ε|) ) ε| log ε|, and hence, for ε sufficiently small, Φε (ρ2,ε ) = ε1−n α(M(1) − C0 ε| log ε|M 0 (1) + o(ε| log ε|)).
If C0 is chosen sufficiently large (C0 > max{2λ(1), 1/(2λ(1))}), the last three equations imply sup Φε > sup Φε . [ρ2,ε ,ρ1,ε ]
∂[ρ2,ε ,ρ1,ε ]
Hence it follows that the reduced functional Φε possesses a critical point (maximum) ρ in the interval (1/ε − C0 | log ε|, 1/ε − (1/C0 )| log ε|). By Proposition 3.2, this concludes the proof of the theorem. Remark 3.5. The preceding arguments can be adapted to produce boundaryN layer solutions, as in [20]. In this case, the operator Iε00 (z1N/ε ) (or Iε00 (za/ε ) when Ω = A), namely for ρ sitting on the boundary of Ω, is invertible for ε small, and there is no need to perform a finite dimensional reduction. One should just apply the standard contraction argument on Cε . Proof of Theorem 1.11. Since the method relies on the above technique, we just give a simple sketch. The approximate solutions zρ and the manifold Z N have to be substituted in the following way. We set Sρ = {(x 0 , x 00 ) : |x 0 | = ρ, x 00 = 0};
ρ≥
3 , 4ε
and
dρ (x) = dist(x, Sρ ),
and we let ϕε denote a cutoff function which vanishes near x 0 = 0. Then we define zρ (x 0 , x 00 ) = Uλ,k (dρ (x));
λ2 = V (ερ);
Z N,k = {ϕε (x) × (zρ (x 0 , x 00 ) + Uλ,k (d2/ε−ρ (x)))}ρ ,
where Uλ,k is given in (1.6), and where we have set, with abuse of notation, Uλ,k (|y|) = Uλ,k (y), y ∈ Rn−k .
Singularly Perturbed Elliptic Equations with Symmetry II
21
The auxiliary potential Mk is obtained from a simple scaling argument, see also the last section in [6]. In the study of the linearized problem at Z N,k , other small eigenvalues appear due to the presence of the variable x 00 . The approximate kernel is constituted by the n − k functions ∂zρ ∂zρ ∂zρ , , ..., . 00 00 ∂ρ ∂(x )1 ∂(x )n−k−1
❐
Since ∂zρ /∂(x 00 )1 , . . . , ∂zρ /∂(x 00 )n−k−1 are odd with respect to x 00 , the problem due tho their presence can be overcome working in the space of functions u (radial in x 0 ) such that u(x 0 , −x 00 ) = u(x 0 , x 00 ). In this way the degeneracy in the variable x 00 is removed. In particular, the counterpart of Proposition 2.1 holds. Then the finite-dimensional reduction can be performed in the radial variable, only, and we obtain the analogue of Proposition 3.2. The rest of the arguments require minor (and obvious) changes. Let us point out that the above procedure yields solutions which are radial in x 0 and even in x 00 . 4. T HE D IRICHLET PROBLEM In this section we treat the Dirichlet problem. Again, we only discuss the case of the unit ball B1 ⊆ Rn , and we set Ωε = (1/ε)B1 . We work in the space H01,r of the radial functions of class H 1 in Ωε vanishing on the boundary. We look for critical points of the functional Jε (u) =
1 2
Z Ωε
(|∇u|2 + V (ε|x|)u2 ) −
1 p+1
Z Ωε
|u|p+1 ,
u ∈ H01,r .
As before, we will perform a finite-dimensional reduction of the functional Jε . 4.1. Abstract setting and first estimates. Letting g be the function given in (2.3), and letting ϕε be the cutoff function defined in (3.1), we define Z D to be the following manifold
−ρ ε × e−λ(1/ε−ρ) e−λ(1/ε−·) = {zρD ≡ ϕε · (zρ − v¯ ρ )}.
(4.1) Z D = ϕε zρ − λ2/(p−1) α1,p + g λ
1
Here αp is the constant in (2.2) for n = 1, zρ (r ) = Uλ (r − ρ) and λ is given by (2.7). The range of ρ will be chosen appropriately later. As in the Neumann case, for the manifold Z D one can prove the following result. Proposition 4.1. There exists a positive constant µ with the following property. For ε sufficiently small and for all ρ ∈ [r0 /ε, 1/ε−µ] there exists a function w(zρD ) = wρD satisfying
22 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I (i) (ii) (iii) (iv)
Jε0 (zρD + w(zρD )) = αρD (∂/∂ρ)zρD ; w(zρD ) ⊥ TzρD Z D ; kw(zρD )k ≤ CkJε0 (zρD )k; kw(zρD )k∞ ≤ C(ε + ε(n−1)/2 kJε0 (zρD )k),
where αρD ∈ R and C is a positive constant depending only on n, p and µ . Moreover, if for some ε 1, ρε is stationary point of Ψε (ρ) = Jε (zρD +wρD ), then u˜ ε = zρDε +wρDε is a critical point of Jε . We estimate the quantity kJε0 (zρD )k in the next lemma. Lemma 4.2. Let Z D be defined in (4.1). Then there holds kJε0 (zρD )k ≤ Cε(1−n)/2 (ε + o(e−λ(1/ε−ρ) )),
for every zρD ∈ Z D .
for some constant C depending only on n and p . Proof. By the definition of g (see Section 2) and of Z D , the function zρD satisfies zρD (1/ε) = 0, and hence we have Jε0 (zD )[u] = =
Z Z
(−∆zD + V (εr )zD − (zD )p )u n−1 (V (εr ) − V (ερ))z u − r Z D
−
ϕε0 (zρ
(2
0
− v¯ ρ ) +
ϕε00 (zρ
Z
(zD )0 u Z
p
− v¯ ρ ))u − ((zD )p − ϕε zρ )u.
Reasoning as in Lemma 3.3 we obtain the estimates Z Z (V (εr ) − V (ερ))zD u ≤ Cε |r − ρ| |zD | |u| ≤ Cεε(1−n)/2 kuk; Z 1 D 0 (1−n)/2 (ε + εe−λ(1/ε−ρ) )kuk; r (z ) u ≤ Cε Z ϕ0 (zr h − v¯ ρ )0 u ≤ Cεε(1−n)/2 e−λr0 /(4ε) kuk; ε Z ϕ00 (zρ − vρ )u ≤ Cε2 ε(1−n)/2 e−λr0 /(4ε) kuk; ε Z ((zD )p − ϕε zρp )u ≤ Cε(1−n)/2
Hence we obtain the desired conclusion.
❐
× (e−(3∧(p+1)/2)λ(1/ε−ρ) + e−λr0 p/(4ε) )kuk.
Singularly Perturbed Elliptic Equations with Symmetry II
23
4.2. Expansion of Jε on Z D . Let us study the expansion of the functional Iε (zρD ). Since zρD satisfies zρD (1/ε) = 0, we have Z
Z
1 1 (|∇zD |2 + (zD )2 ) − |zD |p+1 Jε (z ) = 2 p+1 Z Z 1 1 = (−∆zD + zD )zD − |zD |p+1 2 p+1 Z Z 1 1 p D = ϕε z z − |zD |p+1 2 p+1 Z Z n − 1 (zD )0 zD ¯ 0 − ϕε0 zD (z − v) − 2 r Z Z 1 1 ¯ + − ϕε00 zD (z − v) (V (εr ) − V (ερ))(zD )2 . 2 2 D
(4.2)
The last four terms in the above expression can be estimates as in Subsection 3.2, and give Z (zD )0 zD ≤ Cε2−n ; r Z 0 2−n −λ/(2ε) ϕ0 zD (z − v) ¯ e ; ≤ Cε ε Z ϕ00 zN (z − v) ¯ ≤ Cε3−n e−λr0 /(2ε) ; ε Z Z (V (εr ) − V (ερ))(zD )2 ≤ Cε |r − ρ|(zD )2 ≤ Cε2−n .
(4.3) (4.4) (4.5) (4.6)
To estimate the second and the third term in (4.2), we can write 1 2
Z p D
ϕε z z −
1
Z
p+1 !Z
|zD |p+1 Z
1 1 1 p+1 p+1 ¯ = − (ϕε2 − ϕε )zp (z − v) ϕε zp+1 + 2 p+1 2 Z Z 1 1 p+1 p+1 ¯ p+1 − zp+1 + (p + 1)zp v). ¯ + ϕε zp v¯ − ϕε (|z − v| 2 p+1 Note that in the last equation, if compared to (3.10), some signs are reversed. Anyway, the same method yields the estimate (4.7)
1 2
Z
1
Z
ϕε z z − |zD |p+1 = p+1 Z Z 1 1 1 − U¯ p+1 + α1,p e−2λ(1/ε−ρ) U¯ p er = ε1−n M(ερ) 2 p+1 R 2 R p D
+ O(ε2−n ) + ε1−n o(e−2λ(1/ε−ρ) ).
24 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I From equations (4.3)-(4.7) we deduce the following result. Lemma 4.3. Let zD be defined in (3.2). Then one has Jε (zρD ) = =ε
1−n
1 M(ερ) −
2
1
Z
p+1
1 U¯ p+1 + α1,p e−2λ(ερ)(1/ε−ρ) 2 R
Z R
U¯ p er
+ O(ε2−n ) + ε1−n o(e−2λ(ερ)(1/ε−ρ) )
for all ρ ∈ [r0 /ε, 1/ε]. 4.3. Proof of Theorem 1.5. Properties analogous to (E1)-(E2) (see Lemma 3.1) hold also in this case, hence reasoning as above we find Jε (zD + w(zD )) = Jε (zD ) + Jε0 (zD )[w(zD )] + O(kw(zD )k2 ) = Jε (zD ) + O(kJε0 (zD )k2 ).
From Lemmas 4.2 and 4.3 it turns out that (4.8)
Jε (zρD + wρD ) = ε1−n M(ερ)[α + βe−2λ(1/ε−ρ) ] + O(ε2−n ) + ε1−n o(e−2λ(1/ε−ρ) ),
where α and β are given by (3.18). In this case, we are going to show that the function ρ , Iε (zρD + wρD ) possesses a minimum point for ρ close to 1/ε. The heuristic argument used before leads to solving the equation εαM 0 (ερ) = −2βλ(ερ)M(ερ)e−2λ(1/ε−ρ) .
which, again, admits the solution (3.19). Note that the last equation now can be solved if M 0 (1) < 0, since the sign is reversed with respect to the Neumann case. As before, we are led to reconsider the three numbers: ρ0,ε =
1
1
| log ε|; 2λ(1) 1 1 ρ1,ε = − | log ε|; ε ε
ρ2,ε =
1 ε
−
C0
− C0 | log ε|,
where C0 is large. We set Ψε (ρ) = J(zρD + wρD ), and we look for a critical point of Ψε near ρ0,ε . Using Lemma 4.3 and the fact that ελ(1−ε| log ε|/(2λ(1)))/λ(1) = ε1+O(ε| log ε|) =
Singularly Perturbed Elliptic Equations with Symmetry II
25
O(ε) ε| log ε|, we obtain Ψε (ρ0,ε ) = = ε1−n M(1) −
! ε| log ε| 0 M (1) + o(ε| log ε|) [α + βελ(1−ε| log ε|/(2λ(1)))/λ(1)] 2λ(1) + O(ε2−n ) + ε1−n o(ε2λ(1−ε| log ε|/(2λ(1)))/λ(1) ) !
= ε1−n α M(1) −
ε| log ε| 0 M (1) + o(ε| log ε|) . 2λ(1)
Now we use the estimate ε2λ(1−ε| log ε|/C0 )/C0 = ε2λ(1)/C0 +O(ε| log ε|) = ε2λ(1)/C0 (1 + o(1)) ε| log ε|
(for C0 large), which implies Ψε (ρ1,ε ) = ε1−n M(1)[α + βε2λ(1)/C0 + o(ε2λ(1)/C0 )].
To study the term Ψε (ρ2,ε ), we use the estimate ε2C0 λ(1−C0 ε| log ε|) = O(ε2C0 λ(1)+O(ε| log ε|) ) ε| log ε|,
and hence we get Ψε (ρ2,ε ) = = ε1−n (M(1) − C0 ε| log ε|M 0 (1) + o(ε| log ε|))[α + βε2C0 λ(1−C0 ε| log ε|) ] + O(ε2−n ) + ε1−n o(ε2C0 λ(1−C0 ε| log ε|) ) = ε1−n α(M(1) − C0 ε| log ε|M 0 (1) + o(ε| log ε|)),
for ε sufficiently small. The last three equations imply that, inf
[ρ2,ε ,ρ1,ε ]
Ψε
0 such that (5.5)
the function M is strictly increasing in the interval (0, 2r0 ).
We define a cutoff function ϕε with the following properties
(5.6)
ϕε (r ) = 0 ϕε (r ) = 1 |ϕε0 (r )| ≤ Cε |ϕ00 (r )| ≤ Cε2 ε
for r ∈ [0, r0 /(8ε)]; for r ∈ [r0 /(4ε), 1/ε]; for r ∈ [r0 /(8ε), r0 /(4ε)]; for r ∈ [r0 /(8ε), r0 /(4ε)],
30 A NTONIO A MBROSETTI , A NDREA M ALCHIODI , & W EI -M ING N I and we set Z N = {ϕε · (zρ + α1,p λ2/(p−1) e−λ(1/ε−ρ) e−λ(1/ε−·) )}ρ r0 . = {zρN ≡ ϕε (zρ + vρ )}ρ ; ρ ≥ ε
Now we apply Proposition 3.2, Lemma 3.4, and formula (3.17). If ρ0,ε and ρ1,ε are as in (3.20), condition (5.5) and formula (3.17) yield for C0 large Φε
r0 ε
< Φε
2r0 ε
;
Φε (ρ0,ε ) < Φε (ρ1,ε ).
Therefore Φε must possess a critical point in the interval [r0 /ε, ρ1,ε ], and that concludes the proof of the theorem. Again, we do not use any assumption on M 0 (1). Remark 5.1. (a) We point out that, differently from Theorems 1.2-1.5, in Theorems 1.4-1.6 we are not able to guarantee that the radius of concentration of the solutions approaches the boundary of the domain. In order to do this, we used crucially the assumptions on M 0 . (b) In a quantitative way, the effect of the boundary conditions can be explained as follows. When one considers equation (2.10) in Rn , the energy of an approximate solution concentrated near a sphere of radius ρ is of the form C0 ε1−n M(ερ), see [6, Lemma 5.1]. In the case of Neumann boundary conditions the boundary ”bends down” the energy, see formula (3.17). In the case of Dirichlet boundary conditions the bending is in the opposite direction, see (4.8). 6. O PEN P ROBLEMS AND P ERSPECTIVES In this Section we discuss some possible extensions of the results proved in the previous sections. The Dirichlet problem. The Dirichlet problem could admit some non-symmetric extensions. For simplicity we discuss the two-dimensional case, since the higher dimensional case requires only obvious modifications. Theorem 1.5 suggests that the boundary confines the solution inside the domain, when this is not contractible. As a consequence, we expect that there could be solutions of (D) (in a general domain Ω) concentrating along curves which are injective and minimize the length in some fixed homotopy class of Ω. See Figure 1. The Neumann problem. We wonder whether problem (N) might also admit some non-radial extensions. In [6] we conjectured that if concentration for the stationary Schrodinger equation occurs at a k-dimensional manifold Σ, then the following condition should be verified (6.1)
θk ∇⊥ V = V H,
θk =
p+1 1 − (n − k). p−1 2
Singularly Perturbed Elliptic Equations with Symmetry II
31
uε
Ω
F IGURE 1. the Dirichlet case Ω Ω uε
V0
V =1
F IGURE 2. the Neumann case For the case with Neumann boundary conditions, we suspect that Σ should verify the same equation in the interior of Ω, and intersecting ∂Ω perpendicularly, see Figure 2. Acknowledgements. A. A. and A. M. are supported by MURST, under the project Variational Methods and Nonlinear Differential Equations. A.M. is also supported by NSF under agreement No. DMS-9729992. W.-M. N. is partially supported by the National Science Foundation. Part of this work has been accomplished when the second and the third author visited S.I.S.S.A. in Trieste.
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[21] W.-M. N I, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9–18. [22] W.-M. N I and I. TAKAGI, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819–851. [23] W.-M. N I and I. Y AKAGI, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247–281. [24] W.-M. N I and J. W EI, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math. 48 (1995), 731–768. [25] Y.-G. O H, On positive multibump states of nonlinear Schroedinger equation under multpile well potentials, Comm. Math. Phys. 131 (1990), 223–253. [26] O. R EY , An elliptic Neumann problem with critical nonlinearity in three-dimensional domains, Commun. Contemp. Math. 1 (1999), DOI 10.1142/S0219199799000158, 405–449, http://dx.doi.org/10.1142/S0219199799000158. [27] W.A. S TRAUSS, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149–162. [28] A.-M. T URING, The chemical basis of morphogenesis, Phil. Trans. Royal Soc. London, Series B, Biological Sciences 237 (1952), 37–72. (Antonio Ambrosetti)S.I.S.S.A. 2-4 via Beirut Trieste 34014, ITALY. E- MAIL:
[email protected] (Andrea Malchiodi)School of Mathematics Institute for Advanced Study 1 Einstein Drive Princeton, NJ 08540 U. S. A. E- MAIL:
[email protected] (Wei-Ming Ni)School of Mathematics University of Minnesota Minneapolis, MN 55455, U. S. A. E- MAIL:
[email protected] 2000 M ATHEMATICS S UBJECT C LASSIFICATION: Author: Please supply. K EY WORDS AND PHRASES: singularly perturbed elliptic problems, local inversion. Received : January 2nd, 2003; revised: April 18th, 2003.
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