Ground states of nonlinear Schrodinger equations with potentials vanishing at infinity

Ground states of Nonlinear Schr¨odinger Equations with Potentials Vanishing at Infinity Antonio AMBROSETTI SISSA, via Beirut 2-4, 34014 Trieste - Italy [email protected] Veronica FELLI Univ. Milano Bicocca, Dip. Mat. e Appl., 20126 Milano - Italy [email protected] Andrea MALCHIODI via Beirut 2-4, 34014 Trieste - Italy [email protected] Key words: Nonlinear Schr¨ odinger Equations, Weighted Sobolev spaces.

Abstract. We deal with a class on nonlinear Schr¨ odinger equations (NLS) with potentials V (x) ∼ |x|−α , 0 < α < 2, −β and K(x) ∼ |x| , β > 0. Working in weighted Sobolev spaces, the existence of ground states vε belonging to W 1,2 (RN ) is proved under the assumption that p satisfies (3). Furthermore, it is shown that vε are spikes concentrating at a minimum of A = V θ K −2/(p−1) , where θ = (p + 1)/(p − 1) − 1/2.

1

Introduction

This paper deals with existence of ground state solutions of stationary Nonlinear Schr¨odinger Equations of the form  2 p N   −ε ∆v + V (x)v = K(x)v , x ∈ R , (NLS)   v ∈ W 1,2 (RN ), v(x) > 0, lim|x|→∞ v(x) = 0 . N +2 Hereafter, N ≥ 3 and 1 < p < N −2 . A solution of (N LS) is called a ground state if it is a Mountain-Pass critical point of the corresponding Euler functional, and hence its Morse index is 1. If u is a solution of (N LS), then
ψ(x, t) = exp i λ ε−1 t u(x),

represents a standing wave of the Nonlinear Schr¨odinger Equation (1)

iε

∂ψ = −ε2 ∆ψ + (V (x) − λ)ψ − K(x)|ψ|p−1 ψ, ∂t

where ε(= ~) is the Planck constant and i is the imaginary unit. 1

One of the main purposes of this paper is to look for solutions vε of (NLS) which satisfy the following properties (i) vε ∈ W 1,2 (RN ); (ii) vε is a ground state. As for (i), let us point out that standing waves v which have finite L2 norm are the most relevant from the physical point of view since they correspond to bound states. Moreover, if v ∈ W 1,2 (RN ), one can prove that lim|x|→∞ v(x) = 0 (see the proof of Theorem 16), which implies that solutions are well localized in space. On the other hand, concerning (ii), the interest in searching ground states relies on the fact that a standing wave is possibly orbitally stable provided it corresponds to a ground state of (NLS), in the sense specified before, see e.g. [14, 17]. A lot of work has been devoted to the existence of solutions of (NLS), both for ε = 1 and for ε tending to zero. In the latter case, as a specific feature of the nonlinear (focusing) model, solutions concentrate at points with a soliton profile. We limit ourselves to cite few recent papers [5, 6, 8, 9, 11, 12, 15, 16], referring to their bibliography for a broader list of works, although still not exhaustive. However, at our knowledge, it is everywhere assumed (with the only exception of [18, 20]) that lim inf |x|→∞ V (x) > 0. The main new feature of the present paper is that we will be concerned with potentials V such that lim|x|→∞ V = 0. Our main results are Theorems 1 and 3. The former deals with existence of ground states of (NLS), the latter with concentration. Roughly, (NLS) has a ground state which concentrates at a global minimum of the auxiliary potential A := V θ K −2/(p−1) , where θ = (p + 1)/(p − 1) − N/2, provided (i) V (x) ∼ |x|−α with 0 < α < 2, (ii) K(x) ∼ |x|−β with β > 0, (iii) σ < p < (N + 2)/(N − 2), where σ is a number depending upon α and β, and defined in (2). Some comments on the proof and the preceding assumptions are in order. Dealing with a potential V which decays to zero at infinity, the methods used in the preceding papers cannot be employed. First of all, variational theory in W 1,2 (RN ), as in [11, 12], cannot be used. Nor one can apply perturbation methods, as in [6, 16], since the spectrum of the linear operator −∆ + V is [0, +∞), see [10]. To overcome this difficulty, we frame our problem in a class of weighted Sobolev spaces Hε , discussed in [19], consisting of the functions u on RN for which Z
ε2 |∇u(x)|2 + V (x)u2 (x) dx < ∞. RN

R In these spaces the nonlinear term RN Kup+1 dx is well-defined if (i − ii − iii) hold. Moreover, under these conditions the Euler functional satisfies the Palais-Smale compactness condition on Hε and this allows us to find in a straightforward way a positive Mountain-Pass solution vε ∈ Hε , see Theorem 13. It is worth pointing out that for such a result it suffices to assume that (i) holds with 0 < α ≤ 2. However we are interested in solutions which belong to W 1,2 and which decay to zero at infinity. To achieve these conditions we prove first some careful integral estimates for solutions in Hε . The proof of the concentration phenomenon also relies on some sharp pointwise decay estimates and on appropriate bounds of the energy of the Mountain-Pass solutions vε , uniformly with respect to ε. These estimates require α to be smaller than 2 and represent one of the main novelty of the arguments in the present paper.

2

As for the assumptions, we point out that if V ∼ |x|−α with 0 < α ≤ 2, then (iii) cannot be eliminated if we want to find ground states. For more details concerning this claim, we refer to Proposition 15 in Section 4. Concerning assumption (ii), see also Remark 14-(i). As already pointed out, the only papers dealing with equations on RN with potentials vanishing at infinity are [18] and [20]. The former deals with an eigenvalue problem in the radial case. In the latter, weighted Sobolev spaces have also been used. For more details, see Remark 14-(ii − iii) later on. However, in both the aforementioned papers neither results concerning the fact that the solutions belong to W 1,2 (RN ) are given, nor concentration is proved. The rest of the paper is organized as follows. Section 2 contains our assumptions and main results. Section 3 is devoted to discuss the weighted spaces Hε (including an embedding Theorem from [19]), as well as to prove some uniform integral estimates that are used in the sequel. In Section 4 we deal with the main existence result, Theorem 1. We first prove (see Theorem 13) that in Hε the Mountain-Pass Theorem applies in a direct way for any 0 < α ≤ 2; next, we assume that 0 < α < 2 and prove some exponential decay for the above Mountain-Pass critical points which allows us to show that they give rise to ground states of (NLS), see Theorem 16. Finally in Section 5 we prove that these ground states are spikes concentrating at a minimum of A. This result is achieved by using the preceding decay estimates, jointly with an uniform bound on the energy of the Mountain-Pass critical points found before. Notation. Throughout the paper we will use the following notation: - BR is the sphere {x ∈ RN : |x| < R}; - W k,p (Ω), W k,p (RN ), are the usual Sobolev spaces; - Lp (Ω), Lp (RN ) are the usual Lebesgue spaces. - c, c1 , . . ., C, C1 , . . . denote possibly different constants; - h1 ∼ h2 means that h1 and h2 are of the same order as ε → 0.

2

Assumptions and main results

In order to find solutions of (NLS) we will make the following assumptions on V and K (V ) V : RN → R is smooth and there exist α, a, A > 0 such that a ≤ V (x) ≤ A, 1 + |x|α and, respectively (K) K : RN → R is smooth and ∃ β, k > 0 : 0 < K(x) ≤

k . 1 + |x|β

In order to prove existence of ground states of (NLS) as well as their concentration properties we assume a suitable bound on p involving α and β. Let ( 4β N +2 if 0 < β < α N −2 − α(N −2) , (2) σ = σN,α,β = 1 otherwise. Our main existence result is the following.

3

Theorem 1 Let (V ), (K) hold, with 0 < α < 2 and β > 0, respectively, and suppose p satisfies (3)

σ
N +2 . N −2

Then for every ε > 0 equation (NLS) has a positive classical solution vε ∈ W 1,2 (RN ). Moreover, vε is a ground state of the energy functional corresponding to (NLS). Remarks 2 (i) The ground state found above is obtained as Mountain-Pass of the energy functional associated to (19) or, equivalently, it realizes the following supremum R Kup+1 RN , sup R 2 2 2 (p+1)/2 u∈Hε \{0} RN [ε |∇u| + V u ] where Hε is a suitable weighted Sobolev space defined in Section 3. Such a supremum is +∞ if p < σ as well as if p > (N + 2)/(N − 2). For more details we refer to Proposition 15. (ii) If 0 < β < α, then σ > 1 and the range of p in (3) is smaller than the usual one 1 < p < N +2 (N + 2)/(N − 2). If β = 0, we would have σ = N −2 and the interval of admissible p would be empty. (iii) When α = 2 we can still find a solution in Hε but not in W 1,2 (RN ), see Theorem 13. Concerning semiclassical states of (NLS) we show the following concentration behavior. Theorem 3 Let the same assumptions as in Theorem 1 hold. Then, the preceding ground state vε p+1 − N2 . More precisely, vε has a concentrates at a global minimum x∗ of A = V θ K −2/(p−1) , with θ = p−1 ∗ unique maximum xε for which xε → x as ε → 0, and   x − xε ∗ vε (x) = U + ωε (x), as ε → 0, ε 2 where ωε (x) → 0 in Cloc (RN ) and L∞ (RN ) as ε → 0 and U ∗ is the unique positive radial solution of

−∆U ∗ + V (x∗ )U ∗ = K(x∗ )(U ∗ )p . The proofs of the preceding two theorems will be carried out in the rest of the paper.

3

Some weighted Sobolev spaces

As anticipated in the Introduction, we will work in a class of weighted Sobolev spaces. Precisely, let us set, for all ε > 0, Z  2  1,2 N Hε = {u ∈ D (R ) : ε |∇u(x)|2 + V (x)u2 (x) dx < ∞}. RN

Hε is a Hilbert space with scalar product and norm, respectively Z  2  kuk2ε = ε |∇u(x)|2 + V (x)u2 (x) dx, N ZR  2  (u|v)ε = ε ∇u(x) · ∇v(x) + V (x)u(x)v(x) dx. RN

Set H = Hε=1 with norm k · kH . Remark 4 Since V is positive and uniformly bounded, it follows that W 1,2 (RN ) ⊂ Hε for all ε > 0.

4

Denote by LqK the weighted space of measurable u : RN → R such that  q1 Z K(x)|u(x)|q dx < ∞. |u|q,K = RN

Hε and

LqK

are particular cases of weighted spaces discussed in [19], where the following result is proved.

Theorem 5 Let N ≥ 3 and suppose that (V ), (K) hold with α ∈ (0, 2] and β > 0, respectively. Then for all ε > 0, Hε ⊂ Lp+1 provided K N +2 σ≤p≤ , N −2 and there is Cε > 0 such that |u|q,K ≤ Cε kukε ,

(4)

∀ u ∈ Hε .

Furthermore, the embedding of Hε into LqK is compact if (3) holds. In view of the preceding Theorem we will assume in the sequel that p, α and β always satisfy (3). Remarks 6 (i) If a ≤ V (x) ≤ A, namely when α = 0, we have Hε = W 1,2 (RN ) and Theorem 5 implies provided (β > 0 and) (3) holds. that W 1,2 (RN ) is compactly embedded in Lp+1 K (ii) If V ∼ (1 + |x|α )−1 and K ∼ (1 + |x|β )−1 , with 0 < α ≤ 2 and β > 0, it is proved in [19] that the N +2 N growth restriction σ ≤ p ≤ N −2 is a necessary condition for Hε to be embedded into LK (R ), see also Proposition 15. In the rest of this section we will prove some integral estimates for functions in Hε uniform with respect to ε. We anticipate that, as a byproduct, we will deduce a proof of the embedding result stated in Theorem 5, see Remark 10 below. Proposition 7 Let 0 < α < 2 and let p satisfy (3). Then for all δ > 0 there exists R > 0 such that, for all R ≥ R and all u ∈ Hε with supp{u} ∩ BR = ∅, one has Z N (5) K|u|p+1 ≤ δ ε− 2 (p−1) kukε(p+1 . RN

Proof. proof

The proof is carried out in several steps. First, let us introduce some quantities we need in the

(i) the sequence of radii Rn,ε defined by 

Rn,ε = εRn ,

2−α n Rn = a1/2

(ii) the sequence of continuous functions ψn,ε : R+ 7→ [0, 1]  if  1 r−R1,ε ψ1,ε (r) = − R2,ε −R1,ε + 1 if   0 if

2  2−α

;

satisfying 0 ≤ r ≤ Rn−1,ε , R1,ε ≤ r ≤ R2,ε , r ≥ R2,ε ,

and for n ≥ 2

ψn,ε (r) =

  0      

r−Rn,ε Rn,ε −Rn−1,ε

+1

r−Rn,ε   − Rn+1,ε −R +1  n,ε    0

5

if

0 ≤ r ≤ Rn−1,ε ,

if

Rn−1,ε ≤ r ≤ Rn,ε ,

if

Rn,ε ≤ r ≤ Rn+1,ε ,

if

r ≥ Rn+1,ε ;

(iii) a sequence of sets An,ε An,ε = {x ∈ RN : Rn−1,ε ≤ |x| ≤ Rn+1,ε }. Note that A1,ε is a ball, An,ε is an annulus for n ≥ 2, and that the ψn,ε ’s have been chosen in such a way that X u(x) = ψn,ε (|x|)u(x). n

These cut-off functions are useful to estimate integrals over RN by means of a discrete sum of integrals on the annuli An,ε . Lemma 8 There exists c > 0 such that Z Z ∞ X 1 |ψn,ε u|p+1 , Kup+1 ≤ c β 1 + R A RN n,ε n,ε n=1

∀ u ∈ Hε , ∀ε ∈ (0, 1].

Proof. On An,ε one has u = ψn−1,ε u + ψn,ε u + ψn+1,ε u (with abuse of notation we are taking A0,n = ∅), which implies ! Z Z Z Z p+1 p+1 p+1 p+1 Ku ≤ sup K |ψn,ε u| + |ψn,ε u| + |ψn,ε u| . An,ε

An,ε

An−1,ε

An,ε

An+1,ε

Since the width of An,ε is small with respect to Rn,ε then there exists c2 = c2 (K) such that sup K ≤ c2

(6)

An,ε

The last two formulas imply Z Kup+1 ≤ c2

≤ c2

1+

β Rn+1,ε

1

Z

β 1 + Rn−1,ε Z 1

An,ε

+

1

β 1 + Rn+1,ε

1 1+

β Rn,ε

≤ c2

|ψn,ε u|p+1 +

An−1,ε

1 β 1 + Rn−1,ε

.

Z

1 β 1 + Rn,ε

|ψn,ε u|p+1

An,ε

! |ψn,ε u|p+1

.

An+1,ε

Summing over all the integers n, the Lemma follows. R Next, we estimate each term An,ε |ψn,ε u|p+1 . Let γ satisfy γ(2∗ − 2) = p − 1, or equivalently 2∗ γ = (p − 1)N/2. Then one has Z Z |ψn,ε u|p+1 =

An,ε

|ψn,ε u|2

∗

γ+2−2γ

.

An,ε

Using the H¨older’s inequality we find Z

|ψn,ε u|p+1 ≤ c1

"Z

An,ε

|ψn,ε u|2

∗

#γ "Z

An,ε

#1−γ |ψn,ε u|2

.

An,ε

∗

From the embedding of D1,2 into L2 we infer "Z Z p+1

|ψn,ε u|

(7)

≤ c2

An,ε

2

#2∗ γ/2 "Z

#1−γ 2

|∇(ψn,ε u)|

|ψn,ε u|

An,ε

.

An,ε

From (6) and (7) we get Z (8)

Ku An,ε

p+1

≤ c3

1 β 1 + Rn,ε

"Z

2

#2∗ γ/2 "Z

|∇(ψn,ε u)| An,ε

|ψn,ε u| An,ε

We now show 6

#1−γ 2

.

Lemma 9 There holds

Z

Z

2

  |∇u|2 + ε−2 V u2

|∇(ψn,ε u)| ≤ c4 An,ε

An,ε

Proof. First we evaluate |∇(ψn,ε u)|2 = |u∇ψn,ε + ψn,ε ∇u|2 ≤ 2u2 |∇ψn,ε |2 + 2|∇u|2 .

(9)

From the definition of Rn,ε we get α |Rn+1,ε − Rn,ε |2 = ε2 |Rn+1 − Rn |2 ≥ cε2 Rn+1,ε . −α As above, Rn+1,ε ≤ c5 inf An,ε V , and we deduce

|∇ψn,ε |2 ≤ c6 |Rn+1,ε − Rn,ε |−2 ≤ c7 ε−2 V,

x ∈ An,ε .

Substituting in (9) and integrating over An,ε , the lemma follows. Proof of Proposition 7 completed. Lemma 9 together with (8) yields Z (10)

Ku

p+1

≤c

An,ε

#2∗ γ/2 "Z   |∇u|2 + ε−2 V u2

"Z

1 β 1 + Rn,ε

An,ε

#1−γ 2

|ψn,ε u|

.

An,ε

Let M, s > 0 and let θ, θ0 be any pair of conjugate exponents (M, s, θ, θ0 will be fixed appropriately later). For brevity, let us set Z   S = Sn,ε = |∇u|2 + ε−2 V u2 An,ε

Z T = Tn,ε

|ψn,ε u|2

= An,ε

so that (10) becomes Z

Kup+1 ≤ c

An,ε

1 1+

β Rn,ε

S2

∗

γ/2

· T 1−γ .

Since S2

∗

γ/2

·

T (1−γ) β 1 + Rn,ε

= M εs S 2

∗

γ/2

· M −1 ε−s

0

T (1−γ) β 1 + Rn,ε

≤

0 0 T (1−γ)θ 1 θ sθ 2∗ γθ/2 1 M ε S + 0 M −θ ε−sθ  θ 0 , θ θ β 1 + Rn,ε

we get Z (11)

Kup+1 ≤ c1 M θ εsθ S 2

∗

γθ/2

An,ε

0

T (1−γ)θ

0

+ c2 M −θ ε−sθ 

1+

β Rn,ε

0

θ 0 .

Now we choose s, θ satisfying 2∗ γθ = p + 1,

θ(s − 2∗ γ) = −sθ0 .

Then S

2∗ γθ/2

"Z =

#2∗ γθ/2 "Z   2 −2 2 −(p+1) |∇u| + ε V u =ε

An,ε

An,ε

7

#(p+1)/2  2  2 2 ε |∇u| + V u ,

and hence θ sθ

(12)

M ε S

2∗ γθ/2

θ −(p−1)N/2

#(p+1)/2  2  2 2 . ε |∇u| + V u

"Z

=M ε

An,ε

On the other hand we also have T (1−γ)θ

0

β (1 + Rn,ε )θ0

≤

#(p+1)/2

"Z

1

u

β (1 + Rn,ε )θ0

An,ε

α (p+1)/2 (1 + Rn,ε )

=

"Z

β (1 + Rn,ε )θ0

An,ε

α (p+1)/2 (1 + Rn,ε )

≤

"Z

β Rn,ε )θ0

(1 +

2

#(p+1)/2 u2 α 1 + Rn,ε #(p+1)/2 V u2

.

An,ε

Inserting the above inequality and (12) into (11), and taking into account that −sθ0 = θ(s − 2∗ γ) = −(p − 1)N/2, we infer Z

Kup+1 ≤ c3 ε−(p−1)N/2 ×

An,ε

 M θ

#(p+1)/2  #(p+1)/2 "Z α (p+1)/2  2  0 (1 + Rn,ε ) . V u2 · |ε ∇u|2 + V u2 + M −θ β )θ0 (1 + Rn,ε An,ε An,ε

"Z

Now, let us remark that α (p+1)/2 ) (1 + Rn,ε

(1 +

β )θ0 Rn,ε

0

−βθ +α(p+1)/2 ∼ Rn,ε → 0,

(Rn,ε → ∞),

since p > σ implies that −βθ0 + α(p + 1)/2 < 0. Then, given δ > 0, we can choose M, R > 0 such that Mθ <

δ , 2c3

0

0

and M −θ R−βθ +α(p+1)/2 <

δ 2c3

for R ≥ R,

yielding Z

Kup+1 ≤ δ ε−(p−1)N/2

#(p+1)/2  2  ε |∇u|2 + V u2 ,

"Z

An,ε

An,ε

provided that Rn−1,ε > R. Summing over these annuli An,ε and using the fact that supp{u} ∩ BR = ∅ for all R ≥ R we get Z Ku

p+1

≤ δε

−(p−1)N/2

|x|>R

" X Z

#(p+1)/2  2  2 2 ε |∇u| + V u .

An,ε

n

Setting Z

 2  ε |∇u|2 + V u2 ,

an = an,ε = An,ε

one has (p+1)/2

α en

P

kuk2ε

P < ∞. Letting α en = an / an , we have that 0 < α en ≤ 1 for all n and hence p+1 P P (p+1)/2 (p+1)/2 2 −1 ≤ α en , namely an ≤ ( an ) an . Summing over all n, it follows that an ≤ an <

8

P (p+1)/2 ( an ) < ∞. This implies "

Z Ku

p+1

≤ δε

−(p−1)N/2

XZ

≤ δε

−(p−1)N/2

kukp+1 , ε

|x|>R

#(p+1)/2  2  2 2 ε |∇u| + V u

An,ε

completing the proof of Proposition 7. Remark 10 For ε = 1 the preceding arguments give an alternative proof of the embedding result stated in Theorem 5. To see this, let us write u = χR u + (1 − χR )u, where χR is a cut off function such that χR ≡ 0 on BR , χR ≡ 1 for |x| ≥ R + 1, Rand χR is linear on R < |x| < R + 1.R For σ < p < (N + 2)/(N − 2) we can use inequality (5) to estimate RN K |χR u|p+1 , while the integral |x|≤R+1 K |(1 − χR )u|p+1 can be bounded by using the standard Sobolev embedding theorem. If σ ≤ p ≤ (N + 2)/N − 2), the above method shows that there exists C > 0 and R  1 for which Z (p+1)/2 Kup+1 ≤ CkukH , ∀ u ∈ H. |x|>R

Moreover, modifying the definition of Rn,ε (with a logarithmic dependence on n) we could also recover the embedding in the case α = 2. Proposition 11 Let 0 < α < 2 and let p satisfy (3). Then for all δ > 0 there exists R > 0 such that, for all R ≥ R one has (13) !(p+1)/2 Z Z   K(x)|u|p+1 (x)dx ≤ δ ε−(p−1)N/2 ε2 |∇u(x)|2 + V (x)u2 (x) dx , ∀ u ∈ Hε . |x|>R

|x|>R

Proof. Let ψeR,ε : R+ → [0, 1] be a smooth non-decreasing function such that ( 0 if 0 ≤ r ≤ R − εRα/2 , e ψR,ε (r) = 1 if r ≥ R, 0 satisfying |ψeR,ε (r)| ≤ 2ε−1 R−α/2 . Define, in polar coordinates (r, ϑ) ∈ R+ × S N −1 ,

( ψeR,ε (r)u(2R − r, ϑ) if u eR,ε (r, ϑ) = u(r, ϑ) if

R − εRα/2 ≤ r ≤ R, r > R.

In the annulus AR,ε = {R − εRα/2 < |x| < R} there holds (subscripts denote partial derivatives) 1 0 ∇e uR,ε = −ψeR,ε (r)ur (2R − r, ϑ)er + ψeR,ε (r)uϑ (2R − r, ϑ)eϑ + ψeR,ε (r)u(2R − r, ϑ)er , r where er =

x |x|

and eϑ is a unit vector tangent to the unit sphere {|x| = 1}. Thus, in AR,ε one finds |∇e uR,ε |2 ≤ c1 |∇u(2R − r, ϑ)|2 + c2 ε−2 R−α u2 (2R − r, ϑ).

Let us explicitly point out that here and below the constants ci do not depend upon R, ε. Integrating in AR,ε and performing the change of variable (r, ϑ) 7→ (2R − r, ϑ) we get (14) Z Z Z    2  |∇e uR,ε |2 ≤ c3 |∇u|2 + ε−2 R−α u2 ≤ c4 ε−2 ε |∇u|2 + V (x)u2 . AR,ε

R<|x|
R<|x|
9

In the preceding equation we have taken into account that u eR,ε ≡ u for |x| > R. From (14) we infer Z Z  2  (15) |∇e uR,ε |2 ≤ c5 ε−2 ε |∇u|2 + V (x)u2 . |x|>R

AR,ε

Moreover, similar arguments yield Z Z 2 (16) V (x)e uR,ε ≤ c6

V

R<|x|
AR,ε

(x)e u2R,ε

Z

V (x)u2 .

≤ c6 |x|>R

From (15) and (16) we deduce Z Z  2  2 2 (17) ε |∇e uR,ε | + V (x)e uR,ε ≤ c7

 2  ε |∇u|2 + V (x)u2 .

|x|>R

AR,ε

From the embedding (4) and since u eR,ε = u for r ≥ R, we get Z Z p+1 K(x)|u| ≤ K(x)|e uR,ε |p+1 . RN

|x|>R

From Proposition 7 we have Z

p+1

K|e uR,ε |

≤ δε

−N 2 (p−1)

Z RN

RN

≤ δε

−N 2 (p−1)

 p+1  2  2 2 2 ε |∇e uR,ε | + V u eR,ε

Z AR,ε

 2  ε |∇e uR,ε |2 + V u e2R,ε +

Z |x|>R

 p+1  2  2 2 2 . ε |∇e uR,ε | + V u eR,ε

From this and (17) we finally find (13).

4

Proof of the existence results

This section is devoted to the proof of Theorem 1, which is divided into two parts. First, we show the existence of a least-energy solution in Hε , see Theorem 13 below; in the second part of the section we prove that such a ground state belongs indeed to W 1,2 (RN ). Let us start by introducing the functional set up. If (3) holds, then Theorem 5 applies yielding Z (18) K(x)|u(x)|p+1 dx < ∞, ∀ u ∈ Hε . RN

Define

Iε (u)

= =

Z ε2 |∇u(x)|2 dx + 12 V (x)u2 (x)dx − RN RN Z 2 1 1 K(εx)|u(x)|p+1 dx. 2 kukε − p+1 1 2

Z

1 p+1

Z

K(x)|u(x)|p+1 dx

RN

RN

From (18) and (V ) it follows that Iε is well defined on Hε for all ε > 0. Moreover, Iε is of class C 1 and Z  2  (Iε0 (u)|v) = ε ∇u(x) · ∇v(x) + V (x)u(x)v(x) − K(x)|u(x)|p−1 u(x)v(x) dx, ∀ v ∈ Hε . RN

Hence any critical point uε ∈ Hε of Iε is a weak solution of (NLS).

10

Remark 12 According to Remark 6-(ii), if V ∼ (1 + |x|α )−1 and K ∼ (1 + |x|β )−1 , with 0 < α ≤ 2 and N +2 β > 0, then the growth restriction σ ≤ p ≤ N −2 is necessary in order to work in Hε with the functional Iε . Critical points of Iε can be found by the Mountain Pass Theorem in a straightforward way. Theorem 13 Let (V ), (K) hold with 0 < α ≤ 2, β > 0, respectively, and suppose that p satisfies (3). Then bε = inf max Iε (tu) u∈Hε \{0}

t≥0

is a critical level of Iε . Hence for all ε > 0 the equation −ε2 ∆v + V (x)v = K(x)v p ,

(19)

x ∈ RN ,

has a positive (classical) solution vε ∈ Hε . Moreover, there exists C > 0 such that kvε k2ε ≤ C bε .

(20)

Proof. Let φ be a smooth positive function with compact support in RN . Then (recall that p > 1) one has Iε (tφ) → −∞ as t → +∞. Hence Iε has the M-P geometry. Since Hε is compactly embedded into Lp+1 then standard arguments imply that bε is a M-P critical level carrying a critical point vε ∈ Hε of K Iε which is a weak solution of (NLS). Since V and K are smooth, local regularity implies that vε is in fact a classical solution. It is also standard to see that vε > 0. From −ε2 ∆vε + V (x)vε = K(x)vεp we infer

Z RN

 2  ε |∇vε |2 + V (x)vε2 dx =

Thus bε = Iε (vε ) =



1 2

−

1 p+1

Z RN

Z RN

K(x)vεp+1 dx .

  2  ε |∇vε |2 + V (x)vε2 dx = 21 −

1 p+1



kvε k2ε .

This concludes the proof.

Remarks 14 (i) Assumption (V ) includes potentials which are bounded away from zero (namely 0 < inf RN V ≤ supRN V < +∞). In this case, the space Hε is nothing but W 1,2 (RN ) and in order to recover compactness our approach requires β > 0, see Remark 6-(i). Let us recall that when, in addition, also K is bounded away from zero, (namely 0 < inf RN K ≤ supRN K < +∞), proving the existence of solutions to (19) requires appropriate assumptions on V and/or K, see the papers cited in the Introduction and [4]. On R the other hand, it is well known that if β = 0 a necessary condition for (NLS) to have a solution is that RN ∂xi V (x)u2 (x)dx = 0. Moreover, if (V ) holds with 0 < α ≤ 2 and K is bounded away from zero, then the critical level bε (or the supremum considered in the statement of Proposition 15) is clearly equal to +∞. Of course, different is the story if we look for solutions that are not ground states. For example, it is proved in [6, 24] that if 0 < inf RN V ≤ supRN V < +∞ and 0 < inf RN K ≤ supRN K < +∞, then a solution exists provided ε is sufficiently small and the auxiliary potential A has a stable stationary point. (ii) In [18, Thm. 2] the authors consider the eigenvalue problem ( −∆u + V (|x|)u = λK(|x|)up , x ∈ RN , 2 u ∈ Cloc (RN ), lim|x|→∞ u(x) = 0. ∗

proving the existence of positive solutions (λ, uλ ), uλ ∈ L2 (RN ) and uλ = O(r(2−n)/2 ), r = |x|. It is assumed that V (r) ≥ 0 and that K(r) = O(r−β ), β > 0. In addition, if β < 2, it is required that p > (N + 2 − 2β)/(N − 2). Let us point out that the last condition is stronger than ours (p > σ). (iii) Theorem 13 follows from [20, Thm. 3.1] combined with Theorem 5. Moreover, the case in which N +2 p = σ or p = N −2 is also studied in [20, Thm. 3.2], under some further restrictions on V and K. 11

It is also worth pointing out that if σ < p (here we take 0 < β < α, otherwise σ = 1 and p satisfies the usual growth assumption), Iε has no Mountain-Pass solution. Proposition 15 If either p < σ or p > (N + 2)/(N − 2), then R K|u|p+1 RN sup = +∞.
R 2 |∇u|2 + V u2 ] (p+1)/2 u∈Hε \{0} [ε N R Proof. We can assume for simplicity that ε = 1. Let us consider a function Ψ with compact support, and let (21)

α

|ξ|  1, λ = |ξ|− 2 .

uξ (x) = Ψ(λ(x − ξ)),

From the definition of uξ and the conditions on λ, ξ, see (21), we easily find Z Z Z Z Z Z C −1 C 2 2−N 2 2 2 p+1 (22) |∇uξ | = λ |∇Ψ| ; V uξ ≥ α N Ψ ; K|u| ≤ β N |Ψ|p+1 . |ξ| λ |ξ| λ Hence it follows that R Kup+1 α |ξ|−β λ−N ξ RN 4 [(N +2)−p(N −2)]−β → +∞ R (p+1)/2 ≤ C p+1 = C|ξ| −N −α 2 (λ |ξ| ) [|∇uξ |2 + V u2ξ ] RN

as |ξ| → +∞,

by the fact that p < σ. N +2 On the other hand, also in the case p > N −2 it is standard to see that the above supremum is +∞. It is sufficient for example to consider the family of functions with λ → +∞.

uλ (x) = Ψ(λx);

In the second part of this section we will show that the Mountain-Pass solutions of (NLS) found above belong indeed to W 1,2 (RN ), provided 0 < α < 2. Theorem 16 Let (V ), (K) hold with 0 < α < 2, β > 0, respectively, and suppose that p satisfies (3). Then the Mountain-Pass solution vε found in Theorem 13 is a ground state of (NLS). In particular, vε ∈ W 1,2 (RN ), vε ∈ C 2 (RN ), vε (x) > 0 and lim|x|→∞ vε (x) = 0. The proof of Theorem 16 requires some preliminary decay estimates, based upon the results discussed in Section 3. Let us point out that, for the purpose of the concentration phenomena discussed in Section 5, the decay is proved with estimates which are uniform in ε. In these lemmas is always understood that the assumptions of Theorem 16 hold true. Lemma 17 Let vε be solutions of (19) and suppose there exists Γ > 0 such that kvε k2Hε ≤ Γ εN .

(23)

Then there exists RΓ > 0 such that for all R ≥ RΓ and all Ωn,ε ⊆ RN \ BR one has Z Z  2   2  3 ε |∇vε |2 + V (x)vε2 dx, ε |∇vε |2 + V (x)vε2 dx ≤ 4 Ωn,ε Ωn+1,ε 2

where Ωn,ε = RN \ BRn,ε and Rn,ε = εn 2−α . 12

Proof. Let Rn,ε be as in the statement, and let χn,ε (r) be piecewise affine functions such that χn,ε (r) ≡ 0,

∀ r ≤ Rn,ε ,

χn,ε (r) ≡ 1,

∀ r ≥ Rn+1,ε .

By the definition of Rn,ε it follows that |Rn+1,ε − Rn,ε | ≥ C −1 ε

2−α 2

α/2

α/2

Rn+1,ε ≥ C −1 εRn+1,ε .

Then −α ε2 |Rn+1,ε − Rn,ε |−2 ≤ c1 Rn+1,ε ≤ c2 inf {V (x) : Rn,ε ≤ |x| ≤ Rn+1,ε },

and hence ε2 |∇χn,ε (x)|2 ≤ V (x),

(24)

∀ x ∈ RN .

Let us test (19) on χn,ε vε . Recalling that χn,ε = 0 on BRn,ε and that χn,ε ≤ 1, we get Z Z Z  2  2 2 p+1 2 χn,ε ε |∇vε | + V vε = χn,ε Kvε − ε ∇vε · ∇χn,ε vε Ωn,ε

Ωn,ε

Z ≤ Ωn,ε

Ωn,ε

Z

Kvεp+1 + 21 ε2

  |∇vε |2 + |∇χn,ε |2 vε2 .

Ωn,ε

Using (24) we infer ε2

Z

  |∇vε |2 + |∇χn,ε |2 vε2 ≤

Z

Ωn,ε

 2  ε |∇vε |2 + V (x)vε2 .

Ωn,ε

From the two last equations, it follows that Z Z  2  2 2 ε |∇vε | + V vε dx ≤ Ωn+1,ε

  χn,ε ε2 |∇vε |2 + V vε2

Ωn,ε

Z

Kvεp+1

≤

+

Ωn,ε

1 2

Z

 2  ε |∇vε |2 + V (x)vε2 .

Ωn,ε

Then, from Proposition 11, if δ > 0 is given and R is sufficiently large we deduce Z  2  ε |∇vε |2 + V vε2 dx Ωn+1,ε

≤ δε

−(p−1) N 2

Z

!(p+1)/2  2  ε |∇vε |2 + V (x)vε2 dx

+

Ωn,ε

1 2

Z

 2  ε |∇vε |2 + V (x)vε2 .

Ωn,ε

Now we write Z

!(p+1)/2  2  2 2 ε |∇vε | + V (x)vε dx

Ωn,ε

Z =

!(p−1)/2 Z  2  ε |∇vε |2 + V (x)vε2 dx

Ωn,ε

 2  ε |∇vε |2 + V (x)vε2 dx.

Ωn,ε

From (23) and the last two formulas it follows that  Z Z  2   2  p−1 1 2 2 2 ε |∇vε | + V vε dx ≤ + δΓ ε |∇vε |2 + V (x)vε2 dx. 2 Ωn+1,ε Ωn,ε Choosing δ sufficiently small (and hence for R large), the Lemma follows.

13

Lemma 18 Let vε be solutions of (19), and let Γ, RΓ be as above. Then, for all ρ ≥ 2RΓ there holds   Z  2−α 2−α   2  3 1 , (25) ε |∇vε |2 + V (x)vε2 dx ≤ C Γ εN exp − log ε−1 ρ 2 − RΓ 2 2 4 |x|>ρ for some constant C Γ depending only on Γ. Proof. Given ρ > 2RΓ , let n ˜ > n be positive integers such that Rn,ε ≤ RΓ ≤ Rn+1,ε ; From 17, we deduce Z Z  2  ε |∇vε |2 + V (x)vε2 ≤ |x|>ρ

Rn˜ −1,ε ≤ ρ ≤ Rn˜ ,ε .

 2  ε |∇vε |2 + V (x)vε2 ≤

|x|>Rn,ε ˜

 n˜ −n Z  2  3 ε |∇vε |2 + V (x)vε2 . 4 |x|>RΓ

Then (23) implies Z

  2 ε |∇vε |2 + V (x)vε2 ≤

(26) |x|>ρ

 n˜ −n 3 ΓεN . 4

˜ , there holds By our choices of n, n 2

2

ρ ∼ ε2/(2−α) n ˜ 2−α ;

RΓ ∼ ε2/(2−α) n 2−α ,

which imply 2−α  1 −1  2−α ε ρ 2 − RΓ 2 . 2 The estimate in (26) and the last formula conclude the proof.

n ˜−n≥

Proof of Theorem 16. Here ε > 0 is fixed and can be taken equal to 1 to simplify the notation. Let v ∈ H be any solution of (19) (with ε = 1) and let y ∈ RN be such that |y| > 2. There holds Z Z Z 1 2 2 α v = V (x)v · ≤ c1 |y| V (x)v 2 . V (x) B1 (y) B1 (y) B1 (y) For R = 21 |y| we have

Z

V (x)v 2 ≤

Z RN \B

B1 (y)

V (x)v 2 . R

From the two preceding equations and Lemma 18 we get Z (27) v 2 ≤ C3 |y|α exp{−C4 |y|1−α/2 },

∀ |y|  1.

B1 (y)

Sm Let m ∈ N and yi ∈ RN , i = 1, . . . , m be such that B5 \ B2 ⊂ i=1 B1 (yi ), and let yi,k = 2k yi . Then we get Z ∞ Z X XZ v2 ≤ v2 ≤ v2 . RN \B2

k=0

2k (B5 \B2 )

i,k

B2k (yi,k )

To estimate the right hand side, we use (27) for k  1, which yields Z X v 2 ≤ C3 |yi,k |α exp{−C4 |yi,k |1−α/2 } < ∞, RN \B2

i,k

since 0 < α < 2. This shows that v ∈ L2 (RN ), whence v ∈ W 1,2 (RN ). As already pointed out in Theorem 13, v ∈ C 2 (RN ) and v > 0. Finally, standard arguments show that lim|x|→∞ v(x) = 0, see for example [22]. 14

5

Semiclassical limits for (N LS)

In this section we study the behavior of some solutions of (N LS) as ε tends to 0, and in particular of those obtained in Theorem 13. We always assume that (V ), (K) hold true with 0 < α < 2 and β > 0, and that p satisfies (3). However some results, as Lemma 19 below, hold even if 0 < α ≤ 2. The next Lemma provides an upper bound for the critical values bε in terms of the auxiliary functional A = V θ K −2/(p−1) introduced in Theorem 3. It is worth pointing out explicitly that, since p > σ, A(x) → ∞ as |x| → ∞, and therefore A it has a global minimum on all of RN . Lemma 19 There exists C0 > 0 such that for all ξ ∈ RN and all ε sufficiently small one has ε−N bε = ε−N Iε (vε ) ≤ C0 A(ξ) + o(1)

(28)

as ε → 0+ .

In particular there exists C ∗ > 0 such that bε ≤ C ∗ εN . Proof. For any ξ ∈ RN , let us define the functional Fξ on W 1,2 (RN ) by setting Z Z Z 2 2 1 1 1 |∇u| + 2 V (ξ) u − p+1 K(ξ) |u|p+1 . Fξ (u) = 2 RN

RN

RN

Let f (ξ) denote the Mountain-Pass critical level of Fξ . It is well known that f (ξ) = inf Fξ (u), u∈Nξ

where Nξ is the Nehari manifold  Z 1,2 N Nξ = u ∈ W (R ) \ {0} :

Z

2

Z

2

|∇u| + V (ξ)

|u|

u = K(ξ)

RN

RN

p+1

 .

RN

Let us point out that u ∈ Nξ if and only if 1

1

1

u ˆ(y) := K p−1 (ξ)V − p−1 (ξ)u(V (ξ)− 2 y) ∈ N , R R where N = {u ∈ W 1,2 (RN ) : u 6= 0 and RN |∇u|2 + u2 = RN |u|p+1 }. Hence, with a direct calculation we find Z   1 f (ξ) = inf Fξ = 12 − p+1 K(ξ) inf |u|p+1 dx Nξ u∈Nξ RN Z   p+1 2 − p−1 −N 1 1 p−1 2 (ξ)V (ξ) inf |v|p+1 dy. = 2 − p+1 K v∈N

RN

¯ denote the unique positive radial solution in W 1,2 (RN ) of Let U ¯ +U ¯ =U ¯p −∆U Since inf v∈N (29)

R RN

in RN .

¯ , we get |v|p+1 dy is achieved at U 2 − p−1

f (ξ) = [K(ξ)]

[V (ξ)]

p+1 N p−1 − 2



1 2

−

1 p+1

Z

¯ |p+1 dx = C0 A(ξ). |U

RN

Since f (ξ) is a mountain-pass level of Fξ , for all ν > 0 there exists w ∈ W 1,2 (RN ) such that f (ξ) ≤ max Fξ (tw) ≤ f (ξ) + ν. t>0

Let ϕ ∈ C 2 (RN ) be a cut-off function such that ϕ ≡ 1 in a neighborhood of ξ and define, for any ε > 0, wε ∈ W 1,2 (RN ) by x − ξ  wε (x) = ϕ(x)w . ε 15

Since W 1,2 (RN ) ⊂ Hε , then wε ∈ Hε for any ε; in particular it makes sense to compute Iε (twε ), which yields Z t2 tp+1 Iε (twε ) = kwε k2ε − K(x)|wε |p+1 dx. 2 p + 1 RN By the change of variable y = x−ξ ε , we get Z Z ε−N kwε k2ε = ε2 |∇ϕ(εy + ξ)|2 w2 (y) dy + ε ∇w(y) · ∇ϕ(εy + ξ)w(y)ϕ(εy + ξ) dy RN RN Z Z + ϕ2 (εy + ξ)|∇wε (y)|2 dy + V (εy + ξ)ϕ2 (εy + ξ)w2 (y) dy, RN

as well as ε−N

Z

RN

K(x)|wε (x)|p+1 dx =

Z

RN

K(εy + ξ)|ϕ(εy + ξ)w(y)|p+1 dy.

RN

Putting together the preceding equations we deduce that ε−N Iε (twε ) = Fξ (tw) + o(1)

as ε → 0.

Hence ε−N Iε (vε )

=

inf

v∈Hε \{0}

max ε−N Iε (tv) t>0

≤ max Iε (twε ) ≤ max Fξ (tw) + o(1) t>0

t>0

≤ f (ξ) + ν + o(1) = C0 A(ξ) + ν + o(1). Since ν > 0 is arbitrary, the estimate in (28) is proved. The last statement follows from the fact that A has a global minimum on RN since p > σ. Remark 20 To prove that bε ≤ C ∗ εN one could also argue as follows. Consider the functionals Ieε , Ibε : W 1,2 (RN ) → R, Z Z  2  Ieε (u) = 1 ε |∇u|2 + Au2 dx − 1 K(x)|u|p+1 dx, 2

and Ibε (u) =

1 2

p+1

RN

Z

  |∇u|2 + Au2 dx −

RN

1 p+1

RN

Z

K(εx)|u|p+1 dx.

RN

Clearly, u bε (x) is a critical point of Ibε iff u eε (x) := u bε (x/ε) is a critical point of Ieε ; moreover, Ieε (e uε ) = Nb e b e b uε ). Let bε , resp. bε , denote the Mountain-Pass critical level of Iε , resp. Iε . Since sup V ≤ A ε Iε (b and W 1,2 (RN ) ⊂ Hε , one easily deduces that bε ≤ ebε = εN bbε . On the other hand, critical points of Ibε can be found near those of the unperturbed functional Ib0 ≡ Ibε=0 . Up to translation, we can assume that K(0) = max K. Let U be the unique positive radial solution of −∆U + A U = K(0) U p ,

U ∈ W 1,2 (RN ).

Then, using [1], one infers that Ibε has, for ε > 0 small, a critical point uε such that u bε → U , as ε → 0. b b b In particular, from Iε (b uε ) → I0 (U ) it follows that there exists C > 0 such that Iε (b uε ) ≤ C for all ε > 0 small enough. Moreover, since U is a Mountain Pass critical point of Ib0 , the same holds for uε . This implies that bbε ≤ Ibε (b uε ), and the result follows. From Lemma 19 and (20) it follows Corollary 21 For ε small there exists Γ > 0 such that kvε k2ε ≤ Γ εN , where vε , is given by Theorem 13. 16

The next Lemma provides pointwise uniform decay estimates for the solutions vε . Here 0 < α < 2 is needed. We follow closely the method illustrated in [21], Appendix B. Lemma 22 Let Γ, RΓ and vε be as in Lemma 17. Then there exists a constant C, depending only on Γ, p and N , and a positive number d > 0, depending on N p, α and β, such that    2−α 2−α  3 1 (30) |vε (x)| ≤ C|x|d ε−d exp − log ε−1 |x| 2 − RΓ 2 ; for |x| ≥ 2RΓ + C. 4 4 Proof. The functions vε satisfy the equation −ε2 ∆vε + V (x)vε = K(x)vεp .

(31)

Given x0 ∈ RN , with |x0 | ≥ 2RΓ + 2, we consider a smooth cutoff function η satisfying  for x ∈ B1 (x0 )  1, 0, for x ∈ RN \ B2 (x0 ) (32) η(x) =  |∇η| ≤ 2. Letting for simplicity v = vε , given L > 0 and s ≥ 0, we also define the following function φ = φs,L ≡ v min{|v|2s , L2 }η 2 . Testing (31) on φ we obtain Z Z Z s 2 2 2 2s−2 2 2 2 2s 2 2 η + V (x)v 2 η 2 min{|v|2s , L2 } ∇(|v| ) v ε |∇v| min{|v| , L }η + ε 2 s {|v| ≤L} Z Z 2 2s 2 ≤ −2ε vη min{|v| , L }∇v · ∇η + Kv p+1 η 2 min{|v|2s , L2 } Z Z Z 1 ≤ ε2 |∇v|2 min{|v|2s , L2 }η 2 + Cε2 v 2 min{|v|2s , L2 }|∇η|2 + Kv p+1 η 2 min{|v|2s , L2 }. 2 Hence, if we set w = η v min{|v|s , L},

(33)

from the above inequality we get Z Z Z (34) ε2 |∇w|2 + V (x)w2 ≤ Cε2 v 2 min{|v|2s , L2 } + Kv p+1 η 2 min{|v|2s , L2 }. Next, given M > 0, we divide the last integral into the two regions {v ≤ M } and {v > M } to obtain Z Kv p+1 η 2 min{|v|2s , L2 } Z Z ≤ M p−1 Kη 2 v 2 min{|v|2s , L2 } + Kv p−1 η 2 v 2 min{|v|2s , L2 }. {v>M }∩B2 (x0 )

By the H¨older and the Sobolev inequalities we can write Z

Kv p−1 η 2 v 2 min{|v|2s , L2 } ≤

{v>M }∩B2 (x0 )

(Kv p−1 )

N 2

kvk2L2∗

{v>M }∩B2 (x0 )

Z ≤ C

! N2

Z

(Kv

p−1

)

N 2

! N2 Z

2

|∇v| = Cε

{v>M }∩B2 (x0 )

−2

! N2

Z (Kv

p−1

)

N 2

ε2

Z

|∇v|2 .

{v>M }∩B2 (x0 )

R N If we can make (Kv p−1 ) 2 sufficiently small, then we can bring this term on the left-hand side of (34). We note that, since kvk2L2∗ (RN ) ≤ CεN (by our assumptions), there holds Z ∗ 2N 2N N −2 M |{v > M }| ≤ v 2 ≤ CεN ⇒ |{v > M }| ≤ CεN M − N −2 . 17

Next, from the H¨ older’s inequality we get Z (Kv

p−1

)

N 2

! q1

Z ≤

v

{v>M }∩B2 (x0 )

(p−1) N 2 q

q0 =

2N N −2 ,

q

K

{v>M }

If we choose q in such a way that (p − 1) N2 q =

! 10

Z

N 2

q0

.

{v>M }

namely if

4 , 4 − (p − 1)(N − 2)

from the above equations it follows that Z ε2 |∇w|2 + V (x)w2 Z Z Z 4 − (N −2)q 2 2s+2 p−1 2 2s+2 −2 −β 0 2 ε |∇v|2 ≤ Cε v +M Kη v + Cε |x0 | M B2 (x0 ) B2 (x0 ) Z Z 4−(p−1)(N −2)
− 2 p−1 −β 2s+2 −2 −β 2 (N −2) v ε |∇v|2 . ≤ C ε +M |x0 | + Cε |x0 | M B2 (x0 )

Now we choose M in such a way that ε−2 |x0 |−β M −

4−(p−1)(N −2) (N −2)

M = C ε−2 |x0 |−β

is a small constant, namely we take

−2
4−(NN−2)(p−1)

with C sufficiently large. In this way, choosing s in such a way that 2s + 2 = p + 1, we get Z Z
ε2 |∇w|2 + V (x)w2 ≤ C ε2 + |x0 |−β M p−1 v 2s+2 B2 (x0 ) Z 2(p−1)(N −2) (35) ≤ Cε− 4−(N −2)(p−1) v p+1 . B2 (x0 )

From our assumptions on the functions v = vε and the H¨older’s inequality it follows that !ω !1−ω !ω Z Z Z Z p+1 2 2∗ N (1−ω) α 2 v ≤ v + v ≤ Cε |x0 | Vv , B2 (x0 )

B2 (x0 )

B2 (x0 )

B2 (x0 )

for some ω ∈ (0, 1). By Lemma 18 and the last two equations, we obtain   Z 2−α  2−α p + 1 3 −1  2 2 2 d1 −d1 2 2 ε |∇w| + V (x)w ≤ C|x0 | ε exp − log ε − RΓ |x0 | , 4 4 Rn

|x0 | ≥ 2RΓ + 2,

for some constant C depending on Γ, p and N , and some positive number d1 > 0, depending on N , p, α and β. We note at this point that the last estimate is independent of the number L in the definition of w. 1,2 This implies that |v|s+1 belongs to Wloc (Rn ), with some quantitative estimates on the integrals, which (s+1)2∗ are given in the last formula. Then the Sobolev’s embedding theorem implies v ∈ Lloc . Finally, proceeding in this way and using a bootstrap argument, we obtain the result after a finite number of steps. Remark 23 Although we already proved that lim|x|→∞ vε (x) = 0 for any fixed ε > 0, the preceding Lemma is needed since it gives a pointwise decay uniform in ε. Lemma 24 Let vε be solutions of (19) satisfying (23). Let xε denote any maximum point of vε . Then there exists a constant C > 0, C = C(Γ) such that |xε | ≤ C, for every ε sufficiently small. 18

Proof. Since xε is a maximum of vε , one has ∆vε (xε ) ≤ 0. Therefore, from (19) it follows that V (xε )K −1 (xε ) ≤ vεp−1 (xε ).

(36)

From (V ) and (K) it follows that there exists c > 0 such that c |xε |β−α ≤ V (xε )K −1 (xε ).

(37)

From (36), (37), and (30) we deduce, if |xε | ≥ 2RΓ ,  1 (38) c |xε |β−α ≤ |xε |d(p−1) ε−d(p−1) exp − (p − 1) log 4

 1−α  2−α 3 − 2−α  2 a 2 |x | − R ε . ε Γ 4

This immediately implies that |xε | stays bounded as ε → 0. Lemma 24 is thereby proved. Lemma 25 Let vε be as in Lemma 24. Then there exists a constant C > 0 such that kvε kL∞ ≥ C −1 , for all ε sufficiently small. Proof. From (19) we get Z

kvε k2ε =

(39) Let us fix δ < Γ−

p−1 2

(40)

RN

 2  ε |∇vε |2 + V (x)vε2 =

Z RN

K(x)vεp+1 .

. Then from Proposition 11 there exists R such that Z K(x)vεp+1 dx ≤ δε−N (p−1)/2 kvε kp+1 . ε |x|>R

From (V ) and (K) we have K(x) ≤ hence

Z

k 1 + |x|α k V (x) ≤ (1 + Rα )V (x) β a 1 + |x| a

K(x)vε2

|x|≤R

From this it follows Z (41)

K(x)vεp+1

≤

k ≤ (1 + Rα ) a

kvεp−1 kL∞

Z

V (x)vε2 ≤

|x|≤R

Z

|x|≤R

K(x)vε2 ≤

|x|≤R

for any |x| ≤ R k (1 + Rα )kvε k2ε . a

k (1 + Rα )kvεp−1 kL∞ kvε k2ε . a

From (39), (40), and (41) we get Z k kvε k2ε = K(x)vεp+1 ≤ δε−N (p−1)/2 kvε kp+1 + (1 + Rα )kvεp−1 kL∞ kvε k2ε ε a N R which yields (42)

1 ≤ δε−N (p−1)/2 kvε kp−1 + ε

In view of the estimate kvε kp−1 ≤Γ ε

p−1 2

εN

p−1 2

k (1 + Rα )kvεp−1 kL∞ . a

which follows from (23), (42) implies

k e p−1 2 1 ≤ δC + (1 + Rα )kvεp−1 kL∞ a hence, for our choice of δ, we deduce e kvε kp−1 L∞ ≥ (1 − δ C

p−1 2

)

a >0 k(1 + Rα )

which proves the Lemma. We are now in position to characterize the ground states when ε tends to 0. 19

Theorem 26 Let the same assumptions as in Theorem 16 hold. Then (N LS) has a (classical, positive) p+1 ground state vε concentrating, as ε → 0, at a global minimum x∗ of A = V θ K −2/(p−1) , where θ = p−1 − N2 . ∗ More precisely, vε has a unique maximum xε such that xε → x as ε → 0, and   x − xε vε (x) = U ∗ + ωε (x), as ε → 0, ε 2 where ωε (x) → 0 in Cloc (RN ) and L∞ (RN ) as ε → 0, and U ∗ is the unique positive radial solution of

−∆U ∗ + V (x∗ )U ∗ = K(x∗ )(U ∗ )p . Proof. The proof is based upon the preceding lemmas and is rather standard, see e.g. [12, 24]. However, to keep the paper as self-contained as possible, we will carry out the arguments in details. Let xε denote a global maximum of vε (such a maximum exists since vε (x) → 0 as |x| → ∞). From Lemma 24, we have that, up to a subsequence, xε → x∗ for some x∗ ∈ RN . Set ψε (x) := vε (εx + xε ). Since vε solves (19), then ψε satisfies (43)

−∆ψε (x) + V (εx + xε )ψε (x) = K(εx + xε )ψεp (x),

x ∈ RN .

Using Corollary 21 and assumption (V ) it follows that Z  2  Γ ≥ ε−N kvε k2ε = ε−N ε |∇vε (x)|2 + V (x)vε2 (x) dx RN   Z  Z  a a 2 2 2 v (x) dx = |∇ψ (y)| + ψ (y) dy. ≥ ε−N ε2 |∇vε (x)|2 + ε 1 + |x|α ε 1 + |εy + xε |α ε RN RN From Lemma 24 we infer that |εy + xε | ≤ C(1 + |y|) and therefore we obtain  Z  a 2 ψ (y) dy ≤ C 0 , |∇ψε (y)|2 + 1 + |y|α ε RN ∞ where C 0 is independent of ε. In particular {ψε }ε is bounded in Cloc , uniformly with respect to ε and we 2 N ∗ 2 deduce that ψε converges in Cloc (R ) to some function U ∈ Cloc (RN ). Furthermore, using arguments similar to those carried out in the proof of Lemma 22, one infers that ψε → U ∗ in L∞ (RN ), too. Passing to the limit in equation (43), we find that U ∗ ≥ 0 is a classical solution to equation

(44)

−∆U ∗ (x) + V (x∗ )U ∗ (x) = K(x∗ )(U ∗ )p (x),

x ∈ RN .

Moreover, since ψε attains its maximum at 0 then U ∗ also attains its maximum at 0. Furthermore, Lemma 25 yields that ψε (0) = vε (xε ) = kvε kL∞ ≥ C −1 for some positive constant C, and thus max U ∗ = U ∗ (0) ≥ C −1 > 0. In particular, U ∗ 6≡ 0 (hence U ∗ > 0 by the maximum principle) and is a radial function according to the Gidas-Ni-Nirenberg result [13]. Using again Corollary 21 we get, for any sequence Rn → ∞, Z   (45) |∇ψε (x)|2 + V (εx + xε )ψε2 (x) dx ≤ ε−N kvε k2ε ≤ Γ. B Rn

Since ψε → U ∗ in C 1 (B Rn ), the Dominated Convergence Theorem allows us to pass to the limit in (45) as ε → 0 yielding Z   |∇U ∗ (x)|2 + V (x∗ )(U ∗ )2 (x) dx ≤ Γ. B Rn

Letting now Rn → ∞, we infer that U ∗ ∈ W 1,2 (RN ). To complete the proof of Theorem 26, a further Lemma is in order, which provides a lower bound for bε in terms of U ∗ and x∗ .. 20

Lemma 27 Let Fξ be as in the prof of Lemma 19. Then Fx∗ (U ∗ ) ≤ lim inf ε−N Iε (vε ) = lim inf ε−N bε .

(46)

ε→0

Proof. One has Iε (vε ) = ε

R N RN

ε→0

hε (x)dx where

hε (x) = 21 |∇ψε (x)|2 + 12 V (εx + xε )ψε2 (x) −

(47)

1 p+1 K(εx

+ xε )ψεp+1 (x).

Let R > 0 to be chosen later. In view of the C 1 -convergence of ψε to U ∗ over the compact sets of RN we get Z Z Z Z 1 lim hε dx = 12 |∇U ∗ |2 dx + 12 V (x∗ ) (U ∗ )2 dx − p+1 K(x∗ ) (U ∗ )p+1 dx. ε→0

∗

BR

Since U ∈ W Z (48) lim ε→0

1,2

BR

BR

BR

BR

N

(R ), for any ν > 0 we can choose R > 0 large enough such that Z h i ∗ 2 ∗ ∗ 2 ∗ ∗ p+1 1 1 1 dx − ν = Fx∗ (U ∗ ) − ν. hε dx ≥ |∇U | + V (x )(U ) − K(x )(U ) 2 2 p+1 RN

Let now ηR be a cut-off function such that ηR = 0 in BR−1 , ηR = 1 in RN \ BR , 0 ≤ ηR ≤ 1, |∇ηR | ≤ C, with C independent of R. Testing (43) on ηR ψε we obtain Z  Z 2 2 hε dx + p+1 −1 K(εx + xε )ψεp+1 dx + Eε = 0, RN \BR

where

Z

RN \BR

  ∇ψε · ∇(ηR ψε ) + V (εx + xε )ηR ψε2 − K(εx + xε )ηR ψεp+1 dx.

Eε = BR \BR−1

1 Hence RN \BR hε dx ≥ −Eε /2. Again by the convergence of ψε in Cloc to U ∗ ∈ W 1,2 (RN ), we deduce that for R large enough limε→0 |Eε | ≤ ν and hence Z ν (49) lim inf hε dx ≥ − . ε→0 2 RN \BR

R

From (48) and (49) we conclude Z lim inf ε→0

RN

hε dx ≥ Fx∗ (U ∗ ) − 23 ν

for any ν > 0 and (46) follows. Proof of Theorem 26 completed. Let us first prove that x∗ is a minimum of the function f (ξ) = C0 A(ξ). Arguing by contradiction, we assume that there exists ξ ∗ ∈ RN such that f (x∗ ) > f (ξ ∗ ). From (46) and (28), it follows that Fx∗ (U ∗ ) ≤ lim inf ε−N Iε (vε ) ≤ C0 A(ξ), ∀ ξ ∈ RN . ε→0

∗

On the other hand, since U solves equation (44), Fx∗ (U ∗ ) ≥ inf Fx∗ (u) = f (x∗ ) > f (ξ ∗ ) = C0 A(ξ ∗ ). u∈Nx∗

which yields a contradiction. It remains to show that vε has at most one maximum point. The proofs relies on the arguments carried out above and so we will be sketchy. By contradiction, assume that, up to a subsequence, vε has two distinct maxima xε , zε . From Lemma 24 it follows that there exists x∗ , z ∗ ∈ RN such that xε → x∗ 2 and zε → z ∗ . Let ψε and U ∗ be as above. The convergence of ψε to U ∗ in Cloc and the properties of U ∗ 21

readily imply that there exists r > 0 such that ψε00 (x) < const < 0 for x ∈ Br provided ε is small enough. Noticing that ε−1 (zε − xε ) is a maximum point of ψε , two cases can occur. Case 1: ε−1 (zε − xε ) is bounded and hence, up to a subsequence, it converges to some P ∈ RN . Since ψε (ε−1 (zε − xε )) = max ψε converges to max U ∗ = U ∗ (0), we conclude that P = 0. Therefore for ε sufficiently small ε−1 (zε − xε ) ∈ Br , which is impossible since 0 is the only critical point of ψε in Br . Case 2: ε−1 (zε − xε ) is unbounded, and hence it tends to ∞, up to a subsequence. As above, one shows 2 ˜ ∗ , where ψeε := vε (εx + zε ) and U ˜ ∗ is the unique positive radial solution in that ψeε Cloc -converges to U 1,2 N W (R ) of ˜ ∗ (x) + V (z ∗ )U ˜ ∗ (x) = K(z ∗ )(U ˜ ∗ )p (x), x ∈ RN . −∆U ε

Let us remark that, since |ε−1 (zε − xε )| → ∞, then for any R, the balls B R and B := B R (ε−1 (zε − xε )) are disjoint provided ε is small enough. Using this fact and repeating the arguments carried out above, we readily find that for any ν > 0 it is possible to choose R > 0 large enough such that Z ˜ ∗ ) − ν, (50) lim hε ≥ Fz∗ (U ε→0

B

ε

as well as Z (51)

hε ≥ −ν.

lim inf ε→0

RN \(BR ∪B ε )

From (48), (50) and (51) we conclude Z lim inf ε→0

˜ ∗ ) − 3ν. hε ≥ Fx∗ (U ∗ ) + Fz∗ (U

RN

Since ν is arbitrary we find (52)

˜ ∗ ). lim inf ε−N bε ≥ Fx∗ (U ∗ ) + Fz∗ (U ε→0

˜ ∗ ) ≤ f (x∗ ). Since x∗ and z ∗ are both global minima From (28) and (52) it follows that Fx∗ (U ∗ ) + Fz∗ (U ∗ ∗ of f , we have f (x ) = f (z ) and hence, using the definition of f , we deduce

˜ ∗) ˜ ∗ ) ≤ 1 f (x∗ ) + f (z ∗ ) ≤ 1 Fx∗ (U ∗ ) + Fz∗ (U Fx∗ (U ∗ ) + Fz∗ (U 2 2 which is not possible. The proof is now complete. Acknowledgements. The work has been supported by M.U.R.S.T. under the national project ”Variational methods and nonlinear differential equations”. Part of the work has been carried out during a visit of A.A. and A.M. at the Bernoulli Center of the EPFL, Lausanne, in the frame of the Semester “Nonlinear Analysis and Applications”: Variational Methods and Nonlinear Waves”. They would like to thank the Center for the kind hospitality. A.M. has also been hosted by the Mathematics Department of ETH in Z¨ urich, to which he is very grateful.

References [1] Ambrosetti A., Badiale M.: Homoclinics: Poincar´e-Melnikov type results via a variational approach. Ann. Inst. H. Poincar´e Analyse Non Lin´eaire 15 (1998), 233-252. [2] Ambrosetti A., Badiale M.: Variational perturbative methods and bifurcation of bound states from the essential spectrum. Proc. Royal Soc. Edinburg 128-A (1998), 1131-1161.

22

[3] Ambrosetti, A., Badiale, M., Cingolani, S.: Semiclassical states of nonlinear Schr¨odinger equations. Arch. Rational Mech. Anal. 140 (1997), 285-300. [4] Ambrosetti, A., Garcia Azorero, J., Peral, I.: Remarks on a class of semilinear elliptic equations on Rn , via perturbation methods, Advanced Nonlin. Studies, 1 (2001), 1-13. [5] Ambrosetti, A., Malchiodi, A., Ni, W.-M.: Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I, Comm. Math. Phys. 235 (2003), 427-466. [6] Ambrosetti, A., Malchiodi, A., Secchi, S.: Multiplicity results for some nonlinear singularly perturbed elliptic problems on Rn . Arch. Rat. Mech. Anal. 159, 253-271 (2001). [7] Badiale, M., D’Aprile, T.: Concentration around a sphere for a singularly perturbed Schr¨odinger equation. Nonlin. Anal. T.M.A. 49 (2002), 947-985. [8] Bahri, A., Li, Y.Y.: On a Min-Max Procedure for the Existence of a Positive solution for Certain Scalar Field Equations in RN , Revista Mat. Iberoamericana (6) 1&2 (1990), 1-16. [9] Bahri, A., Lions P.L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. Henri Poincar´e (14) 3 (1997), 365-413. [10] Berezin, F.A., Shubin, M.A.: The Schr¨ odinger equation, Kluwer Acad. Publ. 1991 [11] Byeon, J., Wang, Z-Q. : Standing waves with a critical frequency for nonlinear Schr¨odinger equations, Arch. Rat. Mech. Anal. 165 (2002), 295-316. [12] Del Pino, M., Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. 4, 121-137 (1996). [13] Gidas, B., Ni, W.-M., Nirenberg, L.:Symmetry of positive solutions of nonlinear elliptic equations in Rn . Advances in Mathematics Supplementary Studies, Vol. 7 - A (A. Nachbin, ed.), 369 - 402. [14] Grillakis, M., Shatah, J., Strauss, W. : Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 74 (1987), 160-197. [15] Grossi, M.: On the number of single-peak solutions of the nonlinear Schr¨odinger equation. Ann. Inst. H. Poincar Anal. Non Linaire 19 -3 (2002), 261-280. [16] Li, Y.Y. : On a singurarly perturbed elliptic equation, Adv. Diff. Eqns. 2 (1997), 955-980. [17] McLeod, J. B., Stuart, C. A., Troy, W. C.: Stability of standing waves for some nonlinear Schr¨odinger equations. Diff. Int. Equat. 16-9 (2003), 1025–1038. [18] Noussair, E.S., Swanson, C.A. : Decaying solutions of semilinear elliptic equations in RN , SIAM J. Math. Anal. 20-6 (1989), 1336-1343. [19] Opic, B., Kufner, A.: Hardy-type inequalities, Pitman Res. Notes in Math. Series, 219, Longman Scientific & Technical, Harlow, 1990. [20] Schneider, M.: Entire solutions of semilinear elliptic problems with indefinite nonlinearities, PhD Thesis, Univerist¨ at Mainz, 2001. [21] Struwe, M.: Variational methods : applications to nonlinear partial differential equations and Hamiltonian systems, Springer-Verlag, Berlin , 2000. [22] Stuart, C., An introduction to elliptic equations on RN , in Nonlinear Functional Analysis and Applications to Differential Equations, World Sci., 1988, p. 237-285. [23] Stuart, C. A., Zhou, H. S.: A constrained minimization problem and its application to guided cylindrical TM-modes in an anisotropic self-focusing dielectric. Calc. Var. 16-4 (2003), 335–373. 23

[24] Wang, X., Zeng, B.: On concentration of positive bound states of nonlinear Schr¨odinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), 633-655.

24