0 Introduction - CAPDE

capde.cl

0 Introduction - CAPDE

LOCAL MOUNTAIN PASSES FOR SEMILINEAR

ELLIPTIC PROBLEMS IN UNBOUNDED DOMAINS .

by

Manuel del Pino 1

Mathematics Department, University of Chicago

Chicago, IL, U.S.A.

and

Patricio L. Felmer 2

Departamento de Ingeniería Matemática F.C.F.M.

Universidad de Chile

Casilla 170 Correo 3, Santiago, CHILE.

0 Introduction

This paper has been motivated by some works appeared in recent years concerning

the nonlinear Schrödinger equation

i¯h ∂ψ

∂t = − ¯h2

2m ∆ψ + V (z)ψ − γ | ψ |p−1 ψ. (0.1)

1 Partially supported by Grant

2 Partially supported by FONDECYT Grant No 1212-91 and Fundación Andes.

1


We are interested in solutions of the form ψ(z, t) = exp(−iEt/¯h)v(z) which are called

standing waves. We observe that this ψ satisfies (0.1) if and only if the function v(z)

solves the elliptic equation

¯h 2

2m ∆v − (V (z) − E)v − γ | v |p−1 v = 0. (0.2)

In [3], Floer and Weinstein consider the case N = 1 and p = 3.

For a given

nondegenerate critical point of the potential V , assumed globally bounded, and for

0 < E < inf V , they construct a standing wave provided that ¯h is sufficiently small.

This solution concentrates around the critical point as ¯h → 0.

Their method, based on an interesting Lyapunov-Schmidt reduction, was extended

by Oh in [7], [8] to conclude a similar result in higher dimensions, provided that

1 < p < N+2 . He restricts himself to potentials with “mild oscillation” at infinity,

N−2

namely belonging to a Kato class. In case that V is bounded this restriction is not

necessary as observed by Wang in [12].

In [9], Oh uses this approach to construct solutions with multiple peaks which

concentrate around any prescribed finite set of nondegenerate critical points of V .

The method in the works mentioned above, local in nature, seems to use in a

essential way the nondegeneracy of the critical points. In [11], Rabinowitz uses a

global variational technique to find a solution with “minimal energy” for all small ¯h,

when 1 < p < (N + 2)/(N − 2) and

lim inf

|z|→∞

V (z) > inf V (z), (0.3)

z∈RN see also [2] for related results. A precise concentration statement (around a global

minimum, possibly degenerate) for this solution is established in [12].

A question raised to us by Changfeng Gui, which motivates the present work, is

whether one can find solutions which concentrate around local minima not necessarily

nondegenerate.

As we will see, the answer is affirmative, and moreover the same is true if one

considers the elliptic problem in an arbitrary domain in R N

conditions.

2

with zero boundary


More generally, we deal with a semilinear elliptic problem of the form

{

ε 2 ∆u − V (z)u + f(u) = 0 in Ω

u > 0 in Ω

(0.4)

where Ω is a domain in R N , possibly unbounded, with empty or smooth boundary. V

will be assumed throughout this paper locally Hölder continuous and bounded below

away from zero, say

V (z) ≥ α > 0, for all z ∈ R N . (0.5)

We will also assume that f : R + → R is locally Hölder continuous and satisfies

the following conditions.

(f1) f(ξ) = o(ξ) near ξ ≥ 0.

f(ξ)

(f2) lim = 0 for some 1 < s < N+2.

ξ→∞ ξ s N−2

(f3) For some 2 < θ ≤ s + 1 we have


where F (ξ) = ξ f(τ)aτ.

(f4) The function ξ → f(ξ)

ξ

0

0 ≤ θF (ξ) < f(ξ)ξ for all ξ > 0 (0.6)

in nondecreasing.

Our main result for equation (0.4) is the following.

Theorem 0.1 Assume there is a bounded domain Λ compactly contained in Ω such

that

inf

Λ

V < min V. (0.7)

∂Λ

Then there is an ε 0 > 0 such that for every 0 < ε < ε 0 a positive solution u ε ∈ H 1 0(Ω)

to problem (0.4) exists. Moreover, u ε possesses just one local (hence global) maximum

z ε , which is in Λ. We also have that V (z ε ) → inf Λ V , and

for certain constants α, β.

u ε (z) ≤ α exp(− β ε |z − z ε|), (0.8)

3


We observe that no restriction on the global behavior of V is required other than

(0.5). In particular, V is not required to be bounded or to belong to a Kato class.

On the other hand, the hypotheses on the nonlinearity f are fairly mild, in the

sense that no special nondegeneracy or uniqueness assumptions need to be made on

the “limiting” equation associated to (0.4), namely an equation of the form

∆u − u + f(u) = 0.

The proof of this result is variational, and relies on an elementary idea which

permits to identify “local mountain passes”. The energy functional associated to

equation (0.4) generally satisfies the assumptions of the Mountain Pass Theorem,

except for the P.S. condition. Indeed, a global assumption like (0.3) permits to

recover P.S. at the mountain pass level when ε is small.

What we do in our situation is to build a convenient modification of the energy,

in such a way that the functional satisfies P.S., and then prove that for a sufficiently

small ε the associated mountain pass is indeed a solution to the original equation with

the stated properties. The modification of the functional corresponds to a “penalization

outside Λ”, and this is why no other global assumptions are required. Though

elementary, we believe this idea is flexible enough to be adapted to “catch mountain

passes” in a number of interesting situations.

To ilustrate this point, we will see in the last section how local properties of an

unbounded domain lead to existence and localization of solutions to elliptic problems.

As an example of the situation covered by Theorem 3.1 in §3, let us consider the case

of an axially symmetric domain Ω ⊂ R N , N ≥ 2 given by

Ω = {(t, x) ∈ R N−1 × R / |x| < ρ(t)} (0.9)

where ρ : R → R + is a smooth function. We consider the problem

{

ε 2 u tt + ∆u − u + f(u) = 0 in Ω

u = 0 on ∂Ω and u > 0 in Ω

(0.10)

Here ∆ denotes the Laplacian with respect to the N − 1-dimensional variable x, and

f satisfies (f1)-(f4). We have

4


Theorem 0.2 Assume that for some bounded interval I we have that

sup ρ > ρ| ∂I . (0.11)

I

Then there exists an ε 0 > 0 such that for all 0 < ε < ε 0 there is a positive solution

u ε ∈ H 1 0(Ω) to problem (0.10) such that for some sequence of t ε ∈ I with ρ(t ε ) →

sup I ρ one has

for certain positive constants α and β.

u ε (x, t) ≤ α exp(− β ε |t − t ε|), (0.12)

The organization of this paper is as follows: In §1 we define the modification of the

functional needed for the proof of Theorem 0.1, and prove some preliminary results.

§2 is devoted to the proof of Theorem 0.1, while in §3 we use a similar approach to

study the influence of the domain, and prove a general result from which Theorem

0.2 follows as a particular case.

1 Preliminaries

In this section we establish some preliminary results needed for the proof of Theorem

0.1 and the results of §3.

In the framework of Theorem 0.1, let us consider a function f : R → R satisfying

(f1)-(f4) on R + and defined as zero for negative values.

Associated to equation (0.4) is the “energy” functional

I ε (u) = 1 ∫


|∇u| 2 + V (z)u 2 − F (u) , (1.1)

2

which is well defined for u ∈ H where

Ω

H = {u ∈ H 1 0(Ω) /


Ω

Ω

V (z)u 2 < ∞}.

H becomes a Hilbert space, continuously embedded in H 1 0(Ω), when endowed with

the inner product


< u, v > V =

Ω

ε 2 ∇u · ∇v + V (z)uv (1.2)

5


whose associated norm we denote by ‖ · ‖ H .

Under the regularity assumptions on V and f, it is standard to check that the

nontrivial critical points of J ε correspond exactly to the positive classical solutions in

H 1 0(Ω) of equation (0.4).

We will define a modification of this functional which satisfies the P.S. condition

and to which we can apply the Mountain Pass Theorem. Let θ be a number as given

by (f3), and let us choose k > 0 such that k >

f(a)

a

= α , where α is as in (0.5). Let us set

k

θ . Let a > 0 be the value at which

θ−2

˜f(s) =

{

f(s) if s ≤ a

α

k s if s > a, (1.3)

and define

g(·, s) = χ Λ f(s) + (1 − χ Λ ) ˜f(s), (1.4)

where Λ is a bounded domain as in the assumptions of Theorem 0.1. and χ Λ denotes

its characteristic function. It is easy to check that (f1)-(f4) implies that g defined in

this way is a Caratheodory function and it satisfies the following assumptions:

(g1) g(z, ξ) = o(ξ) near ξ = 0 uniformly in z ∈ Ω.

g(z,ξ)

(g2) lim ξ→∞ = 0 for some 1 < s < N+2 if N ≥ 3 and no restriction on s if

ξ s N−2

N = 1, 2.

(g3) There exist a bounded subset K of Ω, int(K) ≠ ∅, and 2 < θ < s + 1 so that

(i) 0 < θG(z, ξ) ≤ g(z, ξ)ξ for all z ∈ K, ξ > 0.

and

(ii) 0 ≤ 2G(z, ξ) ≤ g(z, ξ)ξ ≤ 1 k V (z)ξ2 for all ξ ∈ R + , z ∉ K with the

number k satisfying k > θ/(θ − 2).

(g4) The function ξ → g(z,ξ)

ξ

in nondecreasing for all z ∈ Ω a.e. and ξ > 0.

Here we have denoted G(z, ξ) = ∫ ξ

0 g(z, τ)dτ.

6


Since a similar approach will be used for the results of §3, we consider the more

general framework of an equation of the form

∆u − V (z)u + g(z, u) = 0 in Ω, (1.5)

with V as before, satisfying (0.5), and g satisfying (g1)-(g4). Here we have set ε = 1

for notational simplicity.

Then we consider the functional J : H → R associated to this equation,

J(u) = 1 ∫


|∇u| 2 + V (z)u 2 − G(z, u) , u ∈ H. (1.6)

2

Ω

Then J is of class C 1 in H. Let us notice that the function f considered in the introduction

satisfies the properties given for g, except for (g3) (ii). This last assumption

implies that J satisfies the Palais Smale condition as we show next.

Lemma 1.1 Let {u n } be a sequence in H such that J(u n ) is bounded and J ′ (u n ) → 0.

Then u n has a convergent subsequence.

Proof. We have that {u n } is bounded in H. In fact, using (g3) we easily see that



|∇u n | 2 + V (z)u 2 n ≥ g(z, u n )u n + o(‖u n ‖ H ). (1.7)

Ω

It also follows from the assumptions that


1

2

Ω

|∇u n | 2 + V (z)u 2 n =



Ω


K

K

Ω

G(z, u n ) + O(1)

G(z, u n ) + 1

2k


Ω\K

V (z)u 2 n + O(1). (1.8)

Thus, from (1.7), (1.8) and (g3) we find

( θ ∫

2 − 1) |∇u n | 2 + V (z)u 2 n ≤ θ

2k

Ω


Ω\K

V (z)u 2 n + o(‖u n ‖ H ) + O(1). (1.9)

Then, it follows from the choice of k in (g3) that {u n } is bounded as desired.

7


Let us choose a subsequence, still denoted by {u n }, weakly convergent to u in H.

This convergence is actually strong. Indeed, it suffices to show that, given δ > 0,

there is an R > 0 such that


lim sup {|∇u n | 2 + V (z)u 2 n} < δ, (1.10)

n→∞

Ω\B R

where B R denotes the ball with center 0 and radius R. We may assume that R is

chosen so that K ⊂ B R/2 .

Let η R be a cut-off function so that η R = 0 on B R/2 , η R = 1 on Ω \ B R , 0 ≤ η R ≤ 1

and |∇η R | ≤ c/R. Since {u n } is a bounded P.S. sequence, we have that

< J ′ (u n ), η R u n >= o(1), (1.11)

so that



{|∇u n | 2 + V (z)u 2 n}η R +

u n ∇u n · ∇η R =


g(z, u n )u n η R + o(1)

Ω

Ω


Ω

1 k


V (z)u 2 nη R + o(1). (1.12)

Ω

We conclude that


from where (1.10) follows. ✷

Ω\B R

|∇u n | 2 + V (z)u 2 n ≤ C R ‖u n‖ L 2 (Ω)‖∇u n ‖ L2 (Ω) + o(1), (1.13)

The previous lemma makes the Mountain Pass Theorem applicable to the functional

J. In fact, on the one hand, it is easily checked using (g1) that

J(u) ≥ c‖u‖ 2 H (1.14)

for ‖u‖ H small enough. Next, choosing φ ∈ H \ {0} non negative, with its support

contained in the set K given in (g3), we see that

J(tφ) → −∞ as t → +∞. (1.15)

8


The Mountain Pass Theorem thus implies that the following is a critical value of J

where P = {γ ∈ C([0, 1], H) / γ(0) = 0 and J(γ(1)) < 0}.

c = inf sup J(γ(t)), (1.16)

γ∈P t∈[0,1]

Condition (g4) implies that the critical value c can be characterized in a simpler

way, as has been essentially established in [11] and [2].

completeness.

Lemma 1.2 If (g1)-(g4) hold then

c = inf

u∈H

u≠0

We provide a proof for

sup J(τu). (1.17)

τ≥0

Proof. Let u ∈ H, u ≠ 0, then the function τ → J(τu) has at most one nontrivial

critical point for τ ≥ 0 and it satisfies


Ω


|∇u| 2 + V (z)u 2 =

Ω

g(z, τu)u

, (1.18)

τ

this is because (g2), (g4) and (g5). We define ϕ(u) as the unique solution of (1.18)

if a solution exists, and we put ϕ(u) = ∞ when no solution exists. If we put C ∗ the

right hand side in (1.17) we see that C ≤ C ∗ . In order to prove the other inequality

we note that

C ∗ = inf J(v) (1.19)

v∈M

where M = {v = ϕ(u)u / u ∈ H, u ≠ 0, ϕ(u) < ∞}. Thus we only have to show that

given γ ∈ P there exist ˜τ ∈ [0, 1] such that γ(˜τ) ∈ M. Assuming the contrary we

have γ(τ) ∉ M for all τ ∈ [0, 1]. In view of (g1) and (1.18)


1

2

Ω

|∇γ(τ)| 2 + γ(τ) 2 > 1 2

Then, using (g3) (i) and (ii) we find for all τ ∈ [0, 1]

J(γ(τ)) > ( θ 2 − 1) ∫

K


Ω

g(z, γ(τ))γ(τ). (1.20)

G(z, γ(τ)) ≥ 0, (1.21)

9


and this contradicts the definition of P. ✷

We end this section with a comparison result between critical values of an “autonomous”

functional which will be used in §3. Let f satisfy assumptions (f1)-(f4)

and let us consider two open subsets of R N , Ω 1 and Ω 2 . Then we write

I(u) = 1 2


R N

|∇u| 2 + u 2 −


R N F (u). (1.22)

Let us consider the mountain pass values c i , i = 1, 2 of I when restricted to H 1 0(Ω i ),

namely

Then we have,

c i =

inf

u∈H 1 0 (Ω i )

u≠0

sup I(τu). (1.23)

τ≥0

Lemma 1.3 If Ω 1 ⊂ Ω 2 , Ω 1 ≠ Ω 2 and c 1 is a critical value of the restriction of I to

H 1 0(Ω 1 ), then

c 2 < c 1 . (1.24)

Proof. A proof of this fact was given in [1], so that we only provide a sketch. Let

u 1 ∈ H 1 0(Ω 1 ) be a critical point associated to c 1 . Then u 1 is a positive solution of an

elliptic equation in Ω 1 , but it does not satisfies the same equation in Ω 2 , by unique

continuation. Thus u 1 is not a critical point of I in H 1 0(Ω 2 ). Then we can deform

the compact set K = {tu 1 /τ ∈ [0, τ 1 ]}, where τ 1 is such that I(τu 1 ) < 0, using a

pseudogradient vector field. This will provide a path with values strictly less that c 1 .


2 Local Mountain Pass: The Potential

We devote this section to the proof of Theorem 0.1, on the existence of solutions of

the equation (0.4). As we already mentioned, in order to localize the mountain passes

we had modified the function f. We consider the functional I defined in (1.6) and

10


the functional

J ε (u) = ε2

2


Ω

|∇u| 2 + 1 2


Ω


V (z)u 2 −

Ω

G(z, u) , u ∈ H,

on H. Since the modified function satisfies the hypotheses (g1)-(g4) of §1 the functional

J ε , associated to the modified equation, have good compactness properties. It

turns out that J ε satisfies the Palais-Smale condition and that one can apply the

Mountain Pass Theorem to it. More precisely, the following result, as shown in §1,

holds.

Lemma 2.1 The functional J ε possesses a positive critical point u ε ∈ H so that

J ε (u ε ) = inf

u∈H

u≠0

sup J ε (τu). (2.1)

τ≥0

Next we use a test function to derive a useful estimate of the number in (2.1). Set

V 0 = min Λ V and let w be a solution to

∆w − V 0 w + f(w) = 0

in H 1 (R N ) which maximizes at zero. Also set

Then w satisfies

I 0 (v) = 1 2


R N

c = I 0 (w) =

|∇v| 2 + V 0 v 2 −

inf

v∈H 1 (R N )

v≠0


R N F (v). (2.2)

sup I(τv). (2.3)

τ≥0

Then, let z 0 ∈ Λ be such that V (z 0 ) = V 0 and set u(z) = η(z)w((z − z 0 )/ε) where η

is a smooth function compactly supported in Ω and with η ≡ 1 in a neighbourhood

of z 0 . A direct computation then shows that

with o(1) → 0 as ε → 0.

J ε (u ε ) ≤ sup J ε (tu) = ε N {c(w) + o(1)}, (2.4)

t>0

11


On the other hand, combining this, the fact that J ε(u ′ ε )u ε = 0 and using the

definitions of g and the number θ we easily obtain that

where C > 0 and

thanks to our choice of k.


γ

Ω

ε 2 | ∇u ε | 2 +V (z)u 2 ε ≤ Cε N , (2.5)

γ = θ 2 (θ − 2

θ

− 1 k ) > 0

We devote what remains of this section to prove that when ε is sufficiently small

u ε actually solves (0.4) and satisfies (0.8). Let a > 0 be the number chosen in the

definition of g. Then the desired result will follow if we show that for all small ε one

has u ε < a on Ω \ Λ. Crucial step in the proof of this fact is the following

Proposition 2.1 If m ε is given by

then

m ε = max

z∈∂Λ u ε(z), (2.6)

lim m ε = 0. (2.7)

ε→o

Moreover, for all ε sufficiently small u ε possesses at most one local maximum z ε ∈ Λ

and we must have

lim V (z ε) = V 0 = min V (z), (2.8)

ε→o

Before proving this fact, let us see how Theorem 0.1 follows from it.

Proof of Theorem 0.1. By Proposition 2.1 there exists ε 0 such that for all 0 < ε <

ε 0 ,

The function u ε ∈ H solves the equation

z∈Λ

u ε (z) < a for all z ∈ ∂Λ. (2.9)

ε 2 ∆u − V (x)u + g(x, u) = 0 in Ω. (2.10)

12


Since u ε ∈ H 1 0(Ω) we can choose (u ε − a) + as a test function in (2.10) so that, after

integration by parts one gets


ε 2 | ∇(u ε − a) + | 2 +c(z)(u ε − a) 2 + + c(z)a(u ε − a) + = 0, (2.11)

where

Ω\Λ

c(z) = V (z) − g(z, u ε(z))

. (2.12)

u ε (z)

The definition of g yields that c(z) > 0 in Ω \ Λ, hence all terms in (2.11) are zero.

We conclude in particular

u ε (z) ≤ a for all z ∈ Ω \ Λ.

Consequently u ε is a solution to equation (0.4). To obtain (0.8), we note that the

maximum value of u ε is achieved at a point z ε ∈ Λ and it is away from zero. Then

we set

From (2.5) we see that

v ε (z) = u ε (z ε + εz).

for some C > 0. Note also that v ε satisfies

‖v ε ‖ H 1 (R N ≤ C (2.13)

∆v ε − V (z ε + εz)v ε + f(v ε ) = 0.

Now, each sequence ε j → 0 possesses a subsequence which we label in the same way

such that z εj → z 0 , with V (¯z) = V 0 . From (2.13), and elliptic estimates, we have that

this subsequence can be chosen in such a way that v εj → v uniformly over compacts,

where v ∈ H 1 (R N ) maximizes at zero and solves

∆v − V 0 v + f(v) = 0 in R N . (2.14)

It is well known that positive solutions to (2.14) decay exponencially and they are

radially symmetric. See the works of Gidas, Ni and Nirenberg [4] and [5]. It follows

from (2.13), (2.14) and the growth assumptions on f that

I 0 (v) ≤ C (2.15)

13


where the constant C does not depend on v, but on the family {v ε }. Then we obtain

the existence of R > 0 so that v(z) < a for all |z| = R.

From these facts, one can easily conclude the existence of a number R > 0 such

that v ε (z) < a if |z| = R, for all ε > 0 sufficiently small.

Since no other local

maxima of v ε exists, as Proposition 2.1 states, we conclude that actually v ε (z) < a

for all |z| > R. Then we can use the maximum principle to conclude that v ε ≤ w 0

for | z |> R, where w is an appropriate multiple of the fundamental solution of

∆w − bw = 0 with b = α − f(a)/a > 0. Since w 0 decays exponentially, relation (0.8)

follows, and the proof of the theorem is complete. ✷

To prove the proposition we will establish the following fact: If ε n ↓ 0 and z n ∈ Λ

are such that u εn (z n ) ≥ b > 0, then

lim V (z n) = V 0 . (2.16)

n→∞

We argue by contradiction. Thus we assume, passing to a subsequence, that z n →

¯z ∈ Λ and

V (¯z) > V 0 . (2.17)

Similarly to the proof of the theorem given above, we consider the sequence

v n (z) = u n (z n + ε n z) (2.18)

and study its behavior as n goes to infinity. The function v n satisfies the equation

{

∆vn − V (x n + ε n z)v n + g(x n + ε n z, v n ) = 0 in Ω n ,

v n = 0 on ∂Ω n

(2.19)

where Ω n = ε −1

n {Ω−x n }. Again from (2.5), we see that v n is bounded in H 1 (R N ), and

from elliptic estimates it can be assumed to converge uniformly on compacts subsets of

R N to a function v ∈ H 1 (R N ). Now, the sequence of functions χ n (z) ≡ χ Λ (z n + ε n z)

can also be assumed to converge weakly in any L p over compacts to a function χ with

0 ≤ χ ≤ 1. Therefore, v satisfies the limiting equation

∆v − V (¯z)v + ḡ(z, s) = 0 in R N . (2.20)

14


where

ḡ(z, s) = χ(z)f(s) + (1 − χ(z)) ˜f(s). (2.21)

as

Associated to the equation (2.20) we have the functional ¯J : H 1 (R N ) → R defined

¯J(u) = 1 2


R N

|∇u| 2 + V (¯z)w 2 −


R N Ḡ(z, u) , u ∈ H 1 (R N ) (2.22)

where Ḡ(z, s) = ∫ s

0 ḡ(z, τ)dτ. Then v is a critical point of ¯J. We also set

J n (u) = 1 ∫


|∇u| 2 + V (z n + ε n z)w 2 − Ḡ(z n + ε n z, u) , u ∈ H

2

0(Ω 1 n ).

Ω n

Ω n

Then J n (v n ) = ε −N

n J εn (u n ), so that the key step in the proof of the proposition is the

following

Lemma 2.2

lim inf

n→∞

J n(v n ) ≥ ¯J(v).

In particular, ¯J(v) ≤ c where c is given by (2.3).

Proof. Write

h n = 1 2 (|∇v n| 2 + V (z n + ε n z)v 2 n) − G(z n + ε n z, v n ).

Then, choose R > 0. Since v n converges in the C 1 -sense over compacts to v we get

lim

n→∞


B R

h n = 1 2


B R


|∇v| 2 + V (¯z)v 2 −

Since v ∈ H 1 (R N ), we have that for each given δ > 0,


lim h n ≥ ¯J(v) − δ,

n→∞

B R

B R

Ḡ(z, v).

provided that R was chosen sufficiently large. Thus it only suffices to check that


lim inf h n ≥ −δ, (2.23)

n→∞

Ω n\B R

15


for large enough R. Let us consider a smooth cut-off function η R such that η R ≡ 0

on B R−1 , η R ≡ 1 on R N \ B R , 0 ≤ η R ≤ 1 and | ∇η N |≤ C, C independent of R.

We use w n = η R v n ∈ H 1 (Ω n ) as a test function for J n(v ′ n ) = 0 to obtain


0 = J n(v ′ n )w n = E n + 2h n + g n (2.24)

Ω n\B R

where g n = 2G(z n + εz) − g(z n + ε n z, v n )v n and E n is given by



E n = ∇v n · ∇(η R v n ) + V (z n + ε n z)η R vn 2 − g(z n + εz, v n )η R v n . (2.25)

B R \B R−1

B R \B R−1

Again, the convergence of v n in the C 1 -sense over compacts to v and the fact that

v ∈ H 1 (R N ) implies that for R > 0 sufficiently large we will have lim n→∞ |E n | ≤ δ.

On the other hand, the definition of g together with the properties of f give that

g n ≤ 0. Using this in (2.24), (2.23) follows, and the proof of the lemma is complete.


Now we are ready to obtain a contradiction with (2.16). Since v is a critical point

of ¯J, and ḡ satisfies hypothesis (g4) in §1, we have that

¯J(v) = max

τ≥0

¯J(τv) (2.26)

Then, since f(s) ≥ ˜f(s) for all s we have that

¯J(v) ≥

inf

u∈H 1 (R N )

u≠0

sup I¯z (τu) ≡ ¯c (2.27)

τ>0

where I¯z is given by


I¯z (u) = 1 2

R N

|∇u| 2 + V (¯z)u 2 −


R N F (u) , u ∈ H 1 (R N ). (2.28)

But, since V (¯z) > V 0 , we clearly have that ¯c > c where c is the number defined in

(2.2). Hence Ī(v) > c. But this contradices the previous lemma, and the proof of the

claim, i.e. (2.16), is thus complete. ✷

16


To conclude the proof of Proposition 2.1, we need to show that u ε possesses at most

one local maximum in Λ. Assume the contrary, namely the existence of a sequence

ε n → 0 such that u εn possesses two local maxima z 1 n, z 2 n ∈ Λ. Then u εn (z i n) ≥ b > 0,

i = 1, 2. From (2.16) we have that these sequences stay away from the boundary of

Λ. Set v n (z) = u εn (z 1 n + ε n z). Then, it turns out, that after passing to a subsequence

v n converges in the C 2 sense over compacts to a solution v in H 1 (R N ) of equation

(2.14), where V 0 is replaced by V (¯z 1 ). Here ¯z 1 = lim z 1 n. The function v has a local

maximum at zero, it is radially symmetric and radially decreasing, as the arguments

in [5] show. Then 0 is a nondegenerate global maximum. This and the local C 2

convergence of v n to v clearly implies that the second local maximum of v n must go

away, namely z n = ε −1

n (zn−z 2 n) 1 satisfies |z n | → +∞. Using this fact, a slight variation

of the argument in the proof of Lemma 2.2 yields

lim inf

n→∞

ε−N n J εn (u εn ) ≥ 2I 0 (w) = 2c.

This is in obvious contradiction with (2.4), and the proof of the proposition is complete.


3 Local Mountain Pass: The Domain.

This section is devoted to the proof of Theorem 0.2. The hypotheses on the behavior

of the domain, given through the function ρ, has certain analogy with the hypotheses

on V in Theorem 0.1. We exploit this analogy, and we obtain results on existence of

positive solution to elliptic equations in unbounded domains.

Theorem 0.2 is a particular case of a more general result we state next. But before

doing that let us see another particular case we think ilustrate the nature of the results

we are about to prove. We consider a domain given by the the graph of a positive

function. Let g be a smooth function g : R N−1 → R with 0 < g(x) ∀x ∈ R N−1 . We

consider

Ω = {(t, x) ∈ R N−1 × R | 0 < x < g(t)} (3.1)

17


and the problem

{

ε 2 ∆ t u + u xx − u + f(u) = 0 in Ω

u = 0 on ∂Ω and u > 0 in Ω

(3.2)

Here ε > 0 is a parameter and ∆ t denotes the Laplacian with respect to the N − 1-

dimensional variable t. We have

Theorem 3.1 Assume there is a bounded neighborhood Λ ⊂ R N−1 containing 0 such

that

max

Λ

g > max g. (3.3)

∂Λ

Then there is an ε 0 > 0 so that for all 0 < ε < ε 0 there is a positive solution

u ε ∈ H 1 (Ω) to problem (3.2). Moreover a property like (0.12) holds.

Theorems 0.2 and 0.3 are consequences of a much more general existence result

which we state next. Assume N ≥ 2 and let 1 ≤ l ≤ N − 1. We identify R N =

R l × R N−l and write z ∈ R N as z = (t, x) with t ∈ R l , x ∈ R N−l .

We consider a smooth domain Ω in R N satisfying the following set of properties:

(Ω1) There is a bounded domain Λ in R l containing 0 R l

with D ≡ (Λ × R N−l ) ∩ Ω

bounded. Aditionally, for each z ∈ ∂D ∩ ∂Ω the normal vector to ∂Ω at z has

a non-zero x-component.

In what follows we denote by Ω t the t-section of Ω, that is

where t ∈ Λ.

Ω t = {x ∈ R N−l / (t, x) ∈ Ω},

(Ω2) For every t ∈ ∂Λ we have

and

Ω t ⊆ Ω 0

Ω t ⊂ Ω 0 for allt ∈ Λ.

18


For a domain Ω satisfying the properties given above we consider the semilinear

elliptic problem

{

ε 2 ∆ t u + ∆ x u − u + f(u) = 0 in Ω

u = 0 on ∂Ω and u > 0 in Ω,

(3.4)

where ∆ t and ∆ x represent the Laplacian operators with respect to the t and x

variables respectively. Note that the situation in theorems 0.2 and 3.1 falls within

this framework with l = 1 and l = N − 1 respectively.

On problem (3.4) we have the following result, which includes Theorems 0.2 and

3.1 as special cases.

Theorem 3.2 Assume (Ω1), (Ω2) and (f1) − (f4) hold.

Then there is an ε 0 > 0 such that for all 0 < ε < ε 0 , problem (3.4) has a positive

solution u ε ∈ H 1 (Ω). Moreover the function u ε concentrates on Λ in the sense that

there is a sequence t ε ∈ Λ with the property that

u ε (t, x) ≤ α exp{− β ε |t − t ε|} (3.5)

for all (t, x) ∈ Ω and some positive constants α, β.

In this section we will prove Theorem 3.2. The proof follows the lines of the

previous section, except that now we rely on a different limiting problem.

Let us consider D = (Λ × R N−l ) ∩ Ω, as given in (Ω1). Then we modify f in the

following way outside D: we set

g(z, s) = χ D (z)f(s) + (1 − χ D (z)) ˜f(s). (3.6)

Thus we search for critical points of the functional

J ε (u) = 1 ∫


|∇ x u| 2 + ε 2 |∇ t u| 2 + u 2 − G(z, u), u ∈ H 1

2

0(Ω). (3.7)

Ω

Then it follows from the results of Section 1, the existence of a critical point u ε of J ε

with a variational characterization like (2.1). To estimate J ε (u ε ) we set

I(u) = 1 ∫


|∇u| 2 + u 2 − F (u). (3.8)

2

Ω

19


Then I possesses a critical point w in H 0 = H 1 0(R l × Ω 0 ) so that

I(w) = ¯c = inf

u∈H 0

u≠0

sup I(τu). (3.9)

τ≥0

Similarly to §2, we can use this w to construct a test function in the variational

characterization of u ε to obtain

The result of the theorem will follow if we prove that

J ε (u ε ) ≤ ε l (¯c + o(1)). (3.10)

max

∂D u ε → 0 as ε → 0, (3.11)

since the same argument given to prove Theorem 0.1 from Proposition 2.1 applies.

To prove (3.11) we argue again by contradiction, assuming the existence of sequences

ε n → 0 and z n = (t n , x n ) ∈ (∂Λ × R N−l ) ∩ Ω with t n → ¯t ∈ ∂Λ and u εn (z n ) ≥ b > 0.

Then we write z = (t, x) ∈ R l × R N−l and set

v n (z) = u εn (t n + ε n t, x).

where, we assume that v n is extended as zero outside its domain of definition Ω n ≡

{(t, x) / (t n + ε n t, x) ∈ Ω}, so that v n is understood as an element of H 1 (R N ).

Then v n satisfies the equation

∆v n − v n + g(t n + ε n t, x, v n ) = 0 for all (t, x) ∈ Ω n , (3.12)

and (3.10) implies that the sequence v n is bounded in H 1 (R N ). This, interior

and boundary elliptic estimates, using the regularity of Ω and (Ω1), imply that

v n

converges uniformly and in H 1 (R N ) over compact sets to a nonzero function

v ∈ H 1 0(R l × Ω¯t ) satisfying

{

∆v − v + ḡ(z, v) = 0 in R l × Ω¯t

u = 0 on R l × ∂Ω¯t (3.13)

where ḡ(z, s) = χ(z)f(s) + (1 − χ(z)) ˜f(s) and χ is some measurable function such

that 0 ≤ χ ≤ 1.

20


Now, let J n be the energy functional associated to (3.12) and ¯J that of (3.13).

Then, exactly the same argument used in the proof of Proposition 2.1 yields

lim inf

n→∞

J n(v n ) ≥ ¯J(v).

Therefore, from (3.10) one gets ¯J(v) ≤ ¯c. But this is impossible, since clearly ¯J(v) is

greater than or equal to the mountain pass value of the functional I given by (3.8)

over H 1 0(R l × Ω¯t ), which is strictly less than ¯c.

This proves that m ε → 0. From here it follows, as in the previous section, that

u ε actually solves equation (3.4) for all small ε and that all local maxima of u ε must

lie in Λ × R N−l . Let z ε = (t ε , x ε ) ∈ Ω with t ε ∈ Λ such that u ε (z ε ) > a is a local

maximum of u ε . We define v ε (t, x) = u ε (t ε + εt, x) for z = (t, x) ∈ R l × R N−l . Next

with essentially the same arguments given above, and recalling that Ω t ⊂ Ω 0 for all

t ∈ Λ \ {0}, we find that t ε → ¯t, where ¯t is such that Ω¯t = Ω 0 , and that there is a

sequence ε n → 0 and v n = v εn → v uniformly and in H 1 (R N ) over compacts. The

function v satisfies the equation

{

∆v − v + f(v) = 0 in R l × Ω 0

v = 0 on R l × ∂Ω 0 (3.14)

and v(z) > 0 for z ∈ R l ×Ω 0 . Using the moving planes argument as in [5], only in the

t variables, one can get that the positive solutions of (3.14) are radially symmetric in

the variable t and increasing along the t-rays. These facts allow us to show that for a

number R > 0 v(z) < a if |z| = R. Here we use that for the solutions v of (3.14) we

are interested in, the functional I given in (3.8) is uniformly bounded. Then, using

the uniform convergence of the sequence v n over compact sets, and the fact that the

local maxima of v n must be at a finite distance, we find that v n (z) < a for all |z| > R.

From here the constants α and β can be found, depending only on the family {v ε },

so that (3.5) holds.

References

[1] M. del Pino and P. Felmer, ‘Least energy solutions for elliptic equations in unbounded

domains’.Preprint.

21


[2] W.Y Ding and W.M. Ni, ‘On the existence of Positive Entire Solutions of a

Semilinear Elliptic Equation’, Arch. Rat. Mech. Anal., 91, 283-308 (1986).

[3] A. Floer and A. Weinstein, ‘Nonspreading Wave Packets

for the Cubic Schrödinger Equation with a Bounded Potential’, Journal of Functional

analysis 69, 397-408 (1986).

[4] Gidas Ni Nirenberg

[5] Gidas Ni Nirenberg

[6] W.C. Lien, S.Y Tzeng and H.C. Wang, ‘Existence of solutions of semilinear

elliptic problems in unbounded domains’. Differential and Integral Equations,

Vol. 6, No 6, 1281-1298 (1993).

[7] Y.J. Oh, ‘Existence of semi-classical bound states of nonlinear Schrödinger equations

with potential on the class (V ) a ’. Comm. Partial Diff. Eq. 13, 1499-1519

(1988).

[8] Y.J. Oh, Corrections to ‘Existence of semi-classical bound states of nonlinear

Schrödinger equations with potential on the class (V ) a ’. Comm. Partial Diff. Eq.

14, 833-834 (1989).

[9] Y.J. Oh, ‘On positive multi-lump bound states nonlinear Schrödinger equations

under multiple well potential’. Comm. Math. Phys., 131, 223-253 (1990).

[10] P. Rabinowitz, ‘Minimax Methods in Critical Point Theory with applications to

Differential Equations’. CBMS Regional Conference Series Math. 65, AMS.

[11] P. Rabinowitz, ‘On a class of nonlinear Schrödinger equations’, Z. angew Math

Phys 43, 270-291(1992).

[12] X. Wang ‘ On concentration of positive bound states of nonlinear Schrödinger

equations’. Preprint.

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