nodal solutions of a nls equation concentrating on lower dimensional ... · presentation nodal...

Presentation

Nodal solutions of a NLS equation concentratingon lower dimensional spheres

Marcos T.O. Pimenta∗ and Giovany J. M. Figueiredo

Unesp / UFPA

Brussels, September 7th, 2015

* Supported by FAPESP - CNPq - Brazil

Marcos T.O. Pimenta and Giovany J. M. Figueiredo

Presentation

Presentation

1 Introduction

2 The penalization and the variational approach

3 Existence of nodal solutions

4 Concentration in spheres

5 Proof of Theorem 1.1

6 Bibliography


IntroductionThe penalization and the variational approach

Existence of nodal solutionsConcentration in spheres

Proof of Theorem 1.1Bibliography

Apresentacao

1 Introduction





6 Bibliography





Introduction

In this work we study some results about existence andconcentration of solutions of the following NLS equation

−ε2∆u + V (x)u = f (u) in RN ,u ∈ H1(RN),

(1)

where ε > 0, f is a subcritical power-type nonlinearity and thepotential V is assumed to be positive and satisfy a local condition.The concentration is going to occur around k dimensional spheres,where 1 ≤ k ≤ N − 1, as ε→ 0.





Introduction

We assume the following conditions on de odd nonlinearity f :

(f1) There exists ν > 1 such that f (|s|) = o(|s|ν) as s → 0;

(f2) There exist c1, c2 > 0 such that |f ′(s)| ≤ c1|s|+ c2|s|p where0 < p < 2∗ − 2;

(f3) There exists θ > 2 such that

0 < θF (s) ≤ f (s)s, for s 6= 0,

where F (s) =∫ s

0 f (t)dt;

(f4) s 7→ f (s)/s is increasing for s > 0 and decreasing for s < 0.





Introduction

Let k ∈ N with1 ≤ k ≤ N − 1 and define a subspace H ⊂ RN

such that dimH = N − k − 1. Hence dimH⊥ = k + 1. Forx ∈ RN , denote x = (x ′, x ′′) where x ′ ∈ H, x ′′ ∈ H⊥ are such thatx = x ′ + x ′′. When u : RN → R, we write u(x ′, x ′′) = u(x ′, |x ′′|)denoting that u(x ′, y) = u(x ′, z) for all y , z ∈ H⊥ such that|y | = |z |. For a ∈ R, let

−∆u + au = f (u) in RN−k . (2)

The auxiliar potential M : RN → (0,+∞] is given by

M(x) = |x ′′|kE(V (x)) where x = (x ′, x ′′), x ′ ∈ H and x ′′ ∈ H⊥,

where E(V (x)) is the ground-state level of (2) with a = V (x).





Introduction

V will be assumed to satisfy the following conditions

(V1) 0 < V0 ≤ V (x), x ∈ RN and V (x) = V (x ′, x ′′) = V (x ′, |x ′′|);

(M1) There exists a bounded and open set Ω ⊂ RN such that(x ′, x ′′) ∈ Ω implies (x ′, y ′′) ∈ Ω for all y ′′ ∈ H⊥ such that|x ′′| = |y ′′|. Moreover,

0 <M0 := infx∈ΩM(x) < inf

x∈∂ΩM(x).





Introducao

The main result is the following

Teorema 1.1

Let f and V satisfying (f1)− (f4) and (V1) e (M1), respectively.Then for all sequence εn → 0, up to a subsequence, (1) (withε = εn) has a nodal solution un such that, un(x ′, x ′′) = un(x ′, |x ′′|)and, if εnP

1n and εnP

2n are respectively, maximum and minimum

points of un, then εnPin ∈ Ω, i = 1, 2 for n sufficiently large and

εnPin → x0, as n→∞, where M(x0) =M0. Moreover

|un(x)| ≤ C(e−

βεn

dk (x ,εnP1n ) + e−

βεn

dk (x ,εnP2n ))

x ∈ RN ,

where C , β > 0 and dk is a distance defined in (10).





Introducao

Layout of solutions to N = 2 and k = N − 1





Introduction

The works which more have influenced us are [1], in which Alvesand Soares prove the existence and concentration of nodal-peaksolutions for (1), and [4] where the authors prove the existence ofpositive solutions concentrating around lower dimensional spheres.

In this work we use the penalization technique developed in [5],which consists in modifying the nonlinearity in such a way torecover the compactness of the functional. However, since we arelooking for nodal solutions, we have to carry more thoroughly thepenalization and as a consequence, make sharper estimates toprove that the solutions of the modified problem, satisfies theoriginal one.





Apresentacao

1 Introduction





6 Bibliography





The penalization and the variational approach

For τ > 2, let rε > 0 be such that f (rε)rε

= ετ and f (−rε)−rε = ετ .

Since rε → 0 as ε→ 0, (f1) implies that ετ = f (|rε|)|rε| ≤ |rε|

ν−1.

Hence ετ

ν−1 ≤ |rε| and we can choose an odd function fε ∈ C 1(R)satisfying

fε(s) =

f (s) if |s| ≤ 1

2ετ

ν−1 ,

ετ s if |s| ≥ ετ

ν−1 ,

|fε(s)| ≤ ετ |s| for all s ∈ R (3)

and0 ≤ f ′ε (s) ≤ 2ετ for all s ∈ R. (4)






Penalization






Let gε(x , s) := χΩ(x)f (s) + (1− χΩ(x))fε(s). By (f1)− (f4) itfollows that g is such that gε(x

′, x ′′, s) = gε(x′, |x ′′|, s) and

satisfies

(g1) gε(x , s) = o(|s|ν), as s → 0, uniformly in compact sets of RN ;

(g2) There exist c1, c2 > 0 such that |gε(x , s)| ≤ c1|s|+ c2|s|pwhere 1 < p < N+2

N−2 ;

(g3) There exists θ > 2 such that:

i) 0 < θGε(x , s) ≤ gε(x , s)s, para x ∈ Ω and s 6= 0,ii) 0 < 2Gε(x , s) ≤ gε(x , s)s, for x ∈ RN\Ω and s 6= 0,

where Gε(x , s) =∫ s

0 gε(x , t)dt.

(g4) s 7→ gε(x ,s)s is nondecreasing for s > 0 and nonincreasing for

s < 0, for all x ∈ RN .






Let us consider the modified problem

−ε2∆u + V (x)u = gε(x , u) in RN , (5)

or equivalently,

−∆v + V (εx)v = gε(εx , u) in RN . (6)






In order to obtain the solutions with the prescribed symmetry,let

H :=

v∈H1(RN);

∫(|∇v |2+V (εx)v2) < +∞ and v(x ′, x ′′) = v(x ′, |x ′′|)

which is a Hilbert space when endowed with the inner product

〈u, v〉ε =

∫(∇u∇v + V (εx)uv),

which gives rise to the norm

‖v‖ε =

(∫(|∇v |2 + V (εx)v2

) 12

.






Let Iε : H1(RN)→ R be the C 2(H1(RN),R) functional definedby

Iε(v) =1

2

∫(|∇v |2 + V (εx)v2)−

∫Gε(εx , v).

Note that

I ′ε(v)ϕ =

∫(∇v∇ϕ+ V (εx)vϕ)−

∫gε(εx , v)ϕ, ∀ϕ ∈ H1(RN),

and then critical points of Iε em H1(RN) are weak solutions of (6).From now on we are going to work with the functional Iε

restricted to H.





Apresentacao

1 Introduction





6 Bibliography





Existence of nodal solutions

Let us consider the Nehari manifold associated to (6) and definedby

Nε = v ∈ H\0; I ′ε(v)v = 0.

Since we are looking for sign-changing solutions, let us considerthe Nehari nodal set

N±ε = v ∈ H; v± 6= 0 and I ′ε(v)v± = 0,

which contains all the sign-changing solutions. Hence,

dε := infN±ε

Iε,

is the nodal ground state level.Let us show that dε is reached by some function in N±ε .






Lemma 3.1

Let v ∈ H such that v± 6= 0. Then there exist t, s > 0 such thattv+ + sv− ∈ N±ε .

Let (wn) be a minimizing sequence for Iε in N±ε . One can provethat (wn) is bounded in H and then wn wε in H, w±n w±ε inH, where w±ε 6= 0.

Let tε, sε > 0 be such that tεw+ε + sεw

−ε ∈ N±ε . Then

‖tεw+ε + sεw

−ε ‖ε ≤ lim inf

n→∞‖tεw+

n + sεw−n ‖ε (7)

and ∫Ωε

F (tεw+ε + sεw

−ε ) = lim inf

n→∞

∫Ωε

F (tεw+n + sεw

−n ). (8)






In this point the penalization in fact plays its role. By (3) and(4), one can see that for sufficiently small ε > 0,

Iε,RN\Ωε(v) :=

1

2

∫RN\Ωε

(|∇v |2 + V (εx)v2

)−∫RN\Ωε

Fε(v)

is strictly convex. Then, it follows that Iε,RN\Ωεis weakly

semicontinuous from below and then

Iε(tεw+ε + sεw

−ε ) ≤ lim inf

n→∞Iε(tεw

+n + sεw

−n )

≤ lim infn→∞

(Iε(w+n ) + Iε(w

−n ))

= bε.






Hence

Iε(tεw+ε + sεw

−ε ) = bε.

Now, arguments similar to those employed in [3] and [2] can beused to, by a contradiction argument, prove that the minimizing ofIε on N±ε is a weak solution of (6).





Apresentacao

1 Introduction





6 Bibliography





Concentration in spheres

Let εn → 0 as n→∞ and, for each n ∈ N, let us denote byvn := vεn the solution of the last section, dn := dεn , ‖ · ‖n := ‖ · ‖εnand In := Iεn .The following result gives us an estimate from above to the energylevels dn.

Lemma 4.1

lim supn→∞

εkndn ≤ 2ωk infΩM

andεkn‖vn‖2

n ≤ C ∀n ∈ N.






Next result implies that solutions (vn) do not present vanishingand neither do (v+

n ) and (v−n ).

Lemma 4.2

Let P1n , P2

n local maximum and minimum of vn, respectively. ThenP in ∈ Ωεn := ε−1

n Ω and

vn(P1n) ≥ a and vn(P2

n) ≤ −a,

where a > 0 is such that f (a)/a = V0/2.






By the last result there exist P1,P2 ∈ Ω such that, up to asubsequence

limn→∞

εnPin = P i , i ∈ 1, 2.

Lemma 4.3

limn→∞

|P1n − P2

n | = +∞.






Lemma 4.4

Let yn = (y ′n, y′′n ) ⊂ RN be a sequence such that

εnyn → (y ′, y ′′) ∈ Ω as n→∞. Denoting vn(x ′, r) := vn(x ′, x ′′)where |x ′′| = r , let us define wn : RN−k−1 × [−|yn|,+∞)→ R by

wn(x ′, r) := vn(x ′, r + |y ′′n |).

Then there exists w ∈ H1(RN−k) such that wn → w inC 2loc(RN−k) and w satisfies the limit problem

−∆w + V (y ′, |y ′′|)w = gn(x ′, r , w) in RN−k , (9)

where gn(x ′, r , s) := χ(x ′, r)f (s) + (1− χ(x ′, r))fεn(s) andχ(x ′, r) = limn→∞ χΩ(εnx

′ + εny′n, εnr + εn|y ′′n |).






Since the concentration set is expected to be a sphere in RN , itis natural to introduce the distance between two k−dimensionalspheres in RN , which gives rise to neighborhoods in which we wantto estimate the mass of solutions. Then let

dk(x , y) =√

(x ′ − y ′)2 + (|x ′′| − |y ′′|)2, (10)

which denotes the distance between k−dimensional spherescentered at the origin, parallel to H⊥ and of radius |x ′′| and |y ′′|,respectively.






Proposition 4.5

Under assumptions of Theorem 1.1, it holds:

i) limn→∞

εkndn = 2ωk infΩM.

ii) limn→∞

M(εnPin) = inf

ΩM, i ∈ 1, 2.

The proof of this result consists in an analysis of the parts of theintegrals of In(vn) in Bk(P i ,R) and in RN\Bk(P i ,R).





Apresentacao

1 Introduction





6 Bibliography





Proof of Theorem 1.1

First of all we have to prove the following result.

Proposicao 5.1

limn→∞

‖vn‖L∞(Ωn\(Bk (P1n ,R)∪Bk (P2

n ,R))) = 0.

By elliptic regularity theory, it follows that vn ∈ C 2(RN). Then,by continuity, it follows that

‖vn‖L∞∂((Bk (P1n ,Rn)∪Bk (P2

n ,Rn))) = on(1) (11)






Let un(x) := vn(ε−1n x). In order to prove the exponential decay

in εn of un, we need to consider the following functions

W (x) = C (e−βdk (x ,P1n ) + e−βdk (x ,P2

n )),

defined in RN\(Bk(P1

n ,R) ∪ Bk(P2n ,R)

), where C > 0 is a

constant to be choosen.For β > 0 sufficiently small, it follows that for all n ∈ N and

x ∈ RN\(Bk(P1

n ,R) ∪ Bk(P1n ,R)

),(

−∆ + V (εnx)− gn(εn, vn)

vn

)(W ± vn) ≥ 0.






Then, by (11), for x ∈ ∂(Bk(P1

n ,R) ∪ Bk(P2n ,R)

),

W (x)± vn(x) = C2e−βR ± vn ≥ 0

for a constant C > 0 sufficiently large and which does not dependon n. Hence by(g4), Maximum Principle applies and

|vn| ≤W (x), em RN\(Bk(P1

n ,R) ∪ Bk(P2n ,R)

),

which implies that





Prova do Teorema 1.1

|un(x)| ≤ C(e−

βεn

dk (x ,εnP1n ) + e−

βεn

dk (x ,εnP2n )), (12)

for x ∈ RN\(Bk(εnP

1n , εnR) ∪ Bk(εnP

2n , εnR)

).

In particular

‖un‖L∞(RN\Ω) ≤ Ce−βεn

and then un satisfies the original problem, proving the theorem.





Apresentacao

1 Introduction





6 Bibliography





Bibliography I

C.O. Alves, S.H.M. Soares On the location and profile ofspyke-layer nodal solutions to nonlinear Schrodinger equations.Journal of Mathematical Analysis and Applications 296(2004), 563 - 577.

C.O. Alves, M. Souto, Existence of least energy nodal solutionfor a Schrodinger-Poisson system in bounded domains, toappear in Z. Angew. Math. Phys.

T. Bartsch, T. Weth, M. Willem, Partial symmetry of leastenergy nodal solutions to some variational problems, J. Anal.Math. 96 (2005), 1 - 18.





Bibliography II

D. Bonheure, J. Di Cosmo, J.V. Schaftingen, NonlinearSchrodinger equation with unbounded or vanishing potentials:solutions concentrating on lower dimensional spheres, Journalof Differential Equations. 252 (2012), 941 - 968.

M. del Pino, P. Felmer, Local mountain pass for semilinearelliptic problems in unbounded domains, Calc. Var. PartialDifferential Equations 4 (1996), 121 - 137.


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