Ground states of nonlinear Schrödinger equations with potentials

www.elsevier.com/locate/anihpc

Ground states of nonlinear Schrödinger equations with potentials Solutions d’énergie minimale des équations de Schrödinger

non-linéaires avec potentiel

Yongqing Li

, Zhi-Qiang Wang

, Jing Zeng

aSchool of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou, 350007, PR China bDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Received 1 September 2005; accepted 31 January 2006 Available online 7 July 2006

Abstract

In this paper we study the nonlinear Schrödinger equation:

−u+V (x)u=f (x, u), u∈H¹

R^N .

We give general conditions which assure the existence ofground state solutions. Under a Nehari type condition, we show that the standard Ambrosetti–Rabinowitz super-linear condition can be replaced by a more natural super-quadratic condition.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

Dans cet article nous étudions l’équation non-linéaire de Schrödinger : −u+V (x)u=f (x, u),

u∈H¹ R^N

.

Nous donnons les conditions générales qui garantissent l’existence de solutions d’énergie minimale. Sous une condition de type Nehari, nous démontrons que la condition super-linéaire d’Ambrosetti–Rabinowitz peut être remplacée par une condition super- quadratique plus naturelle.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:35B05; 35J60

Keywords:Nonlinear Schrödinger equations; Ground state solutions; The Ambrosetti–Rabinowitz condition

* Corresponding author.

E-mail addresses:zhi-qiang.wang@math.usu.edu, wang@math.usu.edu (Z.-Q. Wang).

1 Supported in part by NNSF of China (10161010), Fujian Provincial Natural Science Foundation of China (A0410015), and the Ky and Yu-Fen Fan Fund from AMS.

0294-1449/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2006.01.003

1. Introduction

We study the nonlinear Schrödinger equation with potentials:

−u+V (x)u=f (x, u), u∈H¹

R^N

. (1.1)

We are concerned with the existence ofground state solutions, i.e., solutions corresponding to the least positive critical value of the variational functional:

Φ(u)=1 2

R^N

|∇u|²+V (x)u² dx−

R^N

F (x, u)dx, whereF (x, u)=_u

0 f (x, t )dt.

To establish the existence of ground states, usually besides the growth condition on the nonlinearity and a Nehari type condition, the following superlinear condition due to Ambrosetti–Rabinowitz (e.g., [2,12]) is assumed:

(AR) There isμ >2 such that foru=0 andx∈R^N, 0< μF (x, u)uf (x, u),

whereF (x, u)=u

0 f (x, t )dt.

This condition implies that for someC >0,F (x, u)C|u|^μ.

In this paper we show that a weaker and more natural version suffices to assure the existence of aground state solution. Instead of (AR) we assume the following super-quadratic condition

(SQ) lim_|u|→∞F (x,u)

u² = ∞, uniformly inx.

We always assumeV (x)∈C(R^N,R), inf_RNV (x) >0. We consider two cases of the potentials, one is periodic, i.e., thex-dependence is periodic; the other is whenV has a bounded potential well in the sense that lim_|x|→∞V (x) exists and is equal to sup_RNV. The results will be stated and proved in Sections 2 and 3.

We would like to mention earlier results on existence of entire solutions of Schrödinger type equations with or without potentials which was studied in [3,4,9,10] (see references therein). In recent years there have been intensive studies on semiclassical states for nonlinear Schrödinger equations for which in Eq. (1.1) there is a small parameter corresponding to the Plank constant. We refer [1] for references in this direction. Our results do not require smallness of such a parameter. A recent result in [5] is in similar spirit of our Theorem 3.1; but the conditions in [5] and ours are mutually non-inclusive and the methods are different.

For (1.1) in bounded domains or if the potential functionV (x)possesses certain compactness condition, one can prove (1.1) have certain solutions. In [8] Liu and Wang first used (SQ) to get the bounds of minimizing sequences on the Nehari manifold, and under coercive condition ofV (x)they proved the existence of three solutions: one positive, one negative, and one sign-changing. The results in this current paper are natural generalizations of that in [8] to noncompact cases. In the two cases we do not have compact embedding, which is the main difficulty in this paper.

We shall make use of a combination of the techniques in [8,7] with applications of the concentration-compactness principle of Lions [6,11,12].

2. The periodic case

We consider weak solutions of −u+V (x)u=f (x, u),

u∈H¹ R^N

.

We need the following assumptions:

(V₁) V (x)∈C(R^N,R), inf_RNV (x)V₀>0.V (x)is 1-periodic in each ofx₁, x₂, . . . , x_N.

(f1) f (x, t )∈C¹is 1-periodic in each ofx1, x2, . . . , xN,ft is a Caratheodory function and there existsC >0, such that

f_t(x, t )C

1+ |t|^2∗−2 , lim

|t|→∞

|f (x, t )|

|t|^2∗−1 =0, uniformly inx∈R^N. (f₂) f (x, t )=o(|t|), as|t| →0, uniformly inx.

(f₃) lim_|t|→∞F (x,t )

t² = ∞, uniformly inx. (f₄) ^{f (x,t )}_|t| is strictly increasing int.

Here 2^∗=_N^2N₋₂forN3. ForN=1,2 we assume there isq >2 in the place of 2^∗in(f1). We work in Hilbert space X= {u∈H¹(R^N);

R^NV (x)u²dx <∞}, with normu²=

R^N(|∇u|²+V (x)u²)dx. The functional associated with Eq. (1.1) is

Φ(u)=1 2

R^N

|∇u|²+V (x)u² dx−

R^N

F (x, u)dx, u∈X.

Define γ (u)=

R^N

|∇u|²+V (x)u² dx−

R^N

f (x, u)udx.

Theorem 2.1.Under assumptions(V1),(f1)–(f4)Eq.(1.1)has a weak solutionu∈X, such thatΦ(u)=c >0,cis defined as

c=inf

N Φ(u),

whereN = {u∈X: u=0, γ (u)=0}.

First we need a few lemmas.

Lemma 2.2.Let(u_n)be a minimizing sequence forc. Then (i) There isβ >0such thatlim infn→∞u_nβ.

(ii) (u_n)is bounded inX.

(iii) For a subsequence, up to translations,u_nconverges weakly tou=0.

Proof. (i) The proof is similar to the case with (AR) satisfied. We omit its proofs (see [12]).

(ii) Let(un)be a minimizing sequence ofc. If(un)is not bounded, we definevn=un/un, sovn =1. Passing to a subsequence, we may assume,vn vinX,vn→vinL^p_loc(R^N), 2p <2^∗,vn→va.e. onR^N.

Ifv=0, we have 1

2−

R^N

F (x, un)

u²_n v²_ndx=c+o(1) un² >0.

By Fadou’s lemma and(f₃)we have a contradiction as follows, 1

2lim inf

n→∞

R^N

F (x, u_n) u²_n v²_ndx

R^N

lim inf

n→∞

F (x, u_n)

u²_n v²_ndx= ∞.

Ifv=0, we takey_n=(y¹_n, y_n², . . . , y_n^N)∈N^N with ally_nⁱ (1iN) being integers. Define translations ofv_nby w_n(x)=v_n(x+y_n). SinceV (x)andf (x, u)are periodic, we havew_n = v_n =1,|w_n|p= |v_n|p, andΦ(w_n)= Φ(v_n). Passing to a subsequence, we have w_n winH¹(R^N),w_n→w inL^p_loc(R^N), 2p <2^∗,w_n→w a.e.

onR^N. If there existyn, such thatwn w=0, we will get a contradiction as the case ofv=0. If for anyyn,wn0, we will get a contradiction by provingv_n→0 inL^p(R^N). In this case, we claim for allp∈(2,2^∗),

lim sup

n→∞y∈R^N

B₂(y)

|v_n|^pdx=0.

If this is not true, there existsp∈(2,2^∗),δ >0, lim sup

n→∞y∈R^N

B2(y)

|v_n|^pdxδ >0,

then there existszn∈R^Nsuch that, limn→∞

B₂(z_n)|vn|^pdxδ/2>0. We can chooseyn∈N^N∈B2(zn)such that B₁(y_n)⊂B₂(z_n)and

nlim→∞

B₁(yn)

|v_n|^pdxδ 2 >0, we have limn→∞

B1(0)|w_n|^pdxδ/2>0, that isw_n w=0, a contradiction.

By Lions Lemma (cf. [12, Lemma 1.21]), we getvn→0 inL^p(R^N),p∈(2,2^∗). Fixp∈(2,2^∗). By(f1)and(f2), for anyε >0 there isCε>0 such that|f (x, u)|ε(|u| + |u|^2∗−1)+Cε|u|^p−1. Then|F (x, u)|ε(|u|²+ |u|^2∗)+ C_ε|u|^p. Then fixing anR >√

2c, using Lebesgue Dominated Convergence theorem, we have

nlim→∞

R^N

F (x, Rv_n)dx=

R^N

nlim→∞F (x, Rv_n)dx=0.

Since by(f4),Φ(t un)Φ(un)fort0 we thus have c+o(1)=Φ(u_n)Φ(Rv_n)=1

2R²−

R^N

F (x, Rv_n)dx, which is a contradiction. Thus(u_n)is bounded.

(iii) We can assumeu_nweakly converges tou. To showu=0, again we define translations ofu_nas above, assume y_n=(y_n¹, y_n², . . . , y_n^N)∈N^N, with all y_nⁱ (1iN) being integers. u^y_nⁿ=u_n(x+y_n)are all possible translation ofun. If for someyn⊂N^N,u^y_nⁿ u=0 we are done. If for anyyn⊂N^N,u^y_nⁿ0, by similar argument as above we can proveu_n→0 inL^p(R^N),p∈(2,2^∗). Then asn→ ∞,

R^Nu_nf (x, u_n)dx→0. Thus by (i) we have a con- tradiction:

0< βu_n²=

R^N

u_nf (x, u_n)dx→0, asn→ ∞. 2

Lemma 2.3.For eachu∈X\ {0}, there exists uniquet=t (u) >0, such thatt u∈N. This is similar to the case of assuming (AR), we omit it.

Lemma 2.4.Let(u_n)⊂Xbe a sequence such thatγ (u_n)→0and

R^Nf (x, u_n)u_n→a >0asn→ ∞. Then exist t_n>0such thatt_nu_n∈N,t_n→1, asn→ ∞.

Proof. Sinceu_n=0, by Lemma 2.3, there exists only onet_n>0, such thatt_nu_n∈N, i.e.

t_n²

R^N

|∇un|²+V (x)|un|² dx−

R^N

f (x, tnun)tnundx=0.

By (f₁)and (f₂), |f (x, u)u|ε(|u|²+ |u|^2∗)+C_ε|u|^p, we see t_n cannot go zero, that ist_nt₀>0. By (f₄), f (x, u)u2F (x, u). Ift_n→ ∞, we get

a+o(1)=

R^N

|∇un|²+V (x)|un|² dx=

R^N

f (x, t_nu_n)t_nu_n t_n² dx2

R^N

F (x, t_nu_n) t_n²u²_n u²_ndx.

By the condition, up to translations,un→u=0 a.e. inR^N. We have

R^N

F (x, t_nu_n)

t_n²u²_n u²_ndx→ +∞, asn→ ∞

a contradiction. Thus 0< t₀t_nC. Assumet_n→T, now we claimT =1. Sincet_nu_n∈N, byγ (u_n)→0 we have

R^N

|∇u_n|²+V (x)|u_n|² dx=

R^N

f (x, u_n)u_ndx+o(1).

Sincet_n→T, by(f₁)and(f₃) T²

R^N

|∇u_n|²+V (x)|u_n|² dx−

R^N

f (x, T u_n)T u_ndx=o(1), that is

o(1)=

R^N

f (x, T u_n)

T u_n u²_n−f (x, u_n) u_n u²_ndx=

R^N

f (x, T u_n)

T u_n −f (x, u_n) u_n u²_ndx.

For a subsequenceu_n→uinL^p_loc(R^N)2p <2^∗. Up to translations, we may assumeu=0. Then by (f4) and Fatou’s lemma

R^N

f (x, T u)

T u −f (x, u)

u u²dx=0, by(f₄)we haveT =1. 2

Next we construct a special minimizing sequence along which

R^NF (x, u) is weakly continuous. Consider Eq. (1.1) onB_R(0),

−u+V (x)u=f (x, u), inB_R(0),

u=0, on∂B_R(0). (2.1)

We can similarly defineNR,c_R. By the result of [8],c_Ris achieved by a positive solution of (2.1) calledu_R. It is easy to check thatc_R> candc_R→casR→ ∞. This implies(u_R)asR→ ∞minimizesc. LetR_n→ ∞,u_n:=u_Rn. Fixp∈(2,2^∗).

Lemma 2.5.

(i)

R^N|u_n|^p→A >0.

(ii) There existx_n∈R^Nsuch that∀ε >0,∃R >0,lim inf

BR(xn)|u_n|^pA−ε.

Proof. (i) follows fromγ (u_n)=0 and the fact that for any ε >0 there isC_ε>0, |f (x, u)|ε(|u| + |u|^2∗−1)+ C_ε|u|^p−1.

For (ii) we apply the concentration compactness principle to

R^N|u_n|^p. Then there existα∈(0,1],(x_n)⊂R^N,

∀ε >0,∃R >0,∀r > R,r> R, have lim inf

B_r(x_n)

|u_n|^pαA−ε, lim inf

B^c

r(xn)

|u_n|^p(1−α)A−ε.

Next we claimα=1. Chooseεn→0,rn→ ∞,r_n =4rn. Letξ be a cut-off function such thatξ(s)=0, fors1 or s4,ξ(s)=1, for 2s3, and|ξ(s)|2. Takeφ(x)=ξ(|x−x_n|/r_n)u_n. Using equation

B_Rn

∇u_n∇φ+V (x)u_nφ−f (x, u_n)φ dx=0,

we have,

B_3rn(xn)\B_2rn(xn)

|∇u_n|²+V (x)|u_n|² dx+

B_3rn(xn)\B_2rn(xn)

f (x, u_n)u_ndx=o(1).

Take another cut-off functionηsuch thatη(s)=1, fors2,η(s)=0, fors3, and|η(s)|2, for 2s3. Set w_n(x)=η

|x−x_n|

r_n u_n, v_n(x)=

1−η

|x−x_n| r_n

u_n(x).

Using equation as above we have Φ(u_n)=Φ(w_n)+Φ(v_n)+o(1),

and

Rⁿ

|w_n|^pαA−ε_n,

Rⁿ

|v_n|^p(1−α)A−ε_n.

Finally usingw_nto test the equation for(u_n)we get γ (w_n)=

Φ(u_n), w_n

+o(1)=o(1).

Similarlyγ (v_n)=o(1), by Lemma 2.4,∃t_n→1,s_n→1, such thatt_nw_n∈N,s_nv_n∈N. Then c+o(1)=Φ(un)=Φ(wn)+Φ(vn)+o(1)=Φ(tnwn)+Φ(snvn)+o(1)2c+o(1), which is a contradiction. Thusα=1. 2

Proof of Theorem 2.1. Let(un)⊂N be the minimizing sequence forcgiven above. By Lemma 2.2(un)is bounded inXand weak convergent tou=0. By Lemma 2.5,−

R^NF (x, un)is weakly continuous. Using the weakly lower semi-continuity we haveΦ(u)c. Ifu∈N we haveΦ(u)=c. Ifu /∈N, by Lemma 2.5, there ist >0 such that t un∈N. Then

cΦ(t u)lim inf

n→∞ Φ(t u_n)lim inf

n→∞ Φ(u_n)=c.

SinceN is smooth, the minimizer is a critical point ofΦ. 2 3. The potential well case

We consider weak solutions of −u+V (x)u=f (u),

u∈H¹

R^N (3.1)

for the case where potential functionV (x)has a bounded potential well. Since the nonlinearity is autonomous, the conditions onf needs modified slightly. More precisely, we make the following assumptions.

(V₂) 0<inf_RNV (x)lim_|x|→∞V (x)=sup_RNV (x) <∞.

(f₁) f (t )∈C¹.f_tis a Caratheodory function and there existsC >0, s. t.

f_t(t )C

1+ |t|^2∗−2

, lim

|t|→∞

|f (t )|

|t|^2∗−1 =0.

(f₂) f (t )=o(|t|), as|t| →0.

(f₃) lim_|t|→∞F (t ) t² = ∞.

(f4) ^{f (t )}_|t| is strictly increasing int.

Theorem 3.1.Under assumptions(V2),(f1)–(f4)Eq.(3.1)has a weak solutionu∈X, such thatΦ(u)=c >0,cis defined as

c=inf

N Φ(u),

whereN = {u∈X: u=0, γ (u)=0}.

In this section we denoteV_∞=lim_|x|→∞V (x). There is an associated problem −u+V_∞u=f (u),

u∈H¹ R^N

.

We define the energy functional Φ_∞ by replacing V with V_∞, c_∞ =inf_N∞Φ_∞(u), here N∞= {u ∈X/{0}:

Φ_∞ (u), u =0}. SinceV_∞is a constant, by Theorem 2.1,c_∞>0 is achieved at someu_∞∈N∞. Lemma 3.2.0< c < c_∞.

Proof. It is easy to seec >0. Letu_∞be the minimizer ofc_∞. Thenγ (u_∞) <0, and there ist >0 such thatt u_∞∈N. We have

cΦ(t u_∞) < Φ_∞(t u_∞)Φ_∞(u_∞)=c_∞. 2 We note that with minor changes Lemma 2.3 and 2.4 still hold.

Lemma 3.3.Let(u_n)be a minimizing sequence forc. Then (i) There isβ >0such thatlim infn→∞||u_n||β.

(ii) (u_n)is bounded inX.

(iii) For a subsequence,u_nconverges weakly tou=0.

Proof. (i) The same as Lemma 2.2.

(ii) If not, define v_n =u_n/u_n. Passing to a subsequence, we may assume, v_n v in X. If v_n →0 in L^q(R^N) for 2 q <2^∗, we use the Lebesgue Dominated Convergence theorem to get for any R >0 fixed, limn→∞

R^NF (Rv_n)dx =0. Therefore a contradiction by choosing a largeR >0 inΦ(u_n)Φ(Rv_n)=¹₂R²−

R^NF (Rvn)dx. Thus by the concentration compactness principle there areyn∈R^Nsuch thatwn(x)=vn(yn+x)→ w=0. Then the proof follows from the arguments in Lemma 2.2(ii). Thus(un)is bounded.

(iii) We can assume un uinX,un→u inL^p_loc(R^N). If u=0, we have

R^N(V (x)−V_∞)|un|²dx→0, as n→ ∞. Thus we havec+o(1)=Φ_∞(un)+o(1). Similarly we haveγ (un)=0,γ_∞(un)=o(1). By Lemma 2.4 there existt_n→1 such thatt_nu_n∈N∞. Then we havec+o(1)=Φ_∞(u_n)+o(1)=Φ_∞(t_nu_n)+o(1)c_∞+o(1), a contradiction with Lemma 3.2. 2

Next consider Eq. (3.1) onB_R(0), −u+V (x)u=f (u), inB_R(0),

u=0, on∂B_R(0) (3.2)

we can similarly defineNR=N∩H₀¹(B_R),c_R. By the result of [8],c_Ris achieved by a positive solution calledu_R. It is easy to check thatc_R> candc_R→casR→ ∞.

Lemma 3.4.Let u_R∈NR be a minimizer ofc_R. Assume for a subsequenceR_n→ ∞,

B_Rn|u_n|^p→A∈(0,∞), whereu_n=u_Rn. Then there exists(y_n)⊂R^Ns.t. for anyε >0, existsr_ε>0, for allrr_ε,

lim inf

n→∞

Br(yn)

|u_n|^pA−ε.

Proof. Note thatuRsatisfies (3.3) for allϕ∈H₀¹(BR),

B_Rn

∇uR∇ϕ+V (x)uRϕ dx−

R^N

f (uR)ϕdx=0. (3.3)

Since(un)is bounded inH¹(R^N), by using the concentration compactness principle, existsα∈(0,1]and(yn)⊂R^N s.t. for anyε >0, there existsrε>0, for allrrrε,

lim inf

n→∞

B_r(y_n)

|u_n|^pαA−ε, (3.4)

lim inf

n→∞

R^N\B_r(y_n)

|u_n|^p(1−α)A−ε. (3.5)

Now supposeα <1, then following exactly the same construction as in Lemma 2.5 we have two sequenceswn and v_nsatisfying

lim inf

n→∞

R^N

|wn|^pαA, lim inf

n→∞

R^N

|vn|^p(1−α)A, Φ(un)=Φ(wn)+Φ(vn)+o(1).

Moreover, if we takeϕ=w_n, by (3.3) γ (w_n)=

Φ(u_n), w_n

+o(1)=o(1).

Similarly,γ (v_n)=o(1). By Lemma 2.4, there existt_n→1,s_n→1, s.t.

w_n=t_nw_n∈N, v˜_n=s_nv_n∈N.

If(y_n)is bounded, then lim infn→∞Φ(w_n)cand lim infn→∞Φ(v˜_n)c_∞. If(y_n)is unbounded, then lim inf

n→∞ Φ(wn)c_∞ and lim inf

n→∞ Φ(v˜n)c.

Altogether, we have

Φ(u_n)=Φ(w_n)+Φ(v_n)+o(1)=Φ(t_nw_n)+Φ(s_nv_n)+o(1), and

lim infΦ(un)lim infΦ(tnwn)+lim infΦ(snvn)c+c_∞. A contradiction, so we haveα=1. 2

Proof of Theorem 3.1. Let (un)⊂ N be the minimizing sequence for c given in Lemma 3.4. Let A= limn→∞

R^N|u_n|^pdx. By Lemma 3.3,(u_n)is bounded inX and weakly converges tou=0. By Lemma 3.4, there exists(y_n)⊂R^Ns.t.∀ε >0,∃r >0,

lim inf

n→∞

Br(yn)

|u_n|^pA−ε.

Then(y_n)must be bounded. Otherwise,γ_∞(u_n)=γ (u_n)+o(1). We findt_n→1, s.t.γ (t_nu_n)=0. Then we have c_∞lim infΦ_∞(tnun)=lim infΦ_∞(un)=lim infΦ(un)=c

a contraction withc < c_∞. Now, when(yn)is bounded, we have un→u inL^p(R^N). This gives that along this sequenceΦ(un)is weakly lower semi-continuous, we have

c=inf

N Φ(u)Φ(u)lim infΦ(un)=c. 2

Remark 3.5.Though we assume in this sectionf depends only ont, looking at the proofs we see the arguments can be used with little changes to deal with the following case:f =b(x)f (t ) withb satisfying b∈C¹(R^N,R), b1b(x)b2for someb1, b2>0, andb(x)inf_R^Nb(x)=lim_|x|→∞b(x). The precisely statement is the same.

References

[1] A. Ambrosetti, A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems onRⁿ, Progr. Math., Birkhäuser, in press.

[2] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.

[3] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983) 313–345.

[4] W.-Y. Ding, W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal. 91 (1986) 283–308.

[5] L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schrödinger equation onR^N, Indiana Univ. Math. J. 54 (2005) 443–464.

[6] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I & II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109–145, 223–283.

[7] J. Liu, Y. Wang, Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004) 879–901.

[8] Z. Liu, Z.-Q. Wang, On the Ambrosetti–Rabinowitz superlinear condition, Adv. Nonlinear Stud. 4 (2004) 561–572.

[9] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992) 270–291.

[10] W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) 149–162.

[11] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 2000.

[12] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.