Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2018: 3.18

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

Existence of a bound state solution for quasilinear Schrödinger equations

Yan-Fang Xue
  • School of Mathematics and Statistics, Southwest University, Chongqing 400715; and College of Mathematics and Information Sciences, Xin-Yang Normal University, Xinyang, Henan 464000, P. R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Chun-Lei TangORCID iD: https://orcid.org/0000-0001-6911-3597
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/anona-2016-0244

Abstract

In this article, we establish the existence of bound state solutions for a class of quasilinear Schrödinger equations whose nonlinear term is asymptotically linear in N. After changing the variables, the quasilinear equation becomes a semilinear equation, whose respective associated functional is well defined in H1(N). The proofs are based on the Pohozaev manifold and a linking theorem.

Keywords: Quasilinear Schrödinger equation; asymptotically linear; barycenter; linking; Pohozaev manifold

MSC 2010: 35J62; 35J20; 35B09

1 Introduction and main result

In the present paper, we consider a class of quasilinear Schrödinger equations of the form

iψt=-ψ+W(x)ψ-l(|ψ|2)ψ-κ[ρ(|ψ|2)]ρ(|ψ|2)ψ,(1.1)

where W:N is a given potential, N3, κ is a positive constant, and l, ρ are real functions. Corresponding to various types of nonlinear terms ρ, this problem appears naturally in different mathematical physical models; see [28, 23, 21] for an explanation. Here we focus on the case ρ(s)=s, κ=1. As we all know, a standing wave of (1.1) is a solution of the form ψ(t,x)=exp(-iEt)u(x), E, and consequently u(x) satisfies the following equation:

-Δu+V(x)u-Δ(u2)u=g(u),xN,N3,(1.2)

where V(x)=W(x)-E is the new potential and g(u)=l(u2)u is the new nonlinear term. The main purpose of this paper is to deal with the existence of solutions for equation (1.2). This kind of problem has been studied by many authors; see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31, 24, 32, 27, 9, 7, 25, 13, 8, 30] and the references therein.

In [28, 23], by using a constrained minimization argument, a positive ground state solution has been proved for equation (1.2) with g(u)=λ|u|q-1u, 4q+1<22*, where 2*=2N/(N-2) is the Sobolev critical exponent. The case 4q+1<22* is called subcritical growth; there are many articles that deal with this class of problem (see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31]). In [22], the existence of both positive and sign-changing ground states of soliton-type solutions were established via the Nehari method. Then by a change of variables, the quasilinear problem is transformed into a semilinear one; see [21] for an Orlicz space framework and [6] for a Sobolev space frame. Recently, a perturbation method was developed in [25] to deal with equation (1.2), which can be applied to more general quasilinear Schrödinger equations (see also [20]).

For the critical case, we would like to mention [32, 27, 9, 26, 7, 25, 13, 8, 30] and the references therein. It seems that Moameni [27] first studied the critical case when the potential V is radial and satisfies some geometric conditions. Do Ó, Miyagaki and Soares [9] obtained a positive classical solution by using the concentration compactness principle of Lions [19]. He and Li [13] obtained the existence, concentration and multiplicity of weak solutions by employing the minimax theorems and Ljusternik–Schnirelmann theory.

In recent years, the use of the Pohozaev manifold was shown very effective when treading nonlinearities which do not satisfy the Ambrosetti–Rabinowitz condition and the monotonicity condition; see [16, 5, 14, 15, 4]. Lehrer and Maia [16] employed the minimization methods restricted to the Pohozaev manifold to obtain the existence of positive solutions for the asymptotically linear case. Later in [5], Carrião, Lehrer and Miyagaki extended the result given in [16] to more general quasilinear equations. Motivated by [16, 5, 17], we consider equation (1.2) with the nonlinear term g(u) being nonhomogeneous and asymptotically linear at infinity. We will use the linking theorem together with the barycenter function restricted to the Pohozaev manifold associated to our problem.

The main obstacle in finding a solution of equation (1.2) is due to the influence of the quasilinear and nonconvex term Δ(u2)u. The other difficulty is the possible lack of compactness due to the unboundedness of the domain. We will employ an argument developed in [6] to overcome the first difficult and a splitting lemma to conquer the second one.

We suppose that V satisfies the following assumptions:

  • (V1)

    VC2(N,);

  • (V2)

    lim|x|V(x)=V<1, V(x)>V>0 for all xN;

  • (V3)

    (V(x),x)0 for all xN, and the strict inequality holds on a subset of positive Lebesgue measure of N;

  • (V4)

    NV(x)+(V(x),x)NV for all xN;

  • (V5)

    xHV(x)xN+(V(x),x)0 for all xN, where HV is the Hessian matrix of the function V.

We assume the following conditions on the function g:

  • (g1)

    gC1(+,+) and lims0+g(s)s=0;

  • (g2)

    limsg(s)s2=1;

  • (g3)

    Q(s):=14g(s)s-G(s)>0 and limsQ(s)=+, where G(s)=0sg(t)𝑑t.

We employ an argument developed in [6] to introduce a variational framework associated with equation (1.2). We observe that equation (1.2) is formally the Euler–Lagrange equation associated with the energy functional

J(u)=12N[(1+2u2)|u|2]𝑑x+12NV(x)u2𝑑x-NG(u)𝑑x.

We make a change of variables v:=f-1(u), where f is defined by

f(t)=1(1+2f2(t))1/2,t[0,+),f(t)=-f(-t),t(-,0].

After the change of variables from J, we obtain the following functional:

I(v)=12N|v|2𝑑x+12NV(x)f2(v)𝑑x-NG(f(v))𝑑x.

Then I(v)=J(u)=J(f(v)) and I is well defined on H1(N), IC1(H1(N),) under the hypotheses (V1), (V2) and (g1)–(g3). Moreover, we observe that if v is a critical point of the functional I, then the function u=f(v) is a solution of equation (1.2) (see [6]).

The critical points of I are weak solutions of the problem

-Δv+V(x)f(v)f(v)=g(f(v))f(v).(1.3)

We can demonstrate that

I(v),φ=Nvφdx+NV(x)f(v)f(v)φ𝑑x-Ng(f(v))f(v)φ𝑑x

for all v,φH1(N). Each solution of equation (1.3) satisfies γ(v)=0, where

γ(v)=N-22N|v|2𝑑x+N2NV(x)f2(v)𝑑x+12N(V(x),x)f2(v)𝑑x-NNG(f(v))𝑑x.

We define the Pohozaev manifold associated with equation (1.3) by

𝒫:={vH1(N):v0,γ(v)=0}.

Let c be the min-max mountain pass level for the functional I given by

c=infζΓmaxt[0,1]I(ζ(t)),

where

Γ={ζC([0,1],H1(N)):ζ(0)=0ζ(1),I(ζ(1))<0}.

Under the previous hypotheses on V and g, we have the following nonexistence result.

Theorem 1.1.

Suppose that (V1)(V5) and (g1)(g3) are satisfied. Then p:=infvPI(v) is not a critical level for the functional I. In particular, the infimum p is not achieved.

Consider now also the limiting problem

-Δv+Vf(v)f(v)=g(f(v))f(v).(1.4)

Its associated energy functional is denoted by I.

I(v)=12N|v|2𝑑x+12NVf2(v)𝑑x-NG(f(v))𝑑x.

Each solution of equation (1.4) satisfies the following Pohozaev identity:

N-22N|v|2𝑑x+N2NVf2(v)𝑑x=NNG(f(v))𝑑x.(1.5)

We define the Pohozaev manifold associated with equation (1.4) by

𝒫:={vH1(N):v0,v satisfies (1.5)},

and p:=infv𝒫I(v).

We say that a solution vH1(N) of equation (1.4) is a least energy solution if and only if I(v)=m, where

m=inf{I(v):vH1(N){0} is a solution of (1.4)}.

We define

Γ={ζC([0,1],H1(N)):ζ(0)=0ζ(1),I(ζ(1))<0},

as well as the mountain pass min-max level

c=infζΓmaxt[0,1]I(ζ(t)).

Using a method similar to the one in [15], we can deduce that c=m=p.

Now we can state our main existence result.

Theorem 1.2.

Suppose that (V1)(V5) and (g1)(g3) are satisfied and the following facts hold:

  • (1)

    gLip(+,+) ;

  • (2)

    V(x)-V is sufficiently small;

  • (3)

    the least energy level c of equation ( 1.4 ) is an isolated radial critical level for I or equation ( 1.4 ) admits a unique positive solution which is radially symmetric about some point.

Then equation (1.2) admits a positive solution whose energy is above c.

Remark 1.3.

These results extend the corresponding results in [17] to the more general quasilinear case. The framework employed and ideas of the proofs for our main results are close to those found in [17]. However, some technical details in this paper are different from those in [17].

Remark 1.4.

For the case of nonlinearities g(u)=|u|p-1u, uniqueness properties of ground state solutions of equation (1.2) were recently proved in [1, 12, 29, 2], so assumption of Theorem 1.2 (3) is expected to be fulfilled.

Example 1.5.

Let V(x)=a+b(1+|x|2)-c, where 0<a<1, 0<cN-12 and b>0 is small enough. Then

(V(x),x)=-2bc|x|2(1+|x|2)-c-1

and

xHV(x)x=2bc(1+|x|2)-c-2[2(c+1)|x|4-(1+|x|2)|x|2].

We can see by a computation that V(x) satisfies all conditions in Theorem 1.2.

Remark 1.6.

Conditions (g1) and (g2) imply that given ε>0 and 3q22*, there exists a positive constant Cε=C(ε,q) such that

G(s)ε2|s|2+Cε|s|qfor all s.(1.6)

We also obtain the estimate

g(s)ε|s|+Cε|s|q-1for all s.

Remark 1.7.

Since we are looking for positive solutions, we set g(s)=0 for all s<0. Let v be a critical point of I. Taking φ=-v-, we have

N|v-|2𝑑x+NV(x)f(v)f(v)(-v-)𝑑x=0.

Since f(v)(-v-)0, we get

N|v-|2𝑑x=0

and

NV(x)f(v)(-v-)1+2f2(v)𝑑x=0.

Hence we may conclude that v-=0 a.e. in N and v=v+0. As u=f(v), we conclude that u is a nonnegative solution for equation (1.2).

Notation.

In this paper, we use the following notations:

  • H1(N) is the usual Hilbert space endowed with the norm

    u2=N(|u|2+u2)𝑑x.

  • Ls(N) is the usual Banach space endowed with the norm

    uss=N|u|s𝑑xfor all s[1,+).

  • u=esssupxN|u(x)| denotes the usual norm in L(N).

  • Br(y)={xN:|x-y|<r}, Br={xN:|x|<r}.

  • u+=max{u,0}, u-=max{-u,0}.

  • |Ω| denotes the Lebesgue measure of the set Ω.

  • C,Cε,C1,C2, denote various positive constants whose exact value is inessential.

2 Some preliminary results

In this section, we first summarize the properties of f, which have been proved in [6, 10].

Lemma 2.1.

The function f satisfies the following properties:

  • (1)

    f is uniquely defined, C and invertible;

  • (2)

    |f(t)|1 for all t ;

  • (3)

    |f(t)||t| for all t ;

  • (4)

    f(t)/t1 as t0 ;

  • (5)

    f(t)/t21/4 as t ;

  • (6)

    f(t)/2tf(t)f(t) for all t>0 ;

  • (7)

    |f(t)|21/4|t|1/2 for all t ;

  • (8)

    f(t)1 as t0 ;

  • (9)

    there exists a positive constant C such that |f(t)|C|t| for |t|1 and |f(t)|C|t|1/2 for |t|1 ;

  • (10)

    |f(t)f(t)|1/2 for all t.

Lemma 2.2.

The functional γ and the Pohozaev manifold P satisfy the following properties:

  • (1)

    {v0} is an isolated point of γ-1({0}) ;

  • (2)

    𝒫 is a closed set;

  • (3)

    𝒫 is a C1 manifold;

  • (4)

    there is a σ>0 such that v>σ for all v𝒫.

Proof.

(1) Thanks to Lemma 2.1 (9), we can deduce that there is C1>0 such that

f2(t)C1(t2-|t|2*).(2.1)

By (V4), (1.6), (2.1), Lemma 2.1 (3), (7), and the Sobolev inequality, we have

γ(v)=N-22N|v|2𝑑x+N2NV(x)f2(v)𝑑x+12N(V(x),x)f2(v)𝑑x-NNG(f(v))𝑑xN-22N|v|2𝑑x+12NNVf2(v)𝑑x-NN(ε2|f(v)|2+Cε|f(v)|22*)𝑑xN-22N|v|2𝑑x+NV2NC1(v2-|v|2*)𝑑x-NN(ε2|v|2+22*2Cε|v|2*)𝑑xmin{N-22,NVC12-εN2}v2-(NVC12+22*2CεN)S-2*2v2*=C2v2-C3v2*,

where S is the best Sobolev constant of the embedding 𝒟1,2(N)L2*(N) and

C2=min{N-22,NVC12-εN2}>0,C3=(NVC12+22*2CεN)S-2*2>0

by taking ε>0 sufficiently small. Let 0<ρ<1 such that ρ2*<C2ρ2/(2C3); then if v=ρ, we have

γ(v)C2ρ2-C3ρ2*>C22ρ2>0.

(2) The functional γ(v) is a C1 functional, thus 𝒫{0}=γ-1({0}) is a closed subset. Moreover, {v0} is an isolated point in γ-1({0}) and the assertion follows.

(3) It follows from Lemma 2.1 (6) and (g3) that

12g(f(v))f(v)v-G(f(v))14g(f(v))f(v)-G(f(v))>0.(2.2)

Since v𝒫, we have

N-22N|v|2𝑑x+N2NV(x)f2(v)𝑑x+12N(V(x),x)f2(v)𝑑x=NNG(f(v))𝑑x.(2.3)

Hence we can deduce that

γ(v)v=(N-2)N|v|2𝑑x+NNV(x)f(v)f(v)v𝑑x+N(V(x),x)f(v)f(v)v𝑑x-NNg(f(v))f(v)v𝑑x.=2NN(G(f(v))-12g(f(v))f(v)v)𝑑x+N[NV(x)+(V(x),x)](f(v)f(v)v-f2(v))𝑑x.

Combining this with (2.2), Lemma 2.1 (6) and (V4), we have γ(v)v<0 if v𝒫. This shows that 𝒫 is a C1 manifold.

(4) Since 0 is an isolated point in γ-1({0}), there must be a ball vσ which does not intersect 𝒫 and the assertion is proved. ∎

Lemma 2.3.

Assume (V1), (V5) and (g1) hold. Then P is a nature constraint for the functional I.

Proof.

Let v𝒫 be a critical point of I|𝒫. By the theorem of Lagrange multipliers, there exists a μ such that I(v)+μγ(v)=0. The proof is complete as soon as we show that μ=0. Evaluating the linear functional above at v𝒫, we obtain

I(v)v+μγ(v)v=0,

namely

N|v|2dx+NV(x)f(v)f(v)vdx-Ng(f(v))f(v)vdx+μ[(N-2)N|v|2dx+NNV(x)f(v)f(v)vdx+N(V(x),x)f(v)f(v)vdx-NNg(f(v))f(v)vdx]=0.

This expression is associated with the equation

-Δv+V(x)f(v)f(v)-g(f(v))f(v)+μ[-(N-2)Δv+NV(x)f(v)f(v)+(V(x),x)f(v)f(v)-Ng(f(v))f(v)]=0,

which can be rewritten as

-[1+μ(N-2)]Δv+(1+μN)V(x)f(v)f(v)+μ(V(x),x)f(v)f(v)=(1+Nμ)g(f(v))f(v).(2.4)

Each solution of equation (2.4) satisfies γ~(v)=0, where

γ~(v)=1+μ(N-2)2(N-2)N|v|2𝑑x+1+μN2N[NV(x)+(V(x),x)]f2(v)𝑑x+μN2N(V(x),x)f2(v)𝑑x+μ2N(xHV(x)x)f2(v)𝑑x-(1+μN)NNG(f(v))𝑑x.

Recalling that v𝒫, and substituting γ(v)=0 in the equation above, we get

γ~(v)=-μ(N-2)N|v|2𝑑x+μN2N(V(x),x)f2(v)𝑑x+μ2N(xHV(x)x)f2(v)𝑑x.

Since v is a solution of equation (2.4), it satisfies γ~(v)=0. This yields

μ(N-2)N|v|2𝑑x=μN2N[(V(x),x)+xHV(x)xN]f2(v)𝑑x.

From (V5) we get that, if μ<0, the right-hand side of the above equation is nonnegative, while the left-hand side is negative. If μ>0, one gets the same contradiction. Hence μ=0. ∎

3 Proof of Theorem 1.1

In this section, we apply ideas similar to those employed in [16, 4, 17]. Set k(s):=g(f(s))f(s)-Vf(s)f(s); then equation (1.4) becomes

-Δv=k(v).(3.1)

Since equation (3.1) is a semilinear equation, we can use the conclusions in [16, 17]. Let

K(s):=0sk(t)𝑑t=G(f(s))-12Vf2(s);

then we can get the following lemmas. The proof of these lemmas can be found in [16]; we omit them.

Lemma 3.1.

Assume that vH1(RN) and RNK(v)𝑑x>0. Then there exist unique t1>0 and t2>0 such that v(/t1)P and v(/t2)P.

If vP, then there exists tv>0 such that v(/tv)P and tv<1.

If wP, then there exists tw>0 such that w(/tw)P and tw>1.

Lemma 3.2.

Let Ω={vH1(RN):v0,RNK(v)𝑑x>0}. Then the function t1:ΩR+ given by vt1(v) such that v(/t1(v))P is continuous.

Lemma 3.3.

Let vP. Then for any yRN, we have v(-y)P. Moreover, there exists ty>1 such that

v(-yty)𝒫𝑎𝑛𝑑lim|y|+ty=1.

Lemma 3.4.

There holds supyRNty:=t¯<+ and t¯>1.

Lemma 3.5.

There exists a real number σ^>0 such that infvPv2σ^.

Lemma 3.6.

There holds p=infvPI(v)>0 and p=c.

Proof of Theorem 1.1.

Arguing by contradiction, we suppose that there is vH1(N) such that I(v)=p and I(v)=0. Then v𝒫. By Lemma 3.1, there exists tv>0 such that v(/tv)𝒫 and tv<1. From (2.3) and (V3) we deduce that

p=I(v)=12N|v|2𝑑x+12NV(x)f2(v)𝑑x-NG(f(v))𝑑x=1NN|v|2𝑑x-12NN(V(x),x)f2(v)𝑑x>tvN-2NN|v|2𝑑x=I(v(xtv))c,

which is a contradiction to Lemma 3.6. ∎

4 Proof of Theorem 1.2

This section is dedicated to proving the existence of a positive solution for equation (1.3). By the previous results, we should search for solutions which have energy levels above c. Similarly to what was done in [16, 17], we start by showing that the min-max levels of the mountain pass theorem for the functionals I and I are equal.

Lemma 4.1.

There holds c=c=p.

The proof is analogous to the proofs of [16, Lemmas 4.1 and 4.2]; we omit it.

Lemma 4.2.

For every ζΓ, there exists s(0,1) such that ζ(s) intersects P.

Proof.

By the proof of Lemma 2.2 (1), we learn that there exists 0<ρ<1 such that γ(v)>0 if 0<v<ρ. Furthermore, we observe that

γ(v)=N-22N|v|2𝑑x+N2NV(x)f2(v)𝑑x+12N(V(x),x)f2(v)𝑑x-NNG(f(v))𝑑x=NI(v)-N|v|2𝑑x+12N(V(x),x)f2(v)𝑑x.

It follows from (V3) that

γ(v)<NI(v).

Therefore, if ζΓ, we have γ(ζ(0))=0 and γ(ζ(1))<NI(ζ(1))<0. Since I(ζ(1))<0, we conclude that there exists s(0,1) such that γ(ζ(s))=0 for which ζ(s)>ρ. The function ζ(s) satisfies ζ(s)𝒫, which shows that every path ζΓ intersects 𝒫. ∎

Lemma 4.3.

There exists a (C)c sequence {vn}H1(RN) where

c=infζΓmaxt[0,1]I(ζ(t)).

The proof of Lemma 4.3 can be found in [16] (see also [17, 18]).

Lemma 4.4.

If {vn} is a (C)d sequence with d>0, then it has a bounded subsequence.

Proof.

First of all, we observe that if a sequence {vn}H1(N) satisfies

N|vn|2𝑑x+NV(x)f2(vn)𝑑xC4

for some constant C4>0, then the sequence {vn} is bounded in H1(N). For that, we simply need to demonstrate that Nvn2𝑑x is bounded. In fact, by Lemma 2.1 (9) and (V2), we observe that

{xN:|vn(x)|1}vn2𝑑x1C2{xN:|vn(x)|1}f2(vn)𝑑x1C2VNV(x)f2(vn)𝑑xC4C2V.

Moreover, by the Sobolev inequality and Lemma 2.1 (9), one deduces

{xN:|vn(x)|>1}vn2𝑑x{xN:|vn(x)|>1}vn2*𝑑xC5({xN:|vn(x)|>1}|vn|2𝑑x)2*2C5(N|vn|2𝑑x)2*2C5C42*2.

Hence there is a constant C6>0 such that

Nvn2𝑑x={xN:|vn(x)|1}vn2𝑑x+{xN:|vn(x)|>1}vn2𝑑xC6.

Therefore, it remains to show that

N|vn|2𝑑x+NV(x)f2(vn)𝑑x

is bounded.

Let {vn}H1(N) be an arbitrary Cerami sequence for I at level d>0, that is,

I(vn)dand(1+vn)I(vn)0,

namely

12N|vn|2𝑑x+12NV(x)f2(vn)𝑑x-NG(f(vn))𝑑x=d+on(1),(4.1)

and for any φH1(N),

I(vn),φ=Nvnφdx+NV(x)f(vn)f(vn)φ𝑑x-Ng(f(vn))f(vn)φ𝑑x=on(1).

Choosing

φ=φn=1+2f2(vn)f(vn)=f(vn)f(vn)

from Lemma 2.1 (6), we get φn22vn2 and

|φn|=(1+2f2(vn)1+2f2(vn))|vn|2|vn|.

Thus there exists a constant C7>0 such that φnC7vn. Recalling that {vn}H1(N) is a Cerami sequence, we get

I(vn),φn=N(1+2f2(vn)1+2f2(vn))|vn|2𝑑x+NV(x)f2(vn)𝑑x-Ng(f(vn))f(vn)𝑑x=on(1).(4.2)

Computing (4.1) - 14(4.2), we get

d+on(1)=14N11+2f2(vn)|vn|2𝑑x+14NV(x)f2(vn)𝑑x+N(14g(f(vn))f(vn)-G(f(vn)))𝑑x.

Thanks to (2.2), we get

14N11+2f2(vn)|vn|2𝑑x+14NV(x)f2(vn)𝑑xd+on(1).(4.3)

Denote wn=f(vn); then |vn|2=(1+2wn2)|wn|2. We can rewrite (4.1) and (4.3) as follows:

12N(1+2wn2)|wn|2𝑑x+12NV(x)wn2𝑑x-NG(wn)𝑑x=d+on(1)(4.4)

and

14N|wn|2𝑑x+14NV(x)wn2𝑑xd+on(1).(4.5)

From (4.5) and (V2) we can see that {wn} is bounded in H1(N). It follows from (1.6) that

NG(wn)𝑑xN(ε2|wn|2+Cε|wn|2*)𝑑xε2N|wn|2𝑑x+CεS-2*2(N|wn|2)2*2C8.

By the above inequality and (4.4), one has

12N(1+2wn2)|wn|2𝑑x+12NV(x)wn2𝑑xC8+d+on(1),

namely

12N|vn|2𝑑x+12NV(x)f2(vn)𝑑xC8+d+on(1).

Lemma 4.5 (Splitting (see [33])).

Let {vn}H1(RN) be a bounded sequence such that

I(vn)c𝑎𝑛𝑑(1+vn)I(vn)0.

Then there exists (if necessary, replace {vn} by a subsequence) a solution v¯ of equation (1.3), a number kN{0}, k functions v1,v2,,vk and k sequences of points {ynj}RN, 1jk, satisfying the following properties:

  • (1)

    vnv¯ in H1(N) or

  • (2)

    vj are nontrivial solutions of equation ( 1.4 );

  • (3)

    |ynj| and |ynj-yni|, ij ;

  • (4)

    vn-i=1kvi(x-yni)v¯ ;

  • (5)

    I(vn)I(v¯)+i=1kI(vi).

Corollary 4.6.

If I(vn)c and (1+vn)I(vn)0, then either {vn} is relatively compact or the splitting lemma holds with k=1 and v¯=0.

Let us set

c:=inf{c>c:c is a radial critical value of I}

Then we have the following lemma.

Lemma 4.7.

Assume that c is an isolated radial critical level for I. Then c>c and I satisfies condition (3) at level d(c,min{c,2c}). Assume now that the limiting equation (1.4) admits a unique positive radial solution. Then I satisfies condition (3) at level d(c,2c).

The proof is analogous to [18, Lemma 5.9]; we omit it.

Lemma 4.8.

If I(vn)d>0 and {vn}P, then the sequence {vn} is bounded.

Proof.

Since I(vn)d>0 and {vn}𝒫, we get

d+1>I(vn)=1NN|vn|2𝑑x-12NN(V(x),x)f2(vn)𝑑x1NN|vn|2𝑑x,

where we also used (V3). Therefore vn2 is bounded. By the Sobolev inequality, the sequence vn2* is also bounded.

It follows from (V2), (1.6), (2.1), and Lemma 2.1 (3) and (7) that

d+1I(vn)=12N|vn|2𝑑x+12NV(x)f2(vn)𝑑x-NG(f(vn))𝑑x12vn22+V2Nf2(vn)𝑑x-N(ε2|f(vn)|2+Cε|f(vn)|22*)𝑑x.12vn22+V2NC1(vn2-|vn|2*)𝑑x-N(ε2vn2+22*2Cε|vn|2*)𝑑x.=12vn22+(VC12-ε2)Nvn2𝑑x-(VC12+22*2Cε)N|vn|2*𝑑x.

Since vn2 and vn2* are bounded, vn2 is bounded as well. Hence {vn} is bounded in H1(N). ∎

Definition 4.9.

Define the barycenter function of a given function uH1(N){0} by setting

μ(u)(x)=1|B1|B1(x)|u(y)|𝑑y,

with μ(u)L(N) and μ is a continuous function. Subsequently, take

u^(x)=[μ(u)(x)-12maxμ(u)]+.

It follows that u^C0(N). Now we define the barycenter of u by

β(u)=1u^1Nxu^(x)𝑑xN.

Then β(u) is well defined since u^ has compact support.

The function β(u) satisfies the following properties:

  • (1)

    β is a continuous function in H1(N){0};

  • (2)

    if u is radial, then β(u)=0;

  • (3)

    if yN is given and if we define uy(x):=u(x-y), then β(uy)=β(u)+y.

We shall also need the following lemma.

Lemma 4.10.

Assume that {un},{vn}H1(RN) are such that un-vn0 and I(vn)0 as n, where {vn} is bounded. Then I(un)0 as n.

Proof.

We simply observe that

f(t)=1(1+2f2(t))1/2,

thus by Lemma 2.1 (10) we have

|f′′(t)|=|-2f(t)f(t)(1+2f2(t))3/2||2f(t)f(t)|2.(4.6)

Since gLip(+,+), we find that there is a constant C9>0 such that

|g(s)-g(t)|C9|s-t|for all s,t.

Therefore,

|g(f(un))f(un)-g(f(vn))f(vn)|=|[g(f(un))-g(f(vn))]f(un)-g(f(vn))[f(vn)-f(un)]|C9|f(un)-f(vn)||f(un)|+C9|f(vn)-0||f(vn)-f(un)|C9|f(ξn)||un-vn||f(un)|+C9|vn||f′′(ξn)||vn-un|C9|un-vn|+2C9|vn||vn-un|,

where we also used the mean value theorem, Lemma 2.1 (2) and (3) and (4.6). Hence, by using the Hölder inequality, one has

N|(g(f(un))f(un)-g(f(vn))f(vn))φ|𝑑xC9N|un-vn||φ|𝑑x+2C9N|vn||vn-un||φ|𝑑xC9un-vn2φ2+2C9vnφ2vn-un20(as un-vn0).(4.7)

Since the function f(s)f(s) is continuous and V(x) satisfies (V2), we can conclude that

NV(x)((f(un)f(un)-f(vn)f(vn))φdx0(as un-vn0).(4.8)

We also find that

N(un-vn)φdx0(as un-vn0).(4.9)

It follows from (4.7), (4.8) and (4.9) that

I(un)-I(vn),φ=N(un-vn)φdx+NV(x)((f(un)f(un)-f(vn)f(vn))φdx-N(g(f(un))f(un)-g(f(vn))f(vn))φ𝑑x0(as un-vn0).

Hence I(un)0 as n. ∎

We define

b:=inf{I(v):v𝒫 and β(v)=0}.

It is clear that bc; moreover, we have the following lemma.

Lemma 4.11.

There holds b>c.

Proof.

Suppose b=c. By the definition of b, there is a minimizing sequence

{vn}{vH1(N):v𝒫 and β(v)=0}

such that I(vn)b>0. By Lemma 4.8, {vn} is bounded. Since b=c=p by Lemma 4.1, {vn} is also a minimizing sequence of I on 𝒫. By Ekeland’s variational principle [33, Theorem 8.5], there is another sequence {v~n}𝒫 such that I(v~n)p, I|𝒫(v~n)0 and v~n-vn0 as n. Now we prove that I(v~n)0 as n. Indeed, if I(v~n) does not go to zero, that means there exist ε0>0 and a subsequence denoted also by v~n such that I(v~n)>ε0. Arguing as in the proof of Lemma 4.10, we can deduce that there is a positive constant C10 such that

|I(v~n)-I(v),φ|C10v~n-vφfor all v,φH1(N).

Thus, if

v~n-v<δ~C10:=3δ,

then

I(v~n)-I(v)<δ~.

This yields

ε0-δ~<I(v~n)-δ~<I(v).

For δ~>0 sufficiently small, we have λ:=ε0-δ~>0, and for all vB3δ(v~n), one has I(v)>λ. Now let ε:=min{p/2,(λδ)/8} and S:={v~n}. By [33, Lemma 2.3], there is a deformation η on the level p, taking all the points of Sδ to the level p-ε.

I(η(1,u))I(u)for all uH1(N).

Moreover, for n sufficiently large,

maxt>0I(η(1,v~n(t)))p-ε,

because {v~n} is a minimizing sequence, I(v~n)p+ε/2, for n sufficiently large, and since {v~n}𝒫, we have

maxt>0I(v~n(t))=I(v~n)p.

On the other hand, ζ0(t):=η(1,v~n(/Mt)) is a path in Γ for M and n large enough, hence

cmaxt[0,1]I(η(1,v~n(Mt)))=maxt>0I(η(1,v~n(t)))p-ε<p,

which is a contradiction to p=c, provided by Lemma 4.1; hence I(v~n)0 as n. By Lemma 4.10, we get I(vn)0 as n. Hereafter, the sequence {vn} satisfies the assumptions of Corollary 4.6. Since p=c and p is not attained by Theorem 1.1, the splitting lemma holds with k=1. This yields

vn(x)=v1(x-yn)+on(1),

where ynN, |yn| and v1 is a solution of equation (1.4). Making a translation, we obtain

vn(x+yn)=v1(x)+on(1).

Calculating the barycenter function on both sides, we have

β(vn(x+yn))=β(vn)-yn,β(v1(x)+on(1))=β(v1(x))+on(1),

where the first equality comes from property (3) of the barycenter function β and the second one due to the continuity of β. Since β(vn)=0, |yn| and β(v1(x))=0, we arrive at a contradiction, yielding b>c. ∎

Let us consider the positive, radially symmetric, ground state solution wH1(N) of equation (1.4). We define the operator Π:N𝒫 by

Π[y](x)=w(x-yty),

where ty is the real number t which projects w(-y) onto the Pohozaev manifold 𝒫. Since ty is unique and ty(w(-y)) is a continuous function of w(-y), we have that Π is a continuous function of y.

The following lemma describes some properties of the operator Π; its proof can be found in [16, 17].

Lemma 4.12.

β(Π[y](x))=y and I(Π[y])c as |y|.

Lemma 4.13.

Assume that

  • (V6)

    there holds

    V-V<2min{c,2c}-ct¯Nw22,

    where t¯=supyNty.

Then I(Π[y])<min{c,2c}.

Proof.

Since I is translation invariant, the maximum of tI(w(/t)) is attained at t=1 and ty>1. It follows from (V6) and Lemma 2.1 (3) that

I(Π[y])=I(Π[y])+I(Π[y])-I(Π[y])I(w)+12N(V(x)-V)f2(Π[y])𝑑x<c+min{c,2c}-ct¯Nw22N(Π[y])2𝑑x=c+(min{c,2c}-c)tyNt¯Nw22w22min{c,2c},

which concludes the proof. ∎

Remark 4.14.

Replacing (V6) with

V-V<2ct¯Nw22

yields I(Π[y])<2c.

Definition 4.15.

Let S be a closed subset of a Banach space X and let Q be a submanifold of X with relative boundary Q. We say that S and Q link if the following facts hold:

  • (1)

    SQ=;

  • (2)

    for any hC0(X,X) such that hQ=id there holds h(Q)S.

Moreover, if S and Q are as stated above and B is a subset of C0(X,X), then S and Q link with respect to B if (1) and (2) hold for any hB.

Theorem 4.16 (Linking).

Suppose that IC1(X,R) is a functional satisfying the (C) condition. Consider a closed subset SX and a submanifold QX with relative boundary Q. In addition, suppose the following:

  • (1)

    S and Q link;

  • (2)

    α=infuSI(u)>supuQI(u)=α0 ;

  • (3)

    supuQI(u)<+.

If B={hC0(X,X):hQ=id}, then the real number τ=infhBsupuQI(h(u)) defines a critical value of I with τα.

Now we are ready to prove our main existence result.

Proof of Theorem 1.2.

It follows from (V2) that I(v)<I(v) for all vH1(N){0}, hence I(Π[y])<I(Π[y]) for any yN. From Lemma 4.11 and Lemma 4.12 we have b>c and I(Π[y])c as |y|, hence there is ρ¯>0 such that for all ρρ¯,

c<max|y|=ρI(Π[y])<b.(4.10)

In order to apply the linking theorem, we take

Q:=Π(Bρ¯¯),S:={vH1(N):v𝒫 and β(v)=0}.

We will show that the conditions of Theorem 4.16 are satisfied.

(1) Since β(Π[y](x))=y, from Lemma 4.12 we have that SQ=, because if vS, then β(v)=0, and if vQ, then β(v)=y0 due to the equality |y|=ρ¯. Now we show that h(Q)S, for any h, where

={hC(Q,𝒫):hQ=id}

Given h, let us define T:Bρ¯¯N for T(y)=βhΠ[y]. The function T is continuous, because it is the composition of continuous functions. Moreover, for any |y|=ρ¯, we have that Π[y]Q, thus hΠ[y]=Π[y], because hQ=id. Hence from Lemma 4.12 we have T(y)=y. By the fixed point theorem of Brouwer, we conclude that there exists y~Bρ¯ such that T(y~)=0, which implies h(Π[y~])S. Therefore h(Q)S. From Definition 4.15, we know that S and Q link, i.e. relation (1) is proved.

(2) From the definitions of b and Q and inequality (4.10) we may write

b=infSI>maxQI,

which implies relation (2).

(3) Let us define

d=infhmaxvQI(h(v)).

Then we have db. Indeed, we have already proved that h(Q)S for all h. If h is fixed, then there exists wS such that w belongs to h(Q), which means that w=h(u) for some uΠ(Bρ¯¯)=Q. Therefore,

I(w)infvSI(v),maxvQI(h(v))I(h(u)).

This gives

maxvQI(h(v))I(h(u))=I(w)infvSI(v)=b,

and hence

d=infhmaxvQI(h(v))b.

In particular, it follows that d>c because from Lemma 4.11 we know that b>c. Furthermore, if we take h=id, then using Lemma 4.13, we get

infhmaxvQI(h(v))<maxvQI(v)<min{c,2c}.(4.11)

Hence we can deduce that d<min{c,2c}; furthermore, we have d(c,min{c,2c}). Thanks to Lemma 4.7, we get that the (C) condition is satisfied at level d. Also, inequality (4.11) tells us that relation (3) is satisfied.

From (1), (2) and (3) above we can apply the linking theorem and conclude that d is a critical level for the functional I. Hence there exists a nontrivial solution vH1(N) of equation (1.3). Reasoning as usual, because of the hypotheses on g and f, and using the maximum principle, we conclude that v is positive and Theorem 1.2 is proved. ∎

Acknowledgements

The authors would like to thank the referee for his/her useful suggestions.

References

  • [1]

    S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal. 75 (2012), 819–833.  CrossrefGoogle Scholar

  • [2]

    S. Adachi and T. Watanabe, Asymptotic uniqueness of ground states for a class of quasilinear Schrödinger equations with H1-supercritical exponent, J. Differential Equations 260 (2016), 3086–3118.  Google Scholar

  • [3]

    A. Ambrosetti and Z.-Q. Wang, Positive solutions to a class of quasilinear elliptic equations on , Discrete Contin. Dyn. Syst. 9 (2003), 55–68.  Google Scholar

  • [4]

    A. Azzollini and A. Pomponio, On the Schrodinger equation in RN under the effect of a general nonlinear term, Indiana Univ. Math. J. 58 (2009), no. 3, 1361–1378.  Web of ScienceGoogle Scholar

  • [5]

    P. C. Carrião, R. Lehrer and O. H. Miyagaki, Existence of solutions to a class of asymptotically linear Schrödinger equations in Rn via the Pohozaev manifold, J. Math. Anal. Appl. 428 (2015), 165–183.  CrossrefGoogle Scholar

  • [6]

    M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. 56 (2004), 213–226.  CrossrefGoogle Scholar

  • [7]

    Y. B. Deng, S. J. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys. 54 (2013), Article ID 011504.  Google Scholar

  • [8]

    Y. B. Deng, S. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations 260 (2016), 1228–1262.  CrossrefGoogle Scholar

  • [9]

    J. M. B. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 248 (2010), 722–744.  CrossrefGoogle Scholar

  • [10]

    J. M. B. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal. 8 (2009), 621–644.  Google Scholar

  • [11]

    X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations 254 (2013), 2015–2032.  CrossrefGoogle Scholar

  • [12]

    F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal. 1 (2012), 159–179.  Web of ScienceGoogle Scholar

  • [13]

    Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents, Discrete Contin. Dyn. Syst. 36 (2016), 731–762.  Google Scholar

  • [14]

    L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman–Lazer-type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.  Google Scholar

  • [15]

    L. Jeanjean and K. Tanaka, A remark on least energy solutions in RN, Proc. Amer. Math. Soc. 131 (2003), 2399–2408.  Google Scholar

  • [16]

    R. Lehrer and L. A. Maia, Positive solutions of asymptotically linear equations via Pohozaev manifold, J. Funct. Anal. 266 (2014), 213–246.  CrossrefWeb of ScienceGoogle Scholar

  • [17]

    R. Lehrer, L. A. Maia and R. Ruviaro, Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential, NoDEA Nonlinear Differential Equations Appl. 22 (2015), 651–672.  CrossrefGoogle Scholar

  • [18]

    R. Lehrer, L. A. Maia and M. Squassina, Asymptotically linear fractional Schrödinger equations, Complex Var. Elliptic Equ. 60 (2015), 529–558.  CrossrefGoogle Scholar

  • [19]

    P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Lineairé 1 (1984), 223–283.  CrossrefGoogle Scholar

  • [20]

    J. Q. Liu, X. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations 39 (2014), 2216–2239.  CrossrefGoogle Scholar

  • [21]

    J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations 187 (2003), 473–493.  CrossrefGoogle Scholar

  • [22]

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

  • [23]

    J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc. 131 (2003), 441–448.  CrossrefGoogle Scholar

  • [24]

    X. Q. Liu, J. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 46 (2013), 641–669.  CrossrefGoogle Scholar

  • [25]

    X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.  Google Scholar

  • [26]

    X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013), 102–124.  CrossrefGoogle Scholar

  • [27]

    A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in N, J. Differential Equations 229 (2006), 570–587.  Google Scholar

  • [28]

    M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344.  CrossrefGoogle Scholar

  • [29]

    A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal. 74 (2011), 1731–1737.  CrossrefGoogle Scholar

  • [30]

    H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl. 71 (2016), 849–858.  CrossrefGoogle Scholar

  • [31]

    E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 39 (2010), 1–33.  CrossrefGoogle Scholar

  • [32]

    E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal. 72 (2010), 2935–2949.  CrossrefGoogle Scholar

  • [33]

    M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.  Google Scholar

About the article

Received: 2016-11-10

Revised: 2017-02-13

Accepted: 2017-02-20

Published Online: 2017-05-11


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11471267

This paper was supported by the National Natural Science Foundation of China (No. 11471267).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 323–338, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0244.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Comments (0)

Please log in or register to comment.
Log in