Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity

Claudianor O. Alves 1 , Giovany M. Figueiredo 2  and Minbo Yang 3
  • 1 Departamento de Matemática, Universidade Federal de Campina Grande, CEP: 58429-900, Campina Grande - Pb, Brazil
  • 2 Departamento de Matemática, Universidade Federal do Pará, CEP: 66075-110, Belém - Pa, Brazil
  • 3 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China
Claudianor O. Alves, Giovany M. Figueiredo and Minbo Yang

Abstract

We study the following nonlinear Choquard equation:

-Δu+V(x)u=(1|x|μF(u))f(u)in N,

where 0<μ<N, N3, V is a continuous real function and F is the primitive function of f. Under some suitable assumptions on the potential V, which include the case V()=0, that is, V(x)0 as |x|+, we prove the existence of a nontrivial solution for the above equation by the penalization method.

1 Introduction and main results

In this paper, we study the existence of nontrivial solutions for the following nonlinear Choquard equation:

{-Δu+V(x)u=(1|x|μF(u))f(u)in N,uD1,2(N),u(x)>0for all xN,

where 0<μ<N, N3, Δ is the Laplacian operator, V is a nonnegative continuous real function and F is the primitive function of f.

This problem arises when one looks for standing wave solutions for a nonlinear Schrödinger equation of the form

itΨ=-ΔΨ+W(x)Ψ-(Q(x)|Ψ|q)|Ψ|q-2Ψin N.

Here W is the external potential and Q is the response function that possesses information on the mutual interaction between the bosons. This type of nonlocal equation is known to describe the propagation of electromagnetic waves in plasmas [9] and plays an important role in the theory of Bose–Einstein condensation (see [15]). It is clear that Ψ(x,t)=u(x)e-iEt solves the evolution equation (1.1) if and only if u solves

-Δu+V(x)u=(Q(x)|u|q)|u|q-2uin N,

with V(x)=W(x)-E.

When the response function is the Dirac function, i.e., Q(x)=δ(x), the nonlinear response is local and we are lead to the Schrödinger equation

-Δu+V(x)u=|u|q-2uin N,

which has been studied extensively under various hypotheses on the potentials and the nonlinearities, see [3, 2, 4, 5, 10, 13, 8, 17, 16, 19, 18, 20, 21, 25].

The aim of this paper is to study the existence of solutions for a class of nonlocal Schrödinger equation, that is, the response function Q in (SNE) is of Coulomb type, for example 1|x|μ, then we arrive at the Choquard–Pekar equation,

-Δu+V(x)u=(1|x|μ|u|q)|u|q-2uin N.

The case q=2 and μ=1 goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954 [32] and the modeling of an electron trapped in its own hole (in the work of Choquard in 1976), in a certain approximation to Hartree–Fock theory of one-component plasma [22]. The equation is also known as the Schrödinger–Newton equation, which was introduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanical wave-function. Penrose suggested that the solutions of (1.4), up to reparametrization, are the basic stationary states which do not spontaneously collapse any further, within a certain time scale.

As far as we know, most of the existing papers consider the existence and properties of the solutions for the nonlinear Choquard equation (SNE) with constant potential. In [22], Lieb proved the existence and uniqueness, up to translations, of the ground state to equation (1.4). Later, in [24], Lions showed the existence of a sequence of radially symmetric solutions to this equation. Involving the properties of the ground state solutions, Ma and Zhao [26] considered the generalized Choquard equation (1.4) for q2, and they proved that every positive solution of (1.4) is radially symmetric and monotone decreasing about some point, under the assumption that a certain set of real numbers, defined in terms of N,α and q, is nonempty. Under the same assumption, Cingolani, Clapp and Secchi [14] gave some existence and multiplicity results in the electromagnetic case, and established the regularity and asymptotic decay at infinity of the ground states of (1.4). In [27], Moroz and Van Schaftingen eliminated this restriction and showed the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters, and derived an asymptotic decay at infinity for them as well. Moreover, Moroz and Van Schaftingen in [28, 30] also considered the existence of ground states for critical growth in the sense of the Hardy–Littlewood–Sobolev inequality and under an assumption of Berestycki–Lions type.

Involving the problem with nonconstant potentials, that is,

-Δu+V(x)u=(1|x|μF(u))f(u)in N,

where V is a continuous periodic function with infNV(x)>0, noticing that the nonlocal term is invariant under translation, it is possible to prove easily an existence result by applying the mountain pass theorem, see [1], for example. For a periodic potential V that changes sign and 0 lies in the gap of the spectrum of the Schrödinger operator -Δ+V, the problem is strongly indefinite, and in [12], the authors proved the existence of nontrivial solution with μ=1 and F(u)=u2 by reduction methods. For a general class of response function Q and nonlinearity f, Ackermann [1] proposed an approach to prove the existence of infinitely many geometrically distinct weak solutions.

As far as we know, the only papers that considered the generalized Choquard equation with vanishing potential are [29, 33]. Replacing Δ by ε2Δ, Secchi [33] obtained the existence result for small ε when V>0 and lim inf|x|V(x)|x|γ>0 for some γ[0,1) by a Lyapunov–Schmidt type reduction. In [29], the authors prove the existence and concentration behavior of the semiclassical states for the problem with F(u)=up for small ε. There, they developed a penalization technique by constructing supersolutions to a linearization of the penalized problem in an outer domain, and then estimated the solutions of the penalized problem by a comparison principle. Besides these two papers, in our mind, for constant ε=1 and general nonlinearities, the existence of nontrivial solution for the Choquard equation with vanishing potential is still not known.

Motivated by the above facts and a recent paper due to Alves and Souto [2], in the present paper we intend to study the Choquard equation (SNE) with potentials vanishing at infinity, that is, V()=0. This class of problems goes back to the work by Berestycki and Lions in [8], where the authors studied the Schrödinger equation with zero mass and showed that the problem

{-Δu=f(u)in N,uD1,2(N),

has no ground state solution if f(t)=|t|p-2t. However, they also proved that if f behaves like |t|q-2t for small t, and like |t|p-2t for large t when p<2*<q, then the problem has a ground state solution. For example, these conditions are verified by the function

f(t)={|t|q-2t,t1,h(t),1t2,|t|p-2t,t2,

where h is selected so that f is a C1() function. Later Benci, Grisanti and Micheletti [7, 6] considered problems of the type

{-Δu+V(x)u=f(u)in N,uD1,2(N),

and studied the existence and nonexistence of ground state solutions.

Motivated by the above papers and their hypotheses, we assume the following conditions on the nonlinearity f:

lims0+sf(s)sq<+,

q2*=2NN-2. Assume that 0<μ<N+22 and

lims+sf(s)sp=0,

for some p(1,2(N-μ)N-2). Note that interval (1,2(N-μ)N-2) is not empty because of the above condition on μ. Note that by (f1), and since 2(N-μ)N-2<2N-μN-2<2*, we have that

lims0sf(s)s2N-μN-2=0andlims0sf(s)sp=0.

Thus, by (f1)–(f2) and (1.6), there exists c0>0 such that

|sf(s)|c0|s|2*,|sf(s)|c0|s|q,|sf(s)|c0|s|2N-μN-2and|sf(s)|c0|s|pfor all s.

The nonlinearity f is also supposed to verify the Ambrosetti–Rabinowitz type superlinear condition for the nonlocal problem, that is, there exists θ>2 such that

0<θF(s)2f(s)sfor all s>0,

where F(t)=0tf(s)𝑑s. Without loss of generality, we will assume that θ<4 in the rest of the paper.

Related to the potential V(x), we assume that it is a nonnegative continuous function, and we fix the following notations:

m=max|x|1V(x),

and we define the function 𝒱:(1,+)[0,) by

𝒱(R)=1R(q-2)(N-2)inf|x|R|x|(q-2)(N-2)V(x).

One of the main results of this paper is the following existence result.

Assume that 0<μ<N+22 and also that conditions (f1)–(f3) hold. There exists a constant V0=V0(m,θ,p,μ,c0) such that if V(R)>V0 for some R>1, then problem (SNE) has a positive solution.

The constant 𝒱0 is a technical constant, which will appear in the proof of the main result, see Section 3. Next, we will introduce an example of potential V for Theorem 1.1.

Let

V(x)={ϕ(x)if |x|2,𝒱02(q-2)(N-2)+1|x|-(q-2)(N-2)if |x|2,

where ϕ:B¯2(0)[0,+) is a continuous, which makes V continuous. Then, we know that

𝒱(2)=2𝒱0,

and so

𝒱(2)>𝒱0.

If the potential V is a radial function, that is,

V(x)=V(|x|)for all xN,

then we define the function 𝒲:(1,+)[0,) by

𝒲(R)=inf|x|R|x|4-μ2V(x),

and we can apply the arguments in Theorem 1.1 to treat the homogeneous case with nonlinearity

f(t)=|t|4-μN-2tfor all t,

where 0<μ<min{N+22,4}.

Involving this situation, we have the following existence result.

Assume that 0<μ<min{N+22,4}, V is a radial function and condition (f4) holds. There exists a constant W0=W0(m,μ) such that if W(R)>W0 for some R>1, then problem (SNE) has a positive solution.

As an application, we can use Theorem 1.3 to study

{-Δu+V(x)u=(1|x|u5)u4in 3,u(x)>0in 3.

where V is a potential as below.

Let δ>0 and

V(x)={1if |x|1,|x|-(4-μ)2+δif |x|1.

We have that

𝒲(R)=Rδfor all R>1.

Hence,

limR+𝒲(R)=+,

and so for R large enough, 𝒲(R)>𝒲0.

Let

V(x)={2𝒲0if |x|1,2𝒲0|x|-(4-μ)2if |x|1,

where ϕ:B¯1(0)[0,+) is a continuous, which makes V continuous. For the above potential, we have that

𝒲(R)=2𝒲0for all R>1,

and so 𝒲(R)>𝒲0 for all R>1.

We now briefly outline the organization of the contents of this paper. In Section 2, we introduce an auxiliary problem, by using a penalization method due to del Pino and Felmer [16]. Section 3 is devoted to the proof of Theorem 1.1. Finally, in Section 4, we consider the case where the potential V is a radial function and the nonlinearity is homogeneous. By adjusting some arguments in the proof of Theorem 1.1, we can prove the result in Theorem 1.3.

Notations.

We fix the following notations, which will be used from now on:

  1. C, Ci denote positive constants.
  2. BR denote the open ball centered at the origin with radius R>0.
  3. Ls(N), 1s, denotes the Lebesgue space with the norm
    |u|s:=(N|u|s)1s.
  4. If u is a mensurable function, we denote by u+ and u- its positive and negative part respectively, i.e.,
    u+(x)=max{u(x),0}andu-(x)=max{-u(x),0}.
  5. C0(N) denotes the space of infinitely differentiable functions with compact support in N.
  6. We denote by D1,2(N) the Hilbert space
    D1,2(N)={uL2*(N):uL2()}
    endowed with the inner product
    u,v=uv
    and the norm
    uD1,2:=(N|u|2)12.
  7. S is the best constant that verifies
    |u|2*2SN|u|2for all uD1,2(N).
  8. From the assumptions on V, it follows that the subspace
    E={uD1,2(N):NV(x)|u|2<}
    is a Hilbert space with its norm defined by
    u:=(N(|u|2+V(x)|u|2))12.
  9. Let E be a real Hilbert space and I:E be a functional of class 𝒞1. We say that (un)E is a Palais–Smale sequence at level c for I, (PS)c sequence for short, if (un) satisfies
    I(un)candI(un)0as n.
    Furthermore, we say that I satisfies the (PS)c condition, if any (PS)c sequence possesses a convergent subsequence.

2 The penalized problem

In this section, adapting some arguments introduced by del Pino and Felmer in [16], we are able to obtain the existence of nontrivial solution for (SNE) by studying an auxiliary problem.

We would like to make some comments on the assumptions involving the nonlinearity f. The first point is that since we are looking for positive solutions, we may suppose that

f(s)=0for all s<0.

The second one is that due to the application of variational methods, we must have

|N(1|x|μF(u))F(u)|<for all uE.

To observe the above integrability, it is important to recall the Hardy–Littlewood–Sobolev inequality, which will be frequently used in the paper.

(The Hardy–Littlewood–Sobolev inequality [23, Theorem 4.3])

Let s,r>1 and 0<μ<N with 1s+μN+1r=2. If gLs(RN) and hLr(RN), then there exists a sharp constant C(s,N,μ,r), independent of g,h, such that

NNg(x)h(y)|x-y|μC(s,N,μ,r)|g|s|h|r.

The above inequality guarantees that (2.1) holds, because by (f3) and (1.7),

|F(u)|2c0|u|2N-μN-2for all uE.

Thereby, by the Hardy–Littlewood–Sobolev inequality,

N(1|x|μ|F(u)|)|F(u)|<+

if F(u)Lt(N) for t>1 and

2t+μN=2,

that is,

t=2N2N-μ.

However, by (2.2),

|F(u)|t(2c0)tN|u|2*<for all uE,

shows that F(u)Lt(N) for all uE.

From the above commentaries, the Euler–Lagrange functional I:E, associated to (SNE) and given by

I(u)=12u2-12N(1|x|μF(u))F(u),

is well defined and belongs to 𝒞1(E,) with its derivative given by

I(u)φ=N(uφ+V(x)uφ)-N(1|x|μF(u))f(u)φfor all u,φE.

Thus, it is easy to see that the solutions of (SNE) correspond to critical points of the energy functional I. However, because I does not verify in general the (PS) condition, there are some difficulties to prove the existence of nontrivial critical points for it.

In order to overcome the lack of compactness of I, we adapt the penalization method introduced by del Pino and Felmer in [16]. For >1 and R>1 to be determined later, we set the functions

f^(x,s):={f(s)if f(s)V(x)s,V(x)sif f(s)>V(x)s
g(x,s):=𝒳BR(x)f(s)+(1-𝒳BR(x))f^(x,s),

where 𝒳BR denotes the characteristic function of the ball BR. Using the previous notations, let us introduce the auxiliary problem

{-Δu+V(x)u=(1|x|μG(x,u))g(x,u)in N,uD1,2(N),

where G(x,s):=0sg(x,τ)𝑑τ. A direct computation shows that, for all s, the following hold:

f^(x,s)f(s)for all xN,
g(x,s)V(x)sfor all |x|R,
G(x,s)=F(s)for all |x|R,
G(x,s)V(x)2s2for all |x|R.

It is easy to check that if u is a positive solution of the equation (APE) with f(u(x))V(x)u(x) for all |x|R, then g(x,u)=f(u), and therefore, u is indeed a solution of problem (SNE).

The Euler–Lagrange functional associated to (APE) is given by

Φ(u)=12u2-Ψ(u),

where

Ψ(u)=12N(1|x|μG(x,u))G(x,u).

From (1.7) and (2.4), we know that Φ is well defined and belongs to 𝒞1(E,) with its derivative given by

Φ(u)φ=N(uφ+V(x)uφ)-Ψ(u)φfor all u,φE.

Therefore, it is easy to see that the solutions of (APE) correspond to the critical points of the energy functional Φ.

Assume that condition (f3) holds. Then,

Ψ(u)uθΨ(u)>0for all uE{0}.

Proof.

For all uE{0}, a direct computation gives

1θΨ(u)u-Ψ(u)=|x|R(1|x|μG(x,u))(1θf(u)u-12F(u))+|x|>R(1|x|μG(x,u))(1θg(x,u)u-12G(x,u))
{|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θg(x,u)u-12G(x,u))
+{|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θg(x,u)u-12G(x,u))
={|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θf(u)u-12F(u))
+{|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θg(x,u)u-12G(x,u))
{|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θg(x,u)u-12G(x,u))
{|x|>R}{f(u(x))V(x)u(x)}(1|x|μG(x,u))(1θ-14)V(x)u2.

Since 2<θ<4, the conclusion follows. ∎

Next, we check that Φ verifies the mountain pass geometry.

Assume that 0<μ<N and conditions (f1)–(f3) hold. Then, the following hold:

  1. (1)There exist ρ,δ0>0 such that Φ|Sρδ0>0 for all uSρ={uE:u=ρ}.
  2. (2)There exist r>0 and eH01(B1) with e>r such that Φ(e)<0.

Proof.

(1) By (1.7),

|G(x,u)||F(u)|c0|u|2N-μN-2,

from which it follows that

Φ(u)12u2-C1N(1|x|μ|u|2N-μN-2)|u|2N-μN-2,

and so, by the Hardy–Littlewood–Sobolev inequality,

Φ(u)12u2-C1u2(2N-μ)N-2.

Since (2N-μ)N-2>1, assertion (1) follows if we choose ρ small enough.

(2) Fixing u0H01(BR0) with u0+(x)=max{u0(x),0}0, we set

𝒜(t)=Ψ(tu0u0)>0for t>0.

By direct calculus,

𝒜(t)𝒜(t)θtfor all t>0.

Hence, integrating the above inequality over [1,su0] with s>1u0, we find

Ψ(su0)Ψ(u0u0)u0θsθ.

Therefore,

Φ(su0)C1s2-C2sθfor s>1u0,

and assertion (2) follows for e=su0 with s large enough. ∎

Applying the mountain pass theorem without the (PS) condition [35], we know that there exists a (PS)cV sequence (un)E such that

Φ(un)0andΦ(un)cV,

where cV is the mountain pass level characterized by

0<cV:=infγΓmaxt[0,1]Φ(γ(t))infuE{0}maxt0Φ(tu)

with

Γ:={γ𝒞1([0,1],E):γ(0)=0 and Φ(γ(1))<0}.

Hence, from the proof of Lemma 2.4,

0<cVd:=infuH01(B1){0}maxt0Φ~(tu),

where

Φ~(u)=12B1(|u|2+mu2)-12B1B1F(u(x))F(u(y))|x-y|μ

and m is given in (1.8). Here, it is important to observe that d is independent of the choice of and R.

Assume that 0<μ<N and conditions (f1)–(f3) hold. Then, the (PS)cV sequence (un) is bounded by a constant independent of the choice of and R.

Proof.

By Lemma 2.3,

Φ(un)-1θΦ(un)un(12-1θ)un2,

which means (un) is bounded in E. Moreover, we may assume that

un22θθ-2(cV+1)2θθ-2(d+1)for all n,

which shows the lemma, because d is independent of the choice of and R. ∎

Before proving the next lemma, we need to fix some notations. In what follows,

:={uE:u22θθ-2(d+1)}

and

K(u)(x):=1|x|μG(x,u).

With the above notations, we are able to show the ensuing estimate.

Assume that 0<μ<N+22 and conditions (f1)–(f3) holds. Then, there exists 0>0, which is independent of R, such that

supu|K(u)(x)|L(N)0<12.

Proof.

From the definition of G,

|G(x,u)||F(u)|,

and so

|G(x,u)|c0|s|2*and|G(x,u)|c0|s|p.

Thereby,

|K(u)(x)|=|NG(x,u)|x-y|μdy|
||x-y|1G(x,u)|x-y|μdy|+||x-y|1G(x,u)|x-y|μdy|
c0|x-y|1|u|p|x-y|μdy+c0|x-y|1|u|2*dy
c0|x-y|1|u|p|x-y|μ𝑑y+C1.

Choosing t=2*p>NN-μ , it follows from Hölder’s inequality that

|x-y|1|u|p|x-y|μdy(|x-y|1|u|2*dy)1t(|x-y|11|x-y|tμt-1)t-1tC1(|r|1|r|N-1-tμt-1dr)t-1t.

Once N-1-tμt-1>-1, there exists C2>0 such that

|x-y|1|u|p|x-y|μC2for all xN.

From this, there exists 0>0 verifying

supu|K(u)(x)|L(N)012.

From now on, we take 0 and consider the penalized problem with the nonlinearity defined in (2.3).

Assume that 0<μ<N+22 and conditions (f1)–(f3) hold. Then, the (PS)cV sequence (un) satisfies the following property: For each ε>0, there exists r=r(ε)>R verifying

lim supnNB2r(|un|2+V(x)|un|2)<ε.

Proof.

From Lemma 2.5,

un22θθ-2(d+1)for all n,

where d is independent of the choice of and R. Therefore, we can assume that there exists uE such that unu in E. Thus, for each ε>0, there exists r>R>0 such that

ωN1Nun(r|x|2r|u|2*)12*<ε8,

where ωN is the volume of the unitary ball in N.

Let ηrC(Brc) be such that ηr(x)=1 if xB2r(0) with 0ηr(x)1 and |ηr(x)|2r. Note that

Nηr(|un|2+V(x)|un|2)=Φ(un)(unηr)+N(1|x|μG(x,un))g(x,un)unηr-Nununηr.

Since (unηr) is bounded in E, it follows that Φ(un)(unηr)=on(1). Moreover, recalling that ηr(x)=0 in BR, we obtain

|x|rηr(|un|2+V(x)|un|2)|x|rsupn|K(un)(x)|L(N)0ηrV(x)|un|2-2rr|x|2runun+on(1).

From Lemma 2.6, we have

supn|K(un)(x)|L(N)012,

and so

|x|2r(|un|2+V(x)|un|2)4rr|x|2r|un||un|+on(1).

By Hölder’s inequality,

r|x|2r|un||un||un|L2(N)(r|x|2r|un|2)12.

Since unu in L2(r|x|2r) and (un) is bounded in E, it follows that

lim supnr|x|2r|un||un|un(r|x|2r|u|2)12.

Using Hölder’s inequality again, we get

lim supnr|x|2r|un||un|un(r|x|2r|u|2*)12*|B2r|1N.

Once |B2r|=ωN2NrN, by (2.7) and (2.8),

lim supn|x|2r(|un|2+V(x)|un|2)8ωN1Nun(r|x|2r|u|2*)12*<ε.

Assume that 0<μ<N+22 and conditions (f1)–(f3) hold. Then, Φ satisfies the (PS)cV condition.

Proof.

Since unu in E, Φ(un)un=on(1) and Φ(un)u=on(1), it follows that

un-u2=NK(un)g(x,un)(un-u)+on(1).

Now, our goal is to show that the following limit holds:

NK(un)g(x,un)(un-u)=on(1).

We begin with recalling that by Lemma 2.6, there exists C>0 such that

|K(un)|Cfor all n.

Recall (1.7) and notice that |g(x,s)s||s|p with p<2* for each r>0. Then, the Sobolev compact embedding theorem implies

|BrK(un)g(x,un)(un-u)|CBr|g(x,un)(un-u)|0.

From (1.7), it also holds that |g(x,s)s||s|2*, and so

NBrK(un)|g(x,un)un|CNBr|un|2*.

Combining Lemma 2.7 with the Sobolev embedding theorem we have that given ε>0, there exists r(ε)>0 such that

lim supnNBrK(un)|g(x,un)un|C1ε.

Similarly, applying Hölder’s inequality, we can also prove that

lim supnNBrK(un)|g(x,un)u|C2ε.

In conclusion,

NK(un)g(x,un)(un-u)0.

Applying Lemmas 2.4 and 2.8, we have the following result.

Assume that 0<μ<N+22 and conditions (f1)–(f3) hold. Then, problem (APE) has a positive solution with

u22θdθ-2.

In the following, we will study the L estimate of the solution u. To this end, we adapt some techniques used in [2, 11].

Let u be the solution obtained in Theorem 2.9. Then, there exists a constant M0, which depends only on N,μ,θ,m,p,c0, such that

|u|M0.

Proof.

By hypothesis, u is a solution of

-Δu+V(x)u=K(u)f(u)in N,

where K(u)L(N) and |K(u)|12. Since V(x)0 for all xN and

|f(t)|c0|t|2*-1for all t,

it follows that

-Δu12a(x)(1+|u|)in N,

where a(x)=|u|4N-2LN2(N). By a Trudinger–Brézis–Kato iteration argument, see Struwe’s book [34, Lemma B3, p. 273], we deduce that uLs(N) for all s>1. Moreover, once |a|N2 does not depend on R>0, the norms |u|s also do not depend on R>1. Now, fixing s large enough, the bootstrap arguments implies that there exists M0>0, which is independent of R, such that |u|M0. ∎

3 Proof of Theorem 1.1

From Lemma 2.9, (APE) has a positive solution uRE for each R>1. Thereby, in order to prove the existence of solution for problem (SNE), we must show that there exists R>1 such that uR satisfies the inequality

f(uR)V(x)0uRfor |x|R.

Let v be the C(N{0}) harmonic function

v(x)=RN-2|u||x|N-2.

By Lemma 2.10, we have the inequality

uvon BR,

which implies that the function

w={(u-v)+if |x|R,0if |x|R

belongs to D1,2(N). Since Δv=0 in NBR(0), w=0 on BR and w0, it follows that

N|w|2=N(u-v)w=|x|R(K(u)(x)g(x,u)w-V(x)uw)
|x|R(10K(u)(x)-1)V(x)uw0,

showing that w0, i.e., uv in |x|R, and thus

u(x)RN-2|u||x|N-2RN-2M0|x|N-2for all |x|R.

Using the inequality |f(s)|c0|s|q-1, we have

f(u)uc0|u|q-2c0M0q-2R(q-2)(N-2)|x|(q-2)(N-2)for all |x|R.

Now, fix R>1 such that 𝒱(R)>0. Then, the last inequality combined with definition (1.9) gives

f(u)uc0|u|q-2c00M0q-2V(x)0𝒱(R)for all |x|R.

Thus, setting the number

𝒱0=c00M0q-2,

and if there is R>1 such that

𝒱(R)>𝒱0,

then we obtain the desired result for the R>1 above fixed, that is,

f(uR)V(x)0uRfor |x|R.

4 Proof of Theorem 1.3

In the proof of Theorem 1.3, we replace the space D1,2(N) by Drad1,2(N) and consider

Erad={uDrad1,2(N):NV(x)|u|2<}.

In this case, due to the Hardy–Littlewood–Sobolev inequality, the energy function

I(u)=12u2-(N-2)22(2N-μ)2N(1|x|μ|u|2N-μN-2)|u|2N-μN-2

is well defined and belongs to 𝒞1(E,). Now, repeating the same ideas of the previous sections, we can consider again problem (APE). Here, the reader is invited to check that the functional Φ still verifies, with natural modifications, Lemmas 2.3, 2.4 and 2.5.

Proof of Theorem 1.3.

From Lemmas 2.3, 2.4, 2.5, and Theorem 2.9, (APE) has a positive solution uRErad. Thereby, in order to prove that it is indeed a solution of problem (SNE), we must show that there exists R>1 such that uR satisfies the inequality

f(uR(x))V(x)0uR(x)for |x|R.

We begin by recalling that

|uR(x)|CuRD1,2|x|N-22for all xN{0}

and

uR22θdθ-2,

from where it follows that

uRD1,2A,

where A=2θdθ-2. Then,

uR(x)CA|x|N-22for |x|R.

Since f(t)=|t|4-μN-2t, we have

f(uR)uR=|uR|4-μN-2(CA)4-μN-2|x|4-μ2for |x|R.

Using the definition of 𝒲(R), it follows that

f(uR)uR=|uR|4-μN-20(CA)4-μN-2V(x)0𝒲(R)for |x|R.

Fixing

𝒲0=0(CA)4-μN-2>0

and assuming that there exists R>1 such that

𝒲(R)>𝒲0,

then for the above R>1, we can ensure that

f(uR)uRV(x)0for |x|R,

implying that I(uR)=0 in Erad. Now, using the principle of symmetric criticality due to Palais [31], we can conclude that I(uR)=0 in E, thus finishing the proof. ∎

References

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    Alves C. O., Souto M. and Montenegro M., Existence of solution for two classes of elliptic problems in N ${\mathbb{R}^{N}}$ with zero mass, J. Differential Equations 252 (2012), 5735–5750.

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    Ambrosetti A., Felli V. and Malchiodi A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), 117–144.

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    Benci V., Grisanti C. R. and Micheletti A. M., Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with V ( ) = 0 ${V(\infty)=0}$, Topol. Methods Nonlinear Anal. 26 (2005), 203–219.

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    Benci V., Grisanti C. R. and Micheletti A. M., Existence of solutions for the nonlinear Schrödinger equation with V ( ) = 0 ${V(\infty)=0}$, Progr. Nonlinear Differential Equations Appl. 66 (2005), 53–65.

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    Bergé L. and Couairon A., Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas 7 (2000), 210–230.

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    Bonheure D. and Van Schaftingen J., Ground states for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010), 273–301.

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    Brezis H. and Kato T., Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl. (4) 58 (1979), 137–151.

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    Buffoni B., Jeanjean L. and Stuart C. A., Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), 179–186.

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    Byeon J. and Wang Z. Q., Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations 18 (2003), 207–219.

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    Cingolani S., Clapp M. and Secchi S., Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248.

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    Dalfovo F., Giorgini S., Pitaevskii L. P. and Stringari S., Theory of Bose–Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), 463–512.

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    Ding Y. and Szulkin A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (2007), 397–419.

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    Ding Y. and Tanaka K., Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), 109–135.

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    Floer A. and Weinstein A., Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.

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    Jeanjean L. and Tanaka K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations 21 (2004), 287–318.

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    Lieb E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105.

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    • Export Citation
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    Lieb E. and Loss M., Analysis, Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001.

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    Lions P. L., The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072.

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    • Export Citation
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    Liu C., Wang Z. and Zhou H., Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations 245 (2008), 201–222.

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    • Export Citation
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    Ma L. and Zhao L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467.

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    Moroz V. and Van Schaftingen J., Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184.

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    • Export Citation
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    Moroz V. and Van Schaftingen J., Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579.

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    Moroz V. and Van Schaftingen J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), 199–235.

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    • Export Citation
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    Moroz V. and Van Schaftingen J., Groundstates of nonlinear Choquard equation: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), Article ID 1550005.

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    Palais R. S., The Principle of Symmetric Criticality, Comm. Math. Phys. 69 (1979), 19–30.

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    • Export Citation
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    Pekar S., Untersuchungüber die Elektronentheorie der Kristalle, Akademie, Berlin, 1954.

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    Secchi S., A note on Schrödinger–Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), 3842–3856.

    • Crossref
    • Export Citation
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    Struwe M., Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed., Springer, Berlin, 2007.

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    Willem M., Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Ackermann N., On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), 423–443.

  • [2]

    Alves C. O. and Souto M., Existence of solutions for a class of elliptic equations in N ${\mathbb{R}^{N}}$ with vanishing potentials, J. Differential Equations 254 (2013), 1977–1991.

    • Crossref
    • Export Citation
  • [3]

    Alves C. O., Souto M. and Montenegro M., Existence of solution for two classes of elliptic problems in N ${\mathbb{R}^{N}}$ with zero mass, J. Differential Equations 252 (2012), 5735–5750.

    • Crossref
    • Export Citation
  • [4]

    Ambrosetti A., Felli V. and Malchiodi A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), 117–144.

  • [5]

    Bartsch T., Pankov A. and Wang Z. Q., Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), 549–569.

    • Crossref
    • Export Citation
  • [6]

    Benci V., Grisanti C. R. and Micheletti A. M., Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with V ( ) = 0 ${V(\infty)=0}$, Topol. Methods Nonlinear Anal. 26 (2005), 203–219.

    • Crossref
    • Export Citation
  • [7]

    Benci V., Grisanti C. R. and Micheletti A. M., Existence of solutions for the nonlinear Schrödinger equation with V ( ) = 0 ${V(\infty)=0}$, Progr. Nonlinear Differential Equations Appl. 66 (2005), 53–65.

    • Crossref
    • Export Citation
  • [8]

    Berestycki H. and Lions P. L., Nonlinear scalar field equations, I Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.

    • Crossref
    • Export Citation
  • [9]

    Bergé L. and Couairon A., Nonlinear propagation of self-guided ultra-short pulses in ionized gases, Phys. Plasmas 7 (2000), 210–230.

    • Crossref
    • Export Citation
  • [10]

    Bonheure D. and Van Schaftingen J., Ground states for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010), 273–301.

    • Crossref
    • Export Citation
  • [11]

    Brezis H. and Kato T., Remarks on the Schrödinger operator with regular complex potentials, J. Math. Pures Appl. (4) 58 (1979), 137–151.

  • [12]

    Buffoni B., Jeanjean L. and Stuart C. A., Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993), 179–186.

    • Crossref
    • Export Citation
  • [13]

    Byeon J. and Wang Z. Q., Standing waves with a critical frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differential Equations 18 (2003), 207–219.

    • Crossref
    • Export Citation
  • [14]

    Cingolani S., Clapp M. and Secchi S., Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys. 63 (2012), 233–248.

    • Crossref
    • Export Citation
  • [15]

    Dalfovo F., Giorgini S., Pitaevskii L. P. and Stringari S., Theory of Bose–Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), 463–512.

    • Crossref
    • Export Citation
  • [16]

    del Pino M. and Felmer P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.

    • Crossref
    • Export Citation
  • [17]

    del Pino M. and Felmer P., Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127–149.

    • Crossref
    • Export Citation
  • [18]

    Ding Y. and Szulkin A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (2007), 397–419.

    • Crossref
    • Export Citation
  • [19]

    Ding Y. and Tanaka K., Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), 109–135.

    • Crossref
    • Export Citation
  • [20]

    Floer A. and Weinstein A., Nonspreading wave pachets for the packets for the cubic Schrödinger with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.

    • Crossref
    • Export Citation
  • [21]

    Jeanjean L. and Tanaka K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations 21 (2004), 287–318.

  • [22]

    Lieb E. H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105.

    • Crossref
    • Export Citation
  • [23]

    Lieb E. and Loss M., Analysis, Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001.

  • [24]

    Lions P. L., The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072.

    • Crossref
    • Export Citation
  • [25]

    Liu C., Wang Z. and Zhou H., Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations 245 (2008), 201–222.

    • Crossref
    • Export Citation
  • [26]

    Ma L. and Zhao L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal. 195 (2010), 455–467.

    • Crossref
    • Export Citation
  • [27]

    Moroz V. and Van Schaftingen J., Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153–184.

    • Crossref
    • Export Citation
  • [28]

    Moroz V. and Van Schaftingen J., Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579.

  • [29]

    Moroz V. and Van Schaftingen J., Semi-classical states for the Choquard equation, Calc. Var. Partial Differential Equations 52 (2015), 199–235.

    • Crossref
    • Export Citation
  • [30]

    Moroz V. and Van Schaftingen J., Groundstates of nonlinear Choquard equation: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), Article ID 1550005.

  • [31]

    Palais R. S., The Principle of Symmetric Criticality, Comm. Math. Phys. 69 (1979), 19–30.

    • Crossref
    • Export Citation
  • [32]

    Pekar S., Untersuchungüber die Elektronentheorie der Kristalle, Akademie, Berlin, 1954.

  • [33]

    Secchi S., A note on Schrödinger–Newton systems with decaying electric potential, Nonlinear Anal. 72 (2010), 3842–3856.

    • Crossref
    • Export Citation
  • [34]

    Struwe M., Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed., Springer, Berlin, 2007.

  • [35]

    Willem M., Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.

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