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Advances in Nonlinear Analysis

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


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Nonlocal perturbations of the fractional Choquard equation

Gurpreet Singh
  • Corresponding author
  • School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
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Published Online: 2017-08-05 | DOI: https://doi.org/10.1515/anona-2017-0126

Abstract

We study the equation

(-Δ)su+V(x)u=(Iα*|u|p)|u|p-2u+λ(Iβ*|u|q)|u|q-2uin N,

where Iγ(x)=|x|-γ for any γ(0,N), p,q>0, α,β(0,N), N3, and λ. First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

Keywords: Choquard equation; fractional operator; nonlocal perturbation; groundstate solution, sign-changing solutions

MSC 2010: 35R11; 35J60; 35Q40; 35A15

1 Introduction

In this paper, we are concerned with the equation

(-Δ)su+V(x)u=(Iα*|u|p)|u|p-2u+λ(Iβ*|u|q)|u|q-2uin N,(1.1)

where p,q>0, α,β(0,N), N3, and λ. Here Iγ stands for the Riesz potential of order γ defined as Iγ=|x|γ-N for any γ(0,N).

The operator (-Δ)s is the fractional Laplace operator of order s(0,1), and is defined as follows (see [6, 15]):

(-Δ)su=C(N,s)P.V.Nu(x)-u(y)|x-y|N+2s𝑑y,

where P.V. stands for the principal value of the integral and C(N,s)>0 is a normalizing constant. The operator (-Δ)s is referred to as the infinitesimal generator of the Levy stable diffusion process. The function VC(N) is required to satisfy one (or both) of the following conditions:

  • (V1)

    infNV(x)V0>0.

  • (V2)

    For all M>0, the set {xN:V(x)M} has finite Lebesgue measure.

Note that condition (V2) is weaker than lim|x|V(x)=, as for instance V(x)=|x|4sin2|x| satisfies (V2) but has no limit as |x|.

In the last few decades, problems involving the fractional Laplacian and nonlocal operators have received considerable attention. These kinds of problems arise in various applications such as continuum mechanics, phase transitions, population dynamics, optimization, finance, and many others.

The prototype model of (1.1) is the fractional Choquard equation

(-Δ)su+V(x)u=(Iα*|u|p)|u|p-2uin N,(1.2)

studied by d’Avenia, Siciliano and Squassina in [5] in the case where V is a positive constant. They obtained the existence of groundstate and radially symmetric solutions with diverging norm and diverging energy levels.

The case of the standard Laplace operator in (1.2) has a long history in the literature. For s=1, V1 and p=α=2, equation (1.2) becomes the well-known Choquard or nonlinear Schrödinger–Newton equation

-Δu+u=(I2*u2)uin N.(1.3)

Equation (1.3) for N=3 was first introduced by Pekar [20] in 1954 in quantum mechanics. In 1996, Penrose [21, 22] used equation (1.3) in a different context as a model in self-gravitating matter (see also [11, 16]). Since then, the Choquard equation has been investigated in various settings and in many contexts (see, e.g., [1, 10, 14, 19]). For a most up to date reference on the study of the Choquard equation in a standard Laplace setting, the reader may consult [18].

For s=12, V1, p=α=2, N=3, and λ=0, equation (1.1) becomes

(-Δ)12u+u=(I2*u2)uin 3,

and has been used to study the dynamics of pseudo-relativistic boson stars and their dynamical evolution (see [7, 8, 9, 12]).

In this paper, we shall be interested in the study of groundstate solutions and least energy sign-changing solutions to (1.1). To this aim, we denote by 𝒟2,s(N) the completion of Cc(N) with respect to the Gagliardo seminorm

[u]s,2=[NN|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y]12.

Also, Hs(N) denotes the standard fractional Sobolev space defined as the set of u𝒟2,s(N) satisfying uL2(N) with the norm

uHs=[N|u|2+[u]s,22]12.

Let us define the functional space

XVs(N)={u𝒟2,s(N):NV(x)u2<}

endowed with the norm

uXVs=[NN|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y+NV(x)u2]12.

Throughout this paper, we shall assume that p and q satisfy

N+αN<p<N+αN-2s(1.4)

and

N+βN<q<N+βN-2s.(1.5)

It is not difficult to see that (1.1) has a variational structure. Indeed, any solution of (1.1) is a critical point of the energy functional λ:XVs(N) defined by

λ(u)=12uXVs2-12pN(Iα*|u|p)|u|p-λ2qN(Iβ*|u|q)|u|q.

A crucial tool to our approach is the Hardy–Littlewood–Sobolev inequality

|N(Iγ*u)v|Curvt(1.6)

for γ(0,N), uLr(N) and vLt(N) such that

1r+1t=1+γN.

Using (1.4) and (1.5) together with the Hardy–Littlewood–Sobolev inequality (1.6), the energy functional λ is well defined, and moreover λC1(XVs).

We shall first be concerned with the existence of ground state solutions for equation (1.1) under the assumption that V satisfies (V1). This will be achieved by a minimization method on the Nehari manifold associated with λ, which is defined as

𝒩λ={uXVs(N){0}:λ(u),u=0}.

The groundstate solutions will be obtained as minimizers of

mλ=infu𝒩λλ(u).

Our main result in this sense is stated below.

Theorem 1.1.

Assume p>q>1, λ>0, p, q satisfy (1.4)–(1.5), and V satisfies (V1). Then equation (1.1) has a ground state solution uXVs(RN).

Our approach relies on the analysis of the Palais–Smale sequences for λ|𝒩λ. Using an idea from [3, 4], we show that any Palais–Smale sequence of λ|𝒩λ is either converging strongly to its weak limit or differs from it by a finite number of sequences which further are the translated solutions of (1.2). The novelty of our approach is that we shall rely on several nonlocal Brezis–Lieb results as we present in Section 2.

We now turn to the study of least energy sign-changing solutions of (1.1). In this setting, we require V to fulfill both conditions (V1) and (V2). By the result in [25, Lemma 2.1] (see also [23, 24]), the embedding XVs(N)Lq(N) is compact for q[2,2s*), where 2s*=2NN-2s.

Our approach in the study of least energy sign-changing solutions of (1.1) is based on the minimization method on the Nehari nodal set defined as

λ={uXVs(N):u±0 and λ(u),u±=0}.

The solutions will be obtained as minimizers for

cλ=infuλλ(u).

In this situation, the problem is more delicate as some of the usual properties of the local nonlinear functional do not work. For instance, since

λ(u),u±=u±XVs2-N(Iα*(u±)p)(u±)p-λN(Iβ*(u±)q)(u±)q-NNu±(x)u(y)+u(x)u±(y)|x-y|N+2s𝑑x𝑑y-N(Iα*(u±)p)(u)p-λN(Iβ*(u±)q)(u)q,

we have in general that

λ(u)λ(u+)+λ(u-)and λ(u),u±λ(u±),u±.

Therefore, the standard local methods used to investigate the existence of sign-changing solutions do not apply immediately to our nonlocal setting.

Our second main result in this regard is stated below.

Theorem 1.2.

Assume λR, (N-4s)+<α,β<N, p>q>2 satisfy (1.4) and (1.5), and V satisfies (V1) and (V2). Then equation (1.1) has a least-energy sign-changing solution uXVs(RN).

The remaining of the paper is organized as follows: In Section 2, we collect some nonlocal versions of the Brezis–Lieb lemma, which will be crucial in investigating the groundstate solutions of (1.1). Further, Sections 3 and 4 contain the proofs of our main results.

2 Preliminary results

Lemma 2.1 ([13, Lemma 1.1], [17, Lemma 2.3]).

Let r[2,2s*]. There exists a constant C>0 such that for any uXVs(RN) we have

N|u|rCu(supyNB1(y)|u|r)1-2r.

Lemma 2.2 ([2, Proposition 4.7.12]).

Let r(1,). Assume (wn) is a bounded sequence in Lr(RN) that converges to w almost everywhere. Then wnw weakly in Lr(RN).

Lemma 2.3 (Local Brezis–Lieb lemma).

Let r(1,). Assume (wn) is a bounded sequence in Lr(RN) that converges to w almost everywhere. Then for every q[1,r] we have

limnN||wn|q-|wn-w|q-|w|q|rq=0

and

limnN||wn|q-1wn-|wn-w|q-1(wn-w)-|w|q-1w|rq=0.

Proof.

Fix ε>0. Then there exists C(ε)>0 such that for all a,b we have

||a+b|q-|a|q|rqε|a|r+C(ε)|b|r.(2.1)

Using (2.1), we obtain

|fn,ε|=(||wn|q-|wn-w|q-|wq||rq-ε|wn-w|r)+(1+C(ε))|w|r.

Now using the Lebesgue dominated convergence theorem, we have

Nfn,ε0as n.

Therefore, we get

||wn|q-|wn-w|q-|w|q|rqfn,ε+ε|wn-w|r,

which gives

lim supnN||wn|q-|wn-w|q-|w|q|rqcε,

where c=supn|wn-w|rr<. Further, letting ε0, we conclude the proof. ∎

Lemma 2.4 (Nonlocal Brezis–Lieb lemma [17, Lemma 2.4]).

Let α(0,N) and p[1,2NN+α). Assume (un) is a bounded sequence in L2Np/(N+α)(RN) that converges almost everywhere to some u:RNR. Then

limnN|(Iα*|un|p)|un|p-(Iα*|un-u|p)|un-u|p-(Iα*|u|p)|u|p|=0.

Proof.

For nN, we observe that

N[(Iα*|un|p)|un|p-(Iα*(|un-u|p))(|un-u|p)]=N[Iα*(|un|p-|un-u|p)](|un|p-|un-u|p)+2N[Iα*(|un|p-|un-u|p)]|un-u|p.(2.2)

Using Lemma 2.3 with q=p, r=2NpN+α, we have |un-u|p-|un|p|u|p strongly in L2N/(N+α)(N), and by Lemma 2.2 we get |un-u|p0 weakly in L2N/(N+α)(N). Also, by the Hardy–Littlewood–Sobolev inequality (1.6) we obtain

Iα*(|un-u|p-|un|p)Iα*|u|pin L2NN-α(N).

Using all the above arguments and passing to the limit in (2.2), we conclude the proof. ∎

Lemma 2.5.

Let α(0,N) and p[1,2NN+α). Assume (un) is a bounded sequence in L2Np/(N+α)(RN) that converges almost everywhere to u. Then for any hL2Np/(N+α)(RN) we have

limnN(Iα*|un|p)|un|p-2unh=N(Iα*|u|p)|u|p-2uh.

Proof.

By using h=h+-h-, it is enough to prove our lemma for h0. Denote vn=un-u and observe that

N(Iα*|un|p)|un|p-2unh=N[Iα*(|un|p-|vn|p)](|un|p-2unh-|vn|p-2vnh)+N[Iα*(|un|p-|vn|p)]|vn|p-2vnh+N[Iα*(|un|p-2unh-|vn|p-2vnh)]|vn|p+N(Iα*|vn|p)|vn|p-2vnh.(2.3)

Apply Lemma 2.3 with q=p and r=2NpN+α by taking (wn,w)=(un,u) and then (wn,w)=(unh1/p,uh1/p), respectively. We find

{|un|p-|vn|p|u|p,|un|p-2unh-|vn|p-2vnh|u|p-2uhstrongly in L2NN+α(N).

Using now the Hardy–Littlewood–Sobolev inequality, we obtain

{Iα*(|un|p-|vn|p)Iα*|u|p,Iα*(|un|p-2unh-|vn|p-2vnh)Iα*(|u|p-2uh)strongly in L2NN-α(N).(2.4)

Also, by Lemma 2.2 we have

|un|p-2unh|u|p-2uh,|vn|p0,|vn|p-2vnh0  weakly in L2NN+α(N).(2.5)

Combining (2.4)–(2.5), we find

{limnN[Iα*(|un|p-|vn|p)](|un|p-2unh-|vn|p-2vnh)=N(Iα*|u|p)|u|p-2uh,limnN[Iα*(|un|p-|vn|p)]|vn|p-2vnh=0,limnN[Iα*(|un|p-2unh-|vn|p-2vnh)]|vn|p=0.(2.6)

By Hölder’s inequality and the Hardy–Littlewood–Sobolev inequality, we have

|N(Iα*|vn|p)|vn|p-2vnh|vn2NpN+αp|vn|p-1h2NN+αC|vn|p-1h2NN+α.(2.7)

On the other hand, by Lemma 2.2 we have vn2N(p-1)/(N+α)0 weakly in Lp/(p-1)(N), so

|vn|p-1h2NN+α=(N|vn|2N(p-1)N+α|h|2NN+α)N+α2N0.

Thus, from (2.7) we have

limnN(Iα*|vn|p)|vn|p-2vnh=0.(2.8)

Passing to the limit in (2.3), from (2.6) and (2.8) we reach the conclusion. ∎

3 Proof of Theorem 1.1

In this section, we discuss the existence of groundstate solutions to (1.1) under the assumption λ>0. For u,vXVs(N), we have

λ(u),v=NN(u(x)-u(y))(v(x)-v(y))|x-y|N+2s𝑑x𝑑y+NV(x)uv-N(Iα*|u|p)|u|p-1v-λN(Iβ*|u|q)|u|q-1v.

So, for t>0 we have

λ(tu),tu=t2uXVs2-t2pN(Iα*|u|p)|u|p-λt2qN(Iβ*|u|q)|u|q.

Since p>q>1, the equation

λ(tu),tu=0

has a unique positive solution t=t(u), and the corresponding element tu𝒩λ is called the projection of u on 𝒩λ. The following result presents the main properties of the Nehari manifold 𝒩λ, which we use in this paper.

Lemma 3.1.

  • (i)

    λ|𝒩λ is bounded from below by a positive constant.

  • (ii)

    Any critical point u of λ|𝒩λ is a free critical point.

Proof.

(i) By using the continuous embeddings XVs(N)L2Np/(N+α)(N) and XVs(N)L2Nq/(N+β)(N) together with the Hardy–Littlewood–Sobolev inequality, for any u𝒩λ we have

0=λ(u),u=uXVs2-N(Iα*|u|p)|u|p-λN(Iβ*|u|q)|u|quXVs2-CuXVs2p-CλuXVs2q.

Therefore, there exists C0>0 such that

uXVsC0>0for all u𝒩λ.(3.1)

Using the above fact, we have

λ(u)=λ(u)-12qλ(u),u=(12-12q)uXVs2+(12q-12p)N(Iα*|u|p)|u|p(12-12q)uXVs2(12-12q)C02>0.

(ii) Let (u)=λ(u),u for uXVs(N). Now, for u𝒩λ, from (3.1) we get

(u),u=2u2-2pN(Iα*|u|p)|u|p-2qλN(Iβ*|u|q)|u|q=2(1-q)uXVs2-2(p-q)N(Iα*|u|p)|u|p-2(q-1)uXVs2<-2(q-1)C0.

Assuming that u𝒩λ is a critical point of λ|𝒩λ and using the Lagrange multiplier theorem, there exists μ such that λ(u)=μ(u). In particular, λ(u),u=μ(u),u. As (u),u<0, this implies μ=0, so λ(u)=0. ∎

3.1 Compactness

Define

𝒥:XVs(N),𝒥(u)=12u2-12pN(Iα*|u|p)|u|p.

For all ϕC0(N), we have

𝒥(u),ϕ=NN(u(x)-u(y))(ϕ(x)-ϕ(y))|x-y|N+2s𝑑x𝑑y+NV(x)uϕ-N(Iα*|u|p)|u|p-1ϕ

and

𝒥(u),u=uXVs2-N(Iα*|u|p)|u|p.

Also, consider the Nehari manifold associated with 𝒥 as

𝒩𝒥={uXVs(N){0}:𝒥(u),u=0},

and let

m𝒥=infu𝒩𝒥𝒥(u).

Lemma 3.2.

Let (un)NJ be a (PS) sequence of Eλ|Nλ, that is, (Eλ(un)) is bounded and Eλ|Nλ(un)0 strongly in XV-s(RN). Then there exists a solution uXVs(RN) of (1.1) such that, by replacing (un) with a subsequence, one of the following alternatives holds:

  • (i)

    unu strongly in XVs(N).

  • (ii)

    unu weakly in XVs(N) , and there exists a positive integer k1 and k functions u1,u2,,ukXVs(N) , which are nontrivial weak solutions to ( 1.2 ), and k sequences of points (zn,1), (zn,2),,(zn,k)N such that the following conditions hold:

    • (b)(a)

      |zn,j| and |zn,j-zn,i| if ij, n ;

    • (b)(b)

      un-j=1kuj(+zn,j)u in XVs(N) ;

    • (b)(c)

      λ(un)λ(u)+j=1k𝒥(uj).

Proof.

Since (un) is bounded in XVs(N), there exists uXVs(N) such that, up to a subsequence, we have

{unuweakly in XVs(N),unuweakly in Lr(N), 2r2s*,unua.e. in N.(3.2)

By using (3.2) and Lemma 2.5, it follows that λ(u)=0, so uXVs(N) is a solution of (1.1). Further, if unu strongly in XVs(N), then Lemma 3.2 (i) holds.

Now, assume that (un) does not converge strongly to u in XVs(N), and set wn,1=un-u. Then (wn,1) converges weakly to zero in XVs(N), and

unXVs2=uXVs2+wn,1XVs2+o(1).(3.3)

By Lemma 2.4, we have

N(Iα*|un|p)|un|p=N(Iα*|u|p)|u|p+N(Iα*|wn,1|p)|wn,1|p+o(1).(3.4)

Using (3.3) and (3.4), we get

λ(un)=λ(u)+𝒥(wn,1)+o(1).(3.5)

Further, for any hXVs(N), by Lemma 2.5 we have

𝒥(wn,1),h=o(1).

From Lemma 2.4 we deduce that

0=λ(un),un=λ(u),u+𝒥(wn,1),wn,1+o(1)=𝒥(wn,1),wn,1+o(1).

This implies

𝒥(wn,1),wn,1=o(1).(3.6)

We need the following auxiliary result.

Lemma 3.3.

Define

δ:=lim supn(supwNB1(z)|wn,1|2NpN+α).

Then δ>0.

Proof.

Assume by contradiction δ=0. By Lemma 2.1, we deduce that wn,10 strongly in L2Np/(N+α)(N). Then by the Hardy–Littlewood–Sobolev inequality we get

N(Iα*|wn,1|p)|wn,1|p=o(1).

Using this fact together with (3.6), we get wn,10 strongly in XVs(N). This is a contradiction. Hence, δ>0. ∎

Now, we return to the proof of Lemma 3.2. Since δ>0, we may find zn,1N such that

B1(zn,1)|wn,1|2NpN+α>δ2.(3.7)

Consider the sequence (wn,1(+zn,1)). Then there exists u1XVs(N) such that, up to a subsequence, we have

wn,1(+zn,1)u1weakly in XVs(N),wn,1(+zn,1)u1strongly in Lloc2NpN+α(N),wn,1(+zn,1)u1a.e. in N.

Next, passing to the limit in (3.7), we get

B1(0)|u1|2NpN+αδ2,

therefore u10. Since (wn,1) converges weakly to zero in XVs(N), it follows that (zn,1) is unbounded. Thus, passing to a subsequence, we may assume that |zn,1|. By (3.6), we deduce that 𝒥(u1)=0, so u1 is a nontrivial solution of (1.2).

Further, define

wn,2(x)=wn,1(x)-u1(x-zn,1).

Similarly to before, we have

wn,12=u12+wn,22+o(1).

Then, using Lemma 2.4, we deduce that

N(Iα*|wn,1|p)|wn,1|p=N(Iα*|u1|p)|u1|p+N(Iα*|wn,2|p)|wn,2|p+o(1).

Hence,

𝒥(wn,1)=𝒥(u1)+𝒥(wn,2)+o(1).

So, by (3.5) one can get

λ(un)=λ(u)+𝒥(u1)+𝒥(wn,2)+o(1).

Using the above techniques, we also obtain

𝒥(wn,2),h=o(1)for any hXVs(N)

and

𝒥(wn,2),wn,2=o(1).

Now, if (wn,2) converges strongly to zero, then we finish the proof by taking k=1 in the statement of Lemma 3.2. If wn,20 weakly and not strongly in XVs(N), then we iterate the process. In k steps one could find a set of sequences (zn,j)N, 1jk, with

|zn,j|and |zn,i-zn,j|  as n,ij,

and k nontrivial solutions u1,u2,,ukXVs(N) of (1.2) such that, denoting

wn,j(x):=wn,j-1(x)-uj-1(x-zn,j-1),2jk,

we have

wn,j(x+zn,j)ujweakly in XVs(N)

and

λ(un)=λ(u)+j=1k𝒥(uj)+𝒥(wn,k)+o(1).

As λ(un) is bounded and 𝒥(uj)m𝒥, we can iterate the process only a finite number of times, which concludes our proof. ∎

Corollary 3.4.

For c(0,mJ), any (PS)c sequence of Eλ|Nλ is relatively compact.

Proof.

Assume (un) is a (PS)c sequence of λ|𝒩λ. From Lemma 3.2 we have 𝒥(uj)m𝒥, and hence it follows that, up to a subsequence, unu strongly in XVs(N) and u is a solution of (1.1). ∎

In order to finish the proof of Theorem 1.1 we need the following result.

Lemma 3.5.

mλ<m𝒥.

Proof.

Let QXVs(N) be a groundstate solution of (1.2); we know that such a groundstate exists, and for this we refer the reader to [5]. Denote by tQ the projection of Q on 𝒩λ, that is, t=t(Q)>0 is the unique real number such that tQ𝒩λ. Set

A(Q)=N(Iα*|Q|p)|Q|p,B(Q)=λN(Iβ*|Q|p)|Q|p.

As Q𝒩𝒥 and tQ𝒩λ, we get

Q2=A(Q)

and

t2Q2=t2pA(Q)+t2qB(Q).

From the above equalities one can easily deduce that t<1. Therefore, we have

mλλ(tQ)=12t2Q2-12pt2pA(Q)-12qt2qB(Q)=(t22-t2p2p)Q2-12q(t2Q2-t2pA(Q))=t2(12-12q)Q2+t2p(12q-12p)Q2<(12-12q)Q2+(12q-12p)Q2<(12-12p)Q2=𝒥(Q)=m𝒥,

as desired. ∎

Further, using the Ekeland variational principle, for any n1 there exists (un)𝒩λ such that

λ(un)mλ+1nfor all n1,λ(un)λ(v)+1nv-unfor all v𝒩λ,n1.

Now, one can easily deduce that (un)𝒩λ is a (PS)mλ sequence for λ on 𝒩λ. Further, using Lemma 3.5 and Corollary 3.4, we obtain that, up to a subsequence, (un) converges strongly to some uXVs(N) which is a groundstate of λ.

4 Proof of Theorem 1.2

In this section, we discuss the existence of a least energy sign-changing solution of (1.1).

Lemma 4.1.

Assume p>q>2 and λR. Then for any uXVs(RN) and u±0 there exists a unique pair (τ0,θ0)(0,)×(0,) such that τ0u++θ0u-Mλ. Furthermore, if uMλ, then for all τ,θ0 we have Eλ(u)Eλ(τu++θu-).

Proof.

We shall follow an idea developed in [26]. Denote

a1=u+XVs2,b1=u-XVs2,a2=N(Iα*|u+|p)|u+|p,b2=N(Iβ*|u+|q)|u+|q,a3=N(Iα*|u-|p)|u-|p,b3=N(Iβ*|u-|q)|u-|q,a4=N(Iα*|u+|p)|u-|p,b4=N(Iβ*|u+|q)|u-|q,A=NNu+(x)u-(y)+u-(x)u+(y)|x-y|N+2s𝑑x𝑑y.

Let us define the function Φ:[0,)×[0,) by

Φ(τ,θ)=λ(τ12pu++θ12pu-)=τ1p2a1+θ1p2b1-λτqp2qb2-λθqp2qb3-λτq2pθq2p2qb4-τ2pa2-θ2pa3-τ12θ122pa4-τ12pθ12pA.

Note that Φ is strictly concave. Therefore, Φ has at most one maximum point. Also

limτΦ(τ,θ)=-for all θ0  and    limθΦ(τ,θ)=-for all τ0,(4.1)

and it is easy to check that

limτ0Φτ(τ,θ)=for all θ>0  and  limθ0Φθ(τ,θ)=for all τ>0.(4.2)

Hence, (4.1) and (4.2) rule out the possibility of achieving a maximum at the boundary. Therefore, Φ has exactly one maximum point (τ0,θ0)(0,)×(0,). ∎

Lemma 4.2.

The energy level cλ>0 is achieved by some vMλ.

Proof.

Let (un)λ be a minimizing sequence for cλ. Note that

λ(un)=λ(un)-12qλ(un),un=(12-12q)unXVs2+(12q-12p)N(Iα*|u|q)|u|q(12-12q)unXVs2C1unXVs2,

where C1>0 is a positive constant. Therefore, for some constant C2>0 we have

unXVs2C2λ(un)M,

which implies that (un) is bounded in XVs(N). So, (un+) and (un-) are also bounded in XVs(N), and, by passing to a subsequence, there exists u+,u-Hs(N) such that

un+u+andun-u-  weakly in XVs(N).

Since p, q>2 satisfy (1.4) and (1.5), we deduce that the embeddings XVs(N)L2Np/(N+α)(N) and XVs(N)L2Nq/(N+β)(N) are compact. Thus,

un±u±strongly in L2NpN+α(N)L2NqN+β(N).(4.3)

Moreover, by the Hardy–Littlewood–Sobolev inequality, we estimate

C(un±L2NpN+α2+un±L2NqN+β2)un±XVs2=N(Iα*|un|p)|un±|p+|λ|N(Iβ*|un|q)|un±|qC(un±L2NpN+αp+un±L2NpN+αp)C(un±L2NpN+α2+un±L2NqN+β2)(un±L2NpN+αp-2+un±L2NqN+βq-2).

Since un±0, we can deduce

un±L2NpN+αp-2+un±L2NqN+βq-2C>0for all n1.(4.4)

Hence, by (4.3) and (4.4) it follows that u±0. Further, using (4.3) and the Hardy–Littlewood–Sobolev inequality, we have

N(Iα*|un±|p)|un±|pN(Iα*|u±|p)|u±|p,N(Iα*|un+|p)|un-|pN(Iα*|u+|p)|u-|p,N(Iβ*|un±|q)|un±|qN(Iβ*|u±|q)|u±|q,N(Iβ*|un+|q)|un-|qN(Iβ*|u+|q)|u-|q.

By Lemma 4.1, there exists a unique pair (τ0,θ0) such that τ0u++θ0u-λ. By the weakly lower semi-continuity of the norm XVs, we deduce that

cλλ(τ0u++θ0u-)lim infnλ(τ0u++θ0u-)lim supnλ(τ0u++θ0u-)limnλ(un)=cλ.

Letting now v=τ0u++θ0u-λ, we finish the proof. ∎

Lemma 4.3.

v=τ0u++θ0u-λ is a critical point of Eλ:XVs(RN)R, that is,

λ(v)=0.

Proof.

Assume by contradiction that v is not a critical point of λ. Then there exists φCc(N) such that

λ(v),φ=-2.

Since λ is continuous and differentiable, there exists r>0 small such that

λ(τu++θu-+εv¯),v¯-1if (τ-τ0)2+(θ-θ0)2r2 and 0εr.(4.5)

Let D be the open disc in 2 of radius r>0 centered at (τ0,θ0). We define a continuous function ψ:D[0,1] as

ψ(τ,θ)={1if (τ-τ0)2+(θ-θ0)2r216,0if (τ-τ0)2+(θ-θ0)2r24.

Further, we define a continuous map S:DXVs(N) as

S(τ,θ)=τu++θu-+rψ(τ,θ)v¯for all (τ,θ)D

and L:D2 as

L(τ,θ)=(λ(S(τ,θ)),S(τ,θ)+,λ(S(τ,θ)),S(τ,θ)-)for all (τ,θ)D.

Since the mapping uu+ is continuous in XVs(N), it follows that L is continuous. If (τ-τ0)2+(θ-θ0)2=r2, that is, if we are on the boundary of D, then ψ=0 by definition. Then S(τ,θ)=τu++θu- and, using Lemma 4.1, we get

L(τ,θ)=(λ(τu++θu-),(τu++θu-)+,λ(τu++θu-),(τu++θu-)-)0on D.

Therefore, the Brouwer degree is well defined and deg(L,int(D),(0,0))=1. Then there exists (τ1,θ1)int(D) such that L(τ1,θ1)=(0,0). Thus, S(τ1,θ1)λ and, using the definition of cλ, we get

λ(S(τ1,θ1))cλ.(4.6)

Using equation (4.5), we deduce that

λ(S(τ1,θ1))=λ(τ1u++θ1u-)+01ddtλ(τ1u++θ1u-+rtψ(τ1,θ1)v¯)𝑑t=λ(τ1u++θ1u-)+01λ(τ1u++θ1u-+rtψ(τ1,θ1)v¯),rψ(τ1,θ1)v¯𝑑t=λ(τ1u++θ1u-)-rψ(τ1,θ1).

If (τ1,θ1)=(τ0,θ0), then ψ(τ1,θ1)=1 by definition and we deduce that

λ(S(τ1,θ1))λ(τ1u++θ1u-)-rcλ-r<cλ.

If (τ1,θ1)(τ0,θ0), then, using Lemma 4.1, we have

λ(τ1u++θ1u-)<λ(τ0u++θ0u-)=cλ.

This yields

λ(S(τ1,θ1))λ(τ1u++θ1u-)<cλ,

which is a contradiction to equation (4.6). ∎

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About the article

Received: 2017-05-30

Accepted: 2017-06-19

Published Online: 2017-08-05


This work is part of the author’s PhD thesis and has been carried out with the financial support of the Research Demonstratorship Scheme offered by the School of Mathematics and Statistics, University College Dublin.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 694–706, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0126.

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

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