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Advances in Nonlinear Analysis

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


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Zero mass case for a fractional Berestycki–Lions-type problem

Vincenzo Ambrosio
  • Corresponding author
  • Dipartimento di Matematica e Applicazioni, Università degli Studi “Federico II” di Napoli, via Cinthia, 80126 Napoli, Italy
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Published Online: 2016-09-20 | DOI: https://doi.org/10.1515/anona-2016-0153

Abstract

In this work we study the following fractional scalar field equation:

{(-Δ)su=g(u)in N,u>0,

where N2, s(0,1), (-Δ)s is the fractional Laplacian and the nonlinearity gC2() is such that g′′(0)=0. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in r=|x|.

Keywords: Zero mass case; fractional Laplacian; Nehari manifold; Orlicz spaces

MSC 2010: 35A15; 35J60; 35R11; 45G05

1 Introduction

In this paper we are concerned with the existence of solutions to the problem

{(-Δ)su=g(u)in N,u>0,(1.1)

where N2, s(0,1) and g: is a smooth function such that g′′(0)=0. Here (-Δ)s is the fractional Laplacian and it can be defined via Fourier transform by

(-Δ)sf(k)=|k|2sf(k)

for u belonging to the Schwartz space 𝒮(N).

Problems like (1.1)) are motivated by the study of standing waves solutions ψ(x,t)=u(x)e-ıct of the fractional Schrödinger equation

ıψt=(-Δ)sψ+G(x,ψ)in N.

This equation has been proposed by Laskin [23, 24] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. After that many papers appeared investigating existence, multiplicity and behavior of solutions to fractional Schrödinger equations; see [4, 5, 6, 17, 18, 19, 22, 26, 29] and references therein.

More in general, problems involving fractional operators are receiving a special attention in these last years; indeed, fractional spaces and nonlocal equations have great applications in many different fields, such as optimization, finance, continuum mechanics, phase transition phenomena, population dynamics, multiple scattering, minimal surfaces and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes. The interested reader may consult [16, 27] and references therein, where a more extensive bibliography and an introduction to the subject are given.

In the seminal paper [12], Berestycki and Lions investigated the existence of positive ground state solutions to (1.1)) when s=1, i.e.

-Δu=g(u)in N.(1.2)

Under general assumptions on g, they proved that there are no finite energy solutions to (1.2)) if g′′(0)>0, while if g′′(0)<0 or g′′(0)=0, then it is possible to show the existence of a solution to (1.2)) via constraint minimization. The case g′′(0)=0 is called null mass case and it is related to the Yang–Mills equation; see [20, 21].

Let us note that the case g′′(0)=0 seems to be more intricate than g′′(0)<0, since unless g satisfies the condition |g(u)|c|u|2NN-2, the energy functional associated to (1.2)) may be infinite on a dense set of points in 𝒟1,2(N) and hence cannot be Fréchet-differentiable on 𝒟1,2(N).

The question that naturally arises is whether or not the above classical existence results for equation (1.2)) can be extended in the nonlocal setting. When g′′(0)<0 (in the case of positive mass), the existence of a ground state has been proved in [15] by combining the Struwe–Jeanjean monotonicity trick and the Pohozaev identity for the fractional Laplacian. Now, our aim is to investigate problem (1.1)) when g′′(0)=0, and g(u) behaves like |u|q for u small and |u|p for u large, with 2<p<2NN-2s<q.

In order to state our result, we introduce the basic assumptions on the nonlinearity g. Here we will assume that g: is an odd C2-function verifying the following conditions:

  • (g1)

    0<μg(t)g(t)tg′′(t)t2 for any t0 and for some μ>2.

  • (g2)

    g(0)=g(0)=g′′(0)=0. There exist c0,c2,p,q with 2<p<2s*:=2NN-2s<q such that

    {c0|t|pg(t)if |t|1,c0|t|qg(t)if |t|1,(1.3)

    and

    {|g′′(t)|c2tp-2if |t|1,|g′′(t)|c2tq-2if |t|1.(1.4)

Remark 1.

The assumptions g′′(t)>0 for all t0, and (g2) imply the existence of c1,c3>0 such that

c0|t|pg(t)c3|t|pfor |t|1,c0|t|qg(t)c3|t|qfor |t|1,|g(t)|c1|t|p-1for |t|1,|g(t)|c1|t|q-1for |t|1.

As a model for g we can take the function

g(t)={tqif t1,a+bt+ctpif t1,

where a,b and c are constants which make the function gC2.

Let us point out that, when s=1, the assumptions (g1) and (g2) have been introduced in [11] to study positive solutions to a nonlinear field equation set in exterior domain. The authors studied (1.2)) in the Orlicz space Lp+Lq which seems to be the natural framework for studying “zero mass” problems. Subsequently, their approach has been also used in [7, 9, 10] to study nonlinear Schrödinger equations in N with bounded or vanishing potentials. Further results concerning zero mass problems can be found in [2, 3, 30].

The main result of this paper is the following.

Theorem 1.

Let N2, s(0,1) and g satisfies (g1) and (g2). Then there exists a positive solution to (1.1)) which is spherically symmetric and decreasing in r=|x|.

To deal with problem (1.1)), we develop an energy minimization argument on a Nehari manifold.

More precisely, solutions to (1.1)) will be obtained by minimizing

I(u)=122N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y-Ng(u(x))𝑑x

on the Nehari manifold

𝒩={u𝒟s,2(N){0}:2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y=Ng(u)u𝑑x},

where

𝒟s,2(N)={uL2NN-2s(N):2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y<}.

In order to obtain the smoothness of the functional I, we introduce the Orlicz space Lp+Lq related to the growth assumptions of g at zero and at infinity. Then, we show that IC2(𝒟s,2(N),), and by proving the compactness of the subspace 𝒟~rads,2(N) of nonnegative radial decreasing functions of 𝒟s,2(N) in Lp+Lq, we deduce that the infimum of I on 𝒩 is achieved at some u𝒟~rads,2(N).

As far as we know the result presented here is new.

The paper is organized as follows: in Section 2 we give some preliminaries about the involved functional spaces, and in Section 3 we provide the proof of the main result.

2 Preliminaries

In this section we collect some preliminary results which will be useful in the sequel.

We denote by 𝒟s,2(N) the completion of C0(N) with respect to

[u]2:=N|(-Δ)s2u|2𝑑x=N|k|2s|f(k)|2𝑑k=2N|u(x)-u(y)|2|x-y|N+2s𝑑x𝑑y.

Then

𝒟s,2(N)={uL2(N):[u]<},

where 2s*:=2NN-2s is the fractional Sobolev exponent.

Let us denote by Hs(N) the standard fractional Sobolev space, defined as the set of u𝒟s,2(N) satisfying uL2(N) with the norm

uHs(N)2=[u]2+uL2(N)2.

We recall the following embeddings.

Theorem 2 ([16]).

Let s(0,1) and N>2s. There exists a constant C>0 such that

uL2s*(N)C[u]for any u𝒟s,2(N).

In particular, Hs(RN) is continuously embedded in Lq(RN) for any q[2,2s*], and compactly embedded for any q[2,2s*).

For more details about fractional Sobolev spaces, we refer to [16].

We remark that the symmetric-decreasing rearrangement of a measurable function u:N that vanishes at infinity (that is, |{xN:|u(x)|>a}|< for all a>0) is given by

u*(x)=0χ{|u|>t}*(x)𝑑t,

where χA*=χA* and A*={x:|x|<r} is such that its volume is that of A. For standard properties of rearrangements of functions one can see [25].

Now, we establish the following fractional Polya–Szegö inequality:

Theorem 3.

Let u:RNR be a nonnegative measurable function that vanishes at infinity, and let us denote by u* its symmetric-decreasing rearrangement. Assume that [u]Hs(RN)<. Then

[u*]Hs(N)[u]Hs(N).(2.1)

Proof.

Let

uc(x)=min{max{u(x)-c,0},1c}

for c(0,1). Since u vanishes at infinity, ucL2(N). In particular, ucHs(N) since

|uc(x)-uc(y)||u(x)-u(y)|

for any x,yN. By Monotone Convergence Theorem we have

limc0[uc]Hs(N)=[u]Hs(N)andlimc0[uc*]Hs(N)=[u*]Hs(N).(2.2)

Now, by using the result in [28], we know that

[uc*]Hs(N)[uc]Hs(N).(2.3)

Then taking into account (2.2) and (2.3) we deduce the thesis. ∎

We also prove the following useful lemma.

Lemma 1.

Let uLt(RN), 1t<, be a nonnegative radial decreasing function (that is, 0u(x)u(y) if |x||y|). Then

|u(x)|(NωN-1)1t|x|-NtuLt(N)for any xN{0},(2.4)

where ωN-1 is the Lebesgue measure of the unit sphere in RN.

Proof.

For all R>0 we have, setting R=|x|,

uLt(N)tωN-10R(u(r))trN-1𝑑rωN-1(u(R))tRNN.

Given p<q, we define the space Lp+Lq:=Lp(N)+Lq(N) as the set of functions u:N such that

u=u1+u2

with u1Lp(N) and u2Lq(N). We recall (see [13]) that Lp+Lq is a Banach space with respect to the norm

uLp+Lq=inf{u1Lp(N)+u2Lq(N):u=u1+u2}.

Moreover, Lp+Lq coincides with the dual of LpLq. Then

Lp+Lq=(LpLq)*,

where p and q are the conjugate exponent of p and q, respectively. In particular, the norm

ue=supw0Nu(x)v(x)𝑑xuLp(N)+vLq(N)

is equivalent to Lp+Lq. Actually, Lp+Lq is an Orlicz space with N-function (see [1])

A(u)=max{|u|p,|u|q}.

Now we state some useful lemmas whose proofs can be obtained following those in [8, 11].

Lemma 2.

The following statements hold.

  • (a)

    If uLp+Lq , the following inequalities hold:

    max{uLq(N-Γu)-1,11+|Γu|1ruLp(Γu)}uLp+Lqmax{uLq(N-Γu),uLp(Γu)},

    where r=pqq-p and Γu={xN:|u(x)|>1}.

  • (b)

    Let {uj}Lp+Lq and set Γj={xN:|uj(x)|>1} . Then {uj} is bounded in Lp+Lq if and only if the sequences {|Γj|} and {ujLq(N-Γj)+ujLp(Γj)} are bounded.

  • (c)

    g is a bounded map from Lp+Lq into Lpp-1Lqq-1.

Lemma 3.

The following statements hold.

  • (a)

    If θ,u are bounded in Lp+Lq , then g′′(θ)u is bounded in LpLq.

  • (b)

    g′′ is a bounded map from Lp+Lq into Lpp-2Lqq-2.

  • (c)

    g′′ is a continuous map from Lp+Lq into Lpp-2Lqq-2.

  • (d)

    The map (u,v)uv from (Lp+Lq)2 in Lp2+Lq2 is bounded.

Lemma 4.

The functional H:Lp+LqR defined by

H(u)=Ng(u(x))𝑑x

is of class C2.

Lemma 5.

If the sequence {uj} converges to u in Lp+Lq, then the sequence {RNg(uj)uj𝑑x} converges to RNg(u)u𝑑x.

Remark 2.

By Lemma 2 (a) we have L2s*(N)Lp+Lq when 2<p<2s*<q. In fact, by using p<2s*<q we find for any uL2s*(N),

uLq(N-Γu)quL2s*(N-Γu)2s*

and

uLp(Γu)puL2s*(Γu)2s*

which together with (a) of Lemma 2 imply the claim. Moreover, by the Sobolev inequality 𝒟s,2(N)L2s*(N) (see Theorem 2), we get the continuous embedding

𝒟s,2(N)Lp+Lq.(2.5)

At this point we are ready to prove the following result.

Theorem 4.

Let N2, s(0,1) and 2<p<2s*<q. Let us denote by D~rads,2(RN) the space of nonnegative radial decreasing functions in Ds,2(RN). Then D~rads,2(RN) is compactly embedded in Lp+Lq.

Proof.

We proceed as in the proof of [8, Lemma 3]. Let {uj} be a sequence in 𝒟~rads,2(N) such that as j+

uj0in 𝒟s,2(N).

By (2.4) it follows that there exists a positive constant C=C(s,N) such that

|uj(x)|C|x|-N-2s2for any j,xN{0}.(2.6)

Fix ε>0. By using (2.6) and q>2, for R>0 big enough we get

{|x|R}|uj(x)|q𝑑xCq{|x|R}1|x|q(N-2s2)𝑑x<ε2(2.7)

for all j. Now, we observe that 𝒟s,2(N)Hlocs(N)Llocp(N) since p(2,2s*). In particular, we have

{|x|<R}|uj(x)|p𝑑x<ε2.(2.8)

Taking into account (2.7) and (2.8) we have for j large

{|x|R}|uj(x)|q𝑑x+{|x|<R}|uj(x)|p𝑑x<ε.(2.9)

If R is sufficiently large, by (2.6) it follows that for any j,

|uj(x)|1for |x|>R.

Then for all j,

Γj:={xN:|uj(x)|>1}BR.

Let us observe that

ujLq(Γjc)q+ujLp(Γj)p=ujLq(BRc)q+ujLq(BRΓj)q+ujLp(BR)p-ujLp(BRΓj)p.(2.10)

Here Ac=NA for AN. Since p<q and |uj(x)|1 in BRΓj, we obtain

ujLq(BRΓj)qujLp(BRΓj)p.(2.11)

Putting together (2.9), (2.10) and (2.11) we have for j large enough,

ujLq(Γjc)q+ujLp(Γj)pujLq(BRc)q+ujLp(BR)p<ε

so, in particular,

ujLq(Γjc)<ε1qandujLp(Γj)<ε1p.(2.12)

Then, by Lemma 1 and (2.12), we can infer that for j large

ujLp+Lqmax{ujLq(Γjc),ujLp(Γj)}<max{ε1q,ε1p}.

3 Proof of Theorem 3

This section is devoted to the proof of the main result of this paper. In order to obtain a solution to (1.1), we will look critical points of the following functional:

I(u):=12[u]Hs(N)2-Ng(u(x))𝑑x

constrained on

𝒩={u𝒟s,2(N){0}:J(u)=0},

where

J(u):=I(u),u=[u]2-Ng(u)u𝑑x.

By using the results in Section 2, we can see that I is well defined on 𝒟s,2(N) and I is a C2-functional.

Proof of Theorem 1.

We divide the proof into several steps.

Step 1: 𝒩 is a C1-manifold. By using (g1) we have, for any u𝒩,

2[u]2-N(g(u)u+g′′(u)u2)𝑑x=[u]2-Ng′′(u)u2𝑑x=N(g(u)u-g′′(u)u2)𝑑x<0.

Step 2: Given u0, there exists a unique t=t(u)>0 such that ut(u)𝒩 and I(ut(u)) is the maximum for I(tu) for t0. Fix u0 and let

h(t):=I(tu)=t22[u]2-Ng(tu(x))𝑑x

for t0. Then

h(t)=t[u]2-Ng(tu(x))u𝑑x

and

h′′(t)=[u]2-Ng′′(tu(x))u2𝑑x.

Let us observe that t=0 is a minimum for h since 0=h(0)=h(0) and h′′(0)>0. Moreover, if t0>0 is a critical point of h, then by (g1), we obtain that t0 is a maximum for h because of

h′′(t0)=[u]2-Ng′′(t0u(x))u2𝑑x=N(g(t0u)t0u-g′′(t0u)u2)𝑑x<0.

By using (g2), we get

h(t)t22[u]2-c0tqt|u|<1|u|q𝑑x-c0tpt|u|>1|u|p𝑑xt22[u]2-c0tpt|u|>1|u|p𝑑x-as t+

since p>2.

Step 3: The dependence of t(u) on u is of class C1. Let us define the following operator:

L(t,u):=t[u]2-Ng(tu(x))u(x)𝑑x

for (t,u)+×𝒟s,2(N). By Lemma 2 we can see that LC1 and if (t0,u0) is a point such that L(t0,u0)=0 and t0,u00, then by (g1)

ddtL(t0,u0)=[u]2-Ng′′(t0u0(x))u02𝑑x=N(g(t0u0)t0u0-g′′(t0u0)u02)𝑑x<0.

By invoking the Implicit Function Theorem, we obtain that ut(u) is C1 and

t(u0),v=t02N2t0(-Δ)s2u0(-Δ)s2v-g(t0u0)v-g′′(t0u0)t0u0vdxNg′′(t0u0)t02u02-g(t0u0)t0u0dx,

where t0=t(u0).

Step 4: infv𝒩[v]2>0. Let vj be a minimizing sequence in 𝒩. We assume by contradiction that vj converges to zero in 𝒟s,2(N). We set tj=[vj], hence we can write vj=tjuj, where [uj]=1. Since the embedding 𝒟s,2(N)Lp+Lq is continuous, we deduce that uj is bounded in Lp+Lq. Then, by (vj)𝒩, tj0 and Remark 1, we get

tj=1tj[vj]2=Ng(tjuj)uj𝑑xc1tjq-1{|vj|1}|uj|q𝑑x+c1tjp-1{|vj|>1}|uj|p𝑑xc1tjq-1{|vj|1}|uj|q𝑑x+c1tjp-1{|uj|>1}|uj|p𝑑xc1tjq-1{|uj|1}|uj|q𝑑x+c1tjq-1{|vj|1}{|uj|>1}|vj|ptjq-p𝑑x+c1tjp-1{|uj|>1}|uj|p𝑑xc1tjq-1{|uj|1}|uj|q𝑑x+2c1tjp-1{|uj|>1}|uj|p𝑑x,

that is,

1c1tjq-2{|uj|1}|uj|q𝑑x+2c1tjp-2{|uj|>1}|uj|p𝑑x.(3.1)

Taking into account (3.1), (b) of Lemma 2 and tj0, we have

1c1tjq-2+c2tjp-20as j,

that is, a contradiction. Therefore infv𝒩[v]2>0.

Since we are looking for positive solutions to (1.1), we can assume that g(t)=0 for t0.

Step 5: The infimum

m:=infv𝒩I(v)(3.2)

is achieved. Let {uj}𝒟s,2(N) be a minimizing sequence for (3.2). Then, by (g1), it follows that

(12-1μ)[uj]212[uj]2-Ng(uj)𝑑x=I(uj),(3.3)

that is, {uj} is bounded in 𝒟s,2(N).

We claim that m>0. Indeed, if m=0, the minimizing sequence {uj}𝒩 is such that I(uj)0, and by (3.3) we deduce that [uj]0. This gives a contradiction because of Step 4.

Now, by using Theorem 3, we can note that I(uj*)I(uj), where uj* is the symmetric-decreasing rearrangement of uj. Moreover, by the boundedness of {uj}, we can see that {uj*} is bounded in 𝒟s,2(N).

In virtue of Theorem 4, the embedding 𝒟~rads,2(N)Lp+Lq is compact, so, up to a subsequence, we may assume that uj*u* strongly in Lp+Lq, and weakly in 𝒟s,2(N).

Let us observe that J(uj*)J(uj)=0 thanks to Theorem 3, so we do not know if uj* belongs to the Nehari manifold 𝒩. Then, for any j there exists a unique tj[0,1] such that tjuj*𝒩 and tj converges to some t0. By Step 3 it follows that I(uj) is the maximum for I(tuj) when t0, so we get

0<mI(tjuj*)I(tjuj)I(uj).(3.4)

Since I(uj)m, we obtain I(tjuj*)m. It is clear that t00. Otherwise tjuj*0 in 𝒟s,2(N), and by (3.4) we deduce that m=0, which provides a contradiction in virtue of Step 4.

Now, we show that u*0. If we suppose that u*=0, then uj*0 strongly in Lp+Lq. Putting together J(uj*)0, Remark 1, and by using Hölder inequality and Lemma 2, it follows that by setting uj*=u1j*+u2j* with u1j*Lp and u2j*Lq and Γj={xN:|uj(x)|>1},

[uj*]2Ng(uj*)u1j*𝑑x+Ng(uj*)u2j*𝑑x[c1uj*Lq(N-Γj)q/p+c1uj*Lp(Γj)p-1]u1j*Lp(N)+[c1uj*Lq(N-Γj)q-1+c1|Γj|p-qpquj*Lp(Γj)p-1]u2j*Lq(N)c4uj*Lp+Lq,

that is, uj*0 in 𝒟s,2(N) as j. This and (3.4) yield m=0, which is impossible because m>0. Then, by using I(tjuj*)m and tjuj*𝒩, and by applying Lemma 3 and Lemma 5, we have

m=limjI(tjuj*)=limjN12g(tjuj*)tjuj*-g(tjuj*)dx=N12g(t0u*)t0u*-g(t0u*)dx.(3.5)

Now, we argue by contradiction in order to prove t0u*𝒩. If we assume that t0u*𝒩, by J(uj*)J(uj)=0 and (3.4) we have

I(t0u*)m

and

[t0u*]2<Ng(t0u*)t0u*𝑑x,

so we can find t1[0,1) such that t1t0u*𝒩. As a consequence,

mI(t1t0u*)=N12g(t1t0u*)t1t0u*-g(t1t0u*)dx.

In view of (g1), the map

t>0ψ(t):=N12g(tu)tu-g(tu)dx

is strictly increasing, so by this and (3.5) we get

mψ(t1t0u*)<ψ(t0u*)=m,

which is a contradiction. This concludes the proof of Step 5.

Then, by applying the Lagrange multiplier rule, there exists λ such that

I(t0u*),φ=λJ(t0u*),φ(3.6)

for any φ𝒟s,2(N).

By testing φ=t0u*𝒩 in (3.6), and keeping in mind that J(u),u<0 for all u𝒩 (see Step 1), we deduce that 0=I(t0u*),t0u*=λJ(t0u*),t0u*, that is, λ=0 and t0u* is a nontrivial solution to (1.1). Actually, by the strong maximum principle [14], we argue that t0u* is positive. ∎

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About the article

Received: 2016-07-02

Revised: 2016-08-13

Accepted: 2016-08-14

Published Online: 2016-09-20

Published in Print: 2018-08-01


Citation Information: Advances in Nonlinear Analysis, Volume 7, Issue 3, Pages 365–374, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0153.

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