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Advances in Nonlinear Analysis

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


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On a logarithmic Hartree equation

Federico Bernini / Dimitri Mugnai
Published Online: 2019-08-29 | DOI: https://doi.org/10.1515/anona-2020-0028

Abstract

We study the existence of radially symmetric solutions for a nonlinear planar Schrödinger-Poisson system in presence of a superlinear reaction term which doesn’t satisfy the Ambrosetti-Rabinowitz condition. The system is re-written as a nonlinear Hartree equation with a logarithmic convolution term, and the existence of a positive and a negative solution is established via critical point theory.

Keywords: planar Schrödinger-Poisson system; logarithmic Hartree equation; Hardy-Littlewood-Sobolev inequality; superlinear source

MSC 2010: 35J50; 35Q40

1 Introduction

In recent past years many papers have been devoted to finding solutions of Schrödinger-Poisson systems of the form

iψtΔψ+E(x)ψ+ywψ=f(x,t), in RN×R,Δw=ψ2, in RN,(SP)

and often the main objects were standing wave solutions, i.e. solutions of the form

ψ(x,t)=eiωtu(x),ωR,

so that (SP) reads

Δu+b(x)u+ywu=f~(x,t), in RN×R,Δw=u2, in RN,(1.1)

where b(x) = E(x) + ω and (x, t) = eiωtf(x, t).

Due to the numerous several Physical applications, the most studied case is N = 3 (or N ≥ 3). On the other hand, the 1–dimensional case was considered in [1] when f = 0, and the existence of a unique ground state was established by decreasing symmetric rearrangements tools. The 2–dimensional case when f = 0 was first approached in [2], Section 6, only from a numerical point of view, while the first rigorous existence result was given in [3] by using a shooting method for ODEs.

Moving down to lower dimensions, in particular to N = 2, introduces several complications, the first important one is that in this case the Coulomb potential is not positive.

In [4] the authors studied the eigenvalue problem for the Schrödinger operator in a bounded domain of ℝ3, with electromagnetic field E-H that is not assigned; in this case the unknowns are the wave function ψ(x, t) and the gauge potentials A(x, t), ϕ(x, t) related to E-H. In particular, they considered the problem in which A and ϕ do not depend on the time and ψ(x, t) = eiωtu(x), with ω ∈ ℝ and u real function. With these consideration they assume A = 0, and thus the system reduces to

Δuϕu=ωu, in ΩR3,Δϕ=4πu2, in ΩR3.

Under this hypotheses, they proved the existence of a sequence of solution in a bounded domain of ℝ3.

Later on, in [5] the authors considered the problem

Δuϕuωu=|u|p2u, in R3,Δϕ=4πu2, in R3(1.2)

with p ∈ ℝ+ and they proved the existence of radially symmetric solutions in ℝ3, for p ∈ [4, 6), while in [6] they showed the nonexistence of solutions for p ∈ (0, 2] or p ∈ [6, ∞). After that, system (1.2) has been object of an intensive study, where generalizations of several type where considered; we refer to [7, 8, 9, 10, 11, 12, 13, 14] for other references and improvements on this subject.

All these works have been done in the whole of ℝ3, while the two dimensional case has remained for a long time a quite open field of study. Indeed, a theoretical approach in dimension 2 is harder than in higher dimensions due to the lack of positivity of the Coulomb interaction term: precisely, the Coulomb potential is neither bounded from above nor from below.

However, in 2008 Stubbe [15] bypassed this problem giving a suitable variational framework for the problem

Δu+au12πln1|x||u|2u=0(1.3)

and proving the existence of ground states, which is a positive spherically symmetric strictly decreasing function, by solving an appropriate minimization problem for the energy functional associated to the system (see also [16]).

In some recent works, a local nonlinear terms of the form b|u|p−2u, p > 2 has been added; this kind of nonlinearity are frequently used in Schrödinger equations to model the interaction among particles, like recalled above (see [4]). Thus in [17], they studied a Schrödinger-Poisson system of the type

Δu+a(x)u12πln1|x||u|2u=b|u|p2u in R2,

with b ≥ 0, p ≥ 2 and aL(ℝ2) and they proved that if p ≥ 4 then the problem has a sequence of solution pairs ± un such that I(un) → ∞ as n → ∞.

In this work we are concerned with the integro-differential equation

Δu+au12πln1|x||u|2u=f(x,u) in R2,(P)

where a > 0 and f is a superlinear function. We refer to (P) as the logarithmic Choquard equation. Note that, if compared with (SP), we have chosen y = −1; since y represents the charge of the particle that we are studying, it means that we are considering electrons.

In order to generalize and include the previous cases, on the reaction term f : ℝ2 × ℝ → ℝ we assume that it is a Carathéodory function having superlinear growth and not verifying the Ambrosetti-Rabinowitz condition, from now on (AR).

Our main result has the following flavour (see Theorem 3.1 for the precise statement):

Theorem 1.1

Under suitable hypotheses on f problem (P) has two nontrivial constant sign solutions.

This work is organized as follows. In Section 2 we recall some useful definitions and results that we shall use, we set up an appropriate variational framework and define the energy functional associated to the problem. Moreover, we give an extended results of the estimates given in the Strauss theorem.

In Section 3 we prove the well-posedness and the regularity of our functional and we give a Lemma that plays a fundamental role in the proof of the Cerami condition. Finally, we give the proof of the main existence theorem.

2 Background and Variational Framework

We provide a suitable variational framework for studying (P): indeed, the associated functional is not well defined on the natural Sobolev space H1(ℝ2), and so we need some adjustments taken from [15], see also [17].

We first recall an important result for Lp-spaces.

Theorem 2.1

(Hardy-Littlewood-Sobolev’s inequality, [18]). Let p, q > 1 and 0 < λ < N with 1p+λN+1q=2. Let fLp(ℝN) and gLq(ℝN). Then there exists a sharp constant C(N, λ, p), independent of f and g, such that

RNRN|f(x)g(y)||xy|λdxdyC(N,λ,p)fLp(RN)gLq(RN).(2.1)

The sharp constant satisfies

C(N,λ,p)N(Nλ)|SN1|NλN1pqλ/N11pλN+λ/N11qλN.

If p=q=2N2Nλ, then

C(N,λ,p)=C(N,λ)=πλ2Γ(N/2λ/2)Γ(Nλ/2)Γ(N/2)Γ(N)1+λ/N.

In this case there is equality in (2.1) if and only if gcf with c constant and

f(x)=Ay2+|xx0|2(2Nλ)/2

for some A ∈ ℝ, 0 ≠ y ∈ ℝ and x0 ∈ ℝN. Here |SN−1| denotes the area of the unit sphere inN.

First of all, we endow H1(ℝ2) with the scalar product (recall that a > 0 is a constant)

(u|v)=R2DuDv+auvdx, for u,vH1(R2),

and we introduce the space

X=uH1(R2):R2|u(x)|2ln(1+|x|)dx<

with the norm defined by

uX2=R2|Du|2+|u|2a+ln(1+|x|)dx.

Then, we define the symmetric bilinear forms

B1(u,v)=R2R2ln(1+|xy|)u(x)v(y)dxdy,

B2(u,v)=R2R2ln1+1|xy|u(x)v(y)dxdy,

and

B(u,v)=B1(u,v)B2(u,v)=R2R2ln|xy|u(x)v(y)dxdy,

since for all r > 0 we have

lnr=ln(1+r)ln1+1r.(2.2)

The definitions above are restricted to measurable functions u, v : ℝ2 → ℝ such that the corresponding double integral is well defined in the Lebesgue sense. Finally, for any measurable functions u : ℝ2 → ℝ we consider the seminorm in X

|u|2=R2ln(1+|x|)u2(x)dx.

We note that, since

ln(1+|xy|)ln(1+|x|+|y|)ln(1+|x|)+ln(1+|y|),(2.3)

we have by the Schwarz inequality

|B1(uv,wz)|R2R2ln(1+|x|)+ln(1+|y|)|u(x)v(x)||w(y)z(y)|dxdy|u||v|wL2(R2)zL2(R2)+uL2(R2)vL2(R2)|w||z|(2.4)

for u, v, w, zL2(ℝ2). Next, since 0 ≤ ln(1 + r) ≤ r for all r > 0, we have by Theorem 2.1

|B2(u,v)|R2R21|xy|u(x)v(y)dxdyCuL43(R2)vL43(R2),(2.5)

for u, vL43(ℝ2), for some constant C > 0. In particular, from (2.4) we have

B1(u2,u2)2|u|2uL2(R2)2(2.6)

for all uL2(ℝ2) and from (2.5) we have

B2(u2,u2)CuL83(R2)4(2.7)

for all uL83(ℝ2).

The energy functional I : X → ℝ associated to (P) is

I(u)=12R2|Du|2+a|u|2dx18πR2R2ln1|xy||u(x)|2|u(y)|2dxdyR2F(x,u)dx,

where F(x,u)=0uf(x,s)ds and the Gâteaux derivative of I along vX is

I(u)(v)=R2DuDv+auvdx12πR2R2ln1|xy||u(x)|2u(y)v(y)dxdyR2f(x,u)vdx.

Definition 1

We say that uX is a weak solution of (P) if

I(u)(v)=0 for all vX,

thus if u is a critical point for I.

Of course, these consideration are only formal, since, without any assumption on f, we cannot differentiate I. In Section 3 we will give some sufficient conditions for I to be of class C1 in X, while for the moment we continue with formal computations.

We have the following results, the second statement being new, as far as we know, and extending Strauss’ Radial Lemma [19] to the space X and its N − dimensional version. Indeed, though later on we shall use only the compact embedding in dimension N = 2 presented below, we can prove an asymptotic result which is valid in any space dimension. Since we believe that this property is of independent interest, we present our result in the general case. For this, let us introduce the sets

X=uH1(RN):RN|u(x)|2ln(1+|x|)dx<

and

Xr=uX:u(x)=u(|x|).

Proposition 2.1

The following properties hold true:

  • X is compactly embedded in Ls(ℝ2), for all s ∈ [2, ∞).

  • There exists c ∈ ℝ such that for all u ∈ 𝓧r

    |u(x)|cuXrxN12ln(1+x)4.

Proof

The compact embedding for N = 2 is an application of the Riesz criterion (see [20, Theorem XIII.66]). Indeed, if S is a bounded subset of X, then S is bounded in Lq(ℝ2) for any q ∈ [2, ∞), as well. Moreover, for any R > 0 and any uS we have

{|x|>R}|u|pdxuL2p2(R2){|x|>R}u2dx1/2C{|x|>R}u2dx1/2

for some C > 0, and

{|x|>R}u2dx1ln(1+R){|x|>R}ln(1+|x|)u2dxCln(1+R)

for some C > 0. Finally, working as in [21, Theorem 9.16] we conclude.

As for the estimate in dimension N, let u ∈ 𝓧rCc (ℝN) and r > 0. We have

ln(1+r)u2rN1=12ln(1+r)u2rN11+r+2uurN1ln(1+r)+(N1)rN2u2ln(1+r)2uurN1ln(1+r).

Integrating from r to ∞ we obtain

ln(1+r)u2rN1BrC2uuρN1ln(1+ρ)dρ=BrC2uuln(1+|x|)dxCBrC|u|2ln(1+|x|)12BrC|Du|212CuXr.

Hence

|u(r)|CuXrxN12ln(1+x)4.

The conclusion follows by density. □

We finally introduce the "positive" and "negative" part of the reaction term, namely

f±(x,t)=f(x,t±) for a.e. xR2 and for all tR,

which are Carathéodory functions if f is, and F±(x,t)=0tf±(x,s)ds; moreover, we set

I±(u)=12R2|Du|2+a|u|2dx18πR2R2ln1|xy||u±(x)|2|u±(y)|2dxdyR2F±(x,u)dx

for all uX.

3 The Existence Theorem

We assume the following hypotheses on the reaction term f:

  • H(i)

    let f : ℝ2 × ℝ → ℝ be a Carathéodory function with f(x, 0) = 0 and f(x, ⋅) = f(|x|, ⋅) for a.e. x ∈ ℝ2. Moreover, there exist cLp(ℝ2) for 1 < p < 2, d > 0 and q ∈ (2, ∞) such that |f(x, t)| ≤ c(x) + d|t|q−1, for a.e. x ∈ ℝ2 and for all t ∈ ℝ;

  • H(ii)

    f(x, t) = o(|t|) as t → 0 uniformly for a.e. x ∈ ℝ2;

  • H(iii)

    lim|t|+F(x,t)t2=+ uniformly for a.e. x ∈ ℝ2;

  • H(iv)

    if σ(x, s) = f(x, s)s − 2F(x, s), then there exists 𝓜*L+1(ℝ2) such that σ(x, s) ≤ σ(x, t) + 𝓜*(x) for a.e. x ∈ ℝ2 and for all 0 ≤ st or ts ≤ 0;

  • H(v)

    there exists ũX such that

    limy+R2F(x,y2u~±(yx))dxy4lny=+.

Remark 3.1

  1. Condition H(iv) was introduced in [22] to overcome the necessity of using the Ambrosetti-Rabinowitz condition.

  2. Condition H(v) is trivially satisfied if f(x, t) = |t|p−2t or if F(x, t) ≥ c|t|ζ(x), where ζL+1(ℝ2) and > 4. The very last condition is generally a consequence of the usual Ambrosetti-Rabinowitz condition, which here should be assumed a priori, see [23].

We start proving

Proposition 3.1

If H(i) holds, then the functional I : X → ℝ is well-defined and of class C1 on X. The same is true for I±.

Proof

We do the proof for I, the ones for I± being completely analogous. From hypothesis H(i) we have

R2F(x,u)dxR2c(x)|u(x)|dx+dqR2|u(x)|qdxcLp(R2)uLp(R2)+dquLq(R2)q.(3.1)

From (3.1), (2.6) and (2.7) we have

|I(u)|12uH1(R2)2+14πR2ln(1+|x|)u2(x)R2u2(y)dxdy+C2uL83(R2)4+cLp(R2)uLp(R2)+dquLq(R2)q<,

for some constant C2 > 0, so the associated functional is well-defined.

Now we observe that the Gâteaux derivative of B(u2, u2) is

B(u2,uφ)=2R2R2ln1|xy||u(x)|2u(y)φ(y)dxdy(3.2)

for all φX, so the functional I is the sum of C1 terms and we have the desired regularity follows. □

Our purpose is to prove that both I+ and I satisfy the assumptions of the mountain pass theorem. While the geometric structure is somehow standard and is obtained exploiting H(i) and H(v), the compactness condition is the delicate part: the lack of the Ambrosetti-Rabinowitz condition makes the bound on Cerami sequences more complicated, and, indeed, by using H(iv) we obtain only a bound in H1(ℝ2). Thus we move to radial functions and use Strauss’ Lemma to exploit the compact embedding in Lq(ℝ2): thanks to the principle of symmetric criticality, a critical point for the functional constrained on the subset of radial functions is a free critical point, see [24]. This permits to recover the desired bound of Cerami sequences in X and finally prove that the Cerami condition holds.

Hence, from now on, we consider I : Xr → ℝ, where

Xr=uX:u(x)=u(|x|)

and we look for critical point for I|Xr. For the sake of simplicity we will continue to denote by I the functional I|Xr.

Now we are ready to prove that the (C)d-condition holds. In order to do that, we first give the following definition

Definition 2

We say that a sequence (un)nXr is a (C)d-sequence if

I(un)d and I(un)Xr1+unXr0 as n.

We say that I satisfies the (C)d-condition if any (C)d-sequence admits a convergent subsequence.

We prove that, under suitable hypotheses, a (C)d-sequence in Xr is bounded in Hr1(ℝ2).

Lemma 3.1

Suppose hypotheses H(i), H(iii) and H(iv) hold and let (un)nXr be a (C)d-sequence for I+ (I respectively). Then (un)n is bounded in Hr1(ℝ2).

Proof

We do the proof for the I+, for I being analogous.

Let (un)nXr be a (C)d-sequence. In particular,

|I+(un)|M1for someM1>0and everyn1,(3.3)

1+unXrI+(un)0inXrasn.(3.4)

We recall that for any vXr we have

I+(un)(v)=R2DunDv+aunvdx12πR2R2ln1|xy||un+(x)|2un+(y)v(y)dxdyR2f+(x,un)vdx.

From (3.4) we have

|I+(un)(h)|εnhXr1+hXr(3.5)

for all hXr(ℝ2), where εn → 0 as n → ∞. We choose h = −unXr and we obtain

|I+(un)(un)|εnunXr1+unXrεn,

so that

R2|Dun|)+a|un|2dx=unHr1(R2)2εn,

which means that

un0 in Hr1(R2) as n.(3.6)

From (3.3) we have

12unHr1(R2)214B(un+)2,(un+)2R2F+(x,un)dxM1(3.7)

so that

unHr1(R2)212B(un+)2,(un+)22R2F+(x,un)dxM2 for some M2>0

and using (3.6) we obtain

un+Hr1(R2)212B(un+)2,(un+)22R2F(x,un+)dxM2.(3.8)

We assume by contradiction that (un)n is unbounded in Hr1(ℝ2), then by passing to a subsequence, if necessary, we assume that un+Hr1(R2) as n → ∞.

We set vn=un+un+Hr1(R2), n ≥ 1, so we may assume that, by Strauss’ Theorem,

vnv in Hr1(R2) and vnv in Ls(R2),s(2,),v0.(3.9)

To reach our goal we show that both v ≠ 0 and v = 0 lead to a contradiction. We start with the case v ≠ 0.

We define the set Z(v) = {x ∈ ℝ2 : v(x) = 0}; then meas(ℝ2Z(v)) > 0 and un+(x) → ∞ as n → ∞ for a.e. x ∈ ℝ2Z(v). By H(iii) we have

F(x,un+)un+Hr1(R2)2=F(x,un+)|un+|2vn2 for a.e. xR2Z(v)

and by Fatou’s Lemma we obtain

R2F(x,un+)un+Hr1(R2)2dx as n.(3.10)

But from (3.7) we have

12+14B(un+)2,(un+)2un+Hr1(R2)2+R2F(x,un+)un+Hr1(R2)2dxM1un+Hr1(R2)2

so that

lim supnR2F(x,un+)un+Hr1(R2)2dxM3 for some M3>0.(3.11)

Compairing (3.10) and (3.11) we reach a contradiction.

Now we consider the case v = 0. For every n ∈ ℕ we define the continuous function yn : [0, 1] → ℝ as

yn(t)=I(tun+) for all n1 and all t[0,1],

and let tn ∈ [0, 1] be such that

yn(tn)=maxyn(t):t[0,1].(3.12)

For λ > 0, let wn=(2λ)12vnHr1(R2). Then wn → 0 in Lp(ℝ2) by (3.9). By H(i) and the Krasnoselskii’s Theorem (see [25, Theorem 2.75]), we have

R2F(x,wn)dx0 as n.(3.13)

Since un+Hr1(R2) as n → ∞, we can find n0 ≥ 1 such that (2λ)12un+Hr1(R2) ∈ (0, 1) for all nn0. Then, by (3.12),

y(tn)y(2λ)12un+Hr1(R2) for all nn0.

Hence, by (2.2) and (2.6) we get

I(tnun+)I((2λ)12vn)=λλ22πR2R2ln1|xy||vn(x)|2|vn(y)|2dxdyR2F(x,wn)dx=λ+λ22πR2R2ln1+|xy||vn(x)|2|vn(y)|2dxdyλ22πR2R2ln1+1|xy||vn(x)|2|vn(y)|2dxdyR2F(x,wn)dxλλ22πR2R2ln1+1|xy||vn(x)|2|vn(y)|2dxdyR2F(x,wn)dxλCvnL83(R2)4R2F(x,wn)dx.

Now we observe that vnL83(R2) → 0 as n → ∞ by Strauss’ Theorem, and by (3.13) we have

I(tnun+)λ+o(1)λ2.

Being λ > 0 arbitrary, we finally find

I(tnun+) as n.(3.14)

Since 0 ≤ tnun+un+ for all n ≥ 1, from H(iv) we have

R2σ(x,tnun+)dxR2σ(x,un+)dx+ML1(R2) for all n1.(3.15)

Moreover, by (3.3) and (3.6) there exists M4 > 0 such that

I(un)=I+(un)+o(1)M4 for all n1.(3.16)

Thus, (3.14) and (3.16) imply that tn ∈ (0, 1) for all nn1 ≥ 1. Hence, by (3.12) we obtain that

0=tnddtI(tun+)t=tn=I(tnun+)(tnun+)=R2|D(tnun+)|2+a|tnun+|2dx12πR2R2ln1|xy||tnun+(x)|2|tnun+(y)|2dxdyR2f(x,tnun+)(tnun+)dx=tnun+Hr1(R2)2B(tnun+)2,(tnun+)2R2f(x,tnun+)(tnun+)dx(3.17)

for all n ≥ 1, that is

R2f(x,tnun+)(tnun+)dx=tnun+Hr1(R2)2B(tnun+)2,(tnun+)2(3.18)

for all nn1. Replacing (3.18) in (3.15), we obtain

tnun+Hr1(R2)2B(tnun+)2,(tnun+)22R2F(x,tnun+)dxR2σ(x,un+)dx+ML1(R2)

for all nn1.

Again by H(iv)

f(x,tnun+)(tnun+)2F(x,tnun+)f(x,un+)(un+)2F(x,un+)+M,

so that

2R2F(x,tnun+)dxR2f(x,un+)(un+)f(x,tnun+)(tnun+)dx2R2F(x,un+)dx+R2Mdx.

Using (3.17) the previous inequality reads as

2R2F(x,tnun+)dxI(un+)(un+)+R2|Dun+|2+a|un+|2dxB(un+)2,(un+)2R2|D(tnun+)|2+a|tnun+|2dx+B(tnun+)2,(tnun+)22R2F(x,un+)dx+R2Mdx

and from (3.4)

2R2F(x,tnun+)dxun+Hr1(R2)2B(un+)2,(un+)2tnun+Hr1(R2)2+B(tnun+)2,(tnun+)22R2F(x,un+)dx+R2Mdx+o(1).(3.19)

Now

2I(tnun+)=tnun+Hr1(R2)212B(tnun+)2,(tnun+)22R2F(x,tnun+)dx.(3.20)

Thus, replacing (3.19) in (3.20), since B(tnun+)2,(tnun+)2B(un+)2,(un+)2 being tn < 1, we have

2I(tnun+)tnun+Hr1(R2)212B(tnun+)2,(tnun+)2+un+Hr1(R2)2B(un+)2,(un+)2tnun+Hr1(R2)2+B(tnun+)2,(tnun+)22R2F(x,un+)dx+R2Mdx+o(1)un+Hr1(R2)212B(un+)2,(un+)22R2F(x,un+)dx+R2Mdx+o(1).

This last formula, together with (3.14), tells us that

un+Hr1(R2)212B(un+)2,(un+)22R2F(x,un+)dx as n.(3.21)

Comparing (3.8) and (3.21) we reach a contradiction.

So (un+)n is bounded in Hr1(ℝ2).□

We use this result to finally prove the Cerami compactness condition.

Proposition 3.2

Let (un)nXr be a (C)d-sequence for I+ (I respectively), with d > 0. Then, up to a subsequence,

unuinXrasn

for some nonzero critical point uXr of I+ (I respectively). In particular, the (C)d-condition holds.

Proof

From Lemma 3.1, we know that, up to a subsequence,

unu in Hr1(R2) as n.

Now, we show that

lim infnsupxR2Br(x)un2(y)dy>0(3.22)

for every r > 0. We argue by contradiction, so we suppose that (3.22) is false. Since (un)n is bounded in Hr1(ℝ2), by [26, Lemma I.1], we have that un → 0 as n → ∞ in Ls(ℝ2) for every s ∈ (2, ∞).

By our assumptions

I+(un)un=unHr1(R2)2+12πR2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdy12πR2R2ln1+1|xy||un+(x)|2|un+(y)|2dxdyR2f+(x,un)undx,

and so

unHr1(R2)2+12πR2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdy=I+(un)un+12πR2R2ln1+1|xy||un+(x)|2|un+(y)|2dxdy+R2f+(x,un)undx.

By H(i) and (2.6) we have

unHr1(R2)2+12πR2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdyI+(un)un+12πR2R2ln1+1|xy||un+(x)|2|un+(y)|2dxdy+cLp(R2)unLp(R2)+dun+Lq(R2)qI+(un)un+Cun+L83(R2)4+cLp(R2)unLp(R2)+dun+Lq(R2)q.(3.23)

Since un → 0 in Ls(ℝ2) for s ∈ (2, ∞), and p′ > 2, we have

unHr1(R2)0,

and then

un±Hr1(R2)0,

and also

R2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdy0.

Hence,

I+(un)=12unHr1(R2)2+18πR2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdy18πR2R2ln1+1|xy||un+(x)|2|un+(y)|2dxdyR2F+(x,un)dx12unHr1(R2)2+18πR2R2ln(1+|xy|)|un+(x)|2|un+(y)|2dxdy+cLp(R2)unLp(R2)+dun+Lq(R2)q0,

but I+(un) → d > 0, so we reach a contradiction. Thus (3.22) holds.

This means that vanishing (see [27]) cannot occur. Moreover, since we use radial functions, dichotomy cannot take place, either. Hence, we can conclude that u ≠ 0. By [17, Lemma 2.1] we can conclude that

(un)nis bounded inXr.

Then we can assume that

unu in Xr,

with u ≠ 0, and by Proposition 2.1 we also have that unu in Ls(ℝ2) for every s ∈ [2, ∞).

Finally, we claim that unu in Xr. In (3.5), we take h = unu and, using (2.2), we have

I+(un)(unu)=R2|Dun|2+a|un|2dxR2DunDu+aunudx12πR2R2ln1+1|xy||un+(x)|2un+(y)(unu)(y)dxdy+12πR2R2ln1+|xy||un+(x)|2un+(y)(unu)(y)dxdyR2f+(x,un)(unu)dx.

Hence,

unHr1(R2)2(un|u)=I+(un)(unu)+12πR2R2ln1+1|xy||un+(x)|2un+(y)(unu)(y)dxdy12πR2R2ln1+|xy||un+(x)|2un+(y)(unu)(y)dxdy+R2f+(x,un)(unu)dx.(3.24)

By Theorem 2.1 and the Hölder inequality, H(i) and (2.3) we have

R2R2ln1+1|xy||un+(x)|2|un+(y)(unu)(y)|dxdyCun+Ls(R2)3unuLs(R2)0

with s ∈ (2, ∞),

R2R2ln1+|xy||un+(x)|2|un+(y)(unu)(y)|dxdy=R2ln1+|xy||un+(x)|2R2|un(y)(unu)(y)|dxdy0

and

R2f+(x,un)(unu)dxcLp(R2)un+Lp(R2)unuLp(R2)+dun+Lq(R2)q1unLq(R2)0

Hence, from (3.24)

unu in Xr

and so the (C)d-condition hold. □

Now we are ready to produce two nontrivial solutions of (P) using the Mountain Pass Theorem.

Theorem 3.1

Under hypotheses H(i) - H(v), problem (P) has two nontrivial constant sign solutions.

Proof

We do the proof for the functional I+; for I it is analogous. First, I+(0) = 0. By Proposition 3.1 we have the regularity of I+ and by Proposition 3.2 the (C)d-condition is verified.

Now, take ũ as in H(v), t > 0 and, following [10], we set ut(x) = t2ũ(tx). Then

I+(ut)=t42R2|Du~|2+au~2dx+t48πR2R2ln(|xy|)|u~+(x)|2|u~+(y)|2dxdyt4lnt8πR2|u~+(x)|2dx2R2F+(x,t2u~(tx))dx,

and by H(v),

limt+I+(ut)=.

In order to complete the proof it only remains to show that I+(u) ≥ α ≥ 0 with ∥u∥ = r, for some r > 0.

By H(i) we have |f(x, t)| ≤ c(x) + d|t|q–1 and then

|F(x,t)|c(x)|t|+dq|t|q a.e. xR2 and for all tR.(3.25)

Hypothesis H(ii) says that for all ε > 0 there exists δ = δ(ε) > 0 such that for |t| < δ we have

|f(x,t)||t|ε a.e. xR2,

thus

|F(x,t)|ε2t2 a.e. xR2 and |t|δ.(3.26)

On the other hand, when |t| ≥ δ

|F(x,t)|c(x)|t|δq1δq1+dq|t|qc(x)δq1+dq|t|q.

Combining the inequality above with (3.26) we get that for a.e. x ∈ ℝ2 and for all t ∈ ℝ

|F(x,t)|c(x)δq1+dq|t|q+ε2t2.

Hence,

R2F(x,u)dxR2c(x)δ|u|q+dqR2|u|q+ε2R2|u|21δcLp(R2)uLqp(R2)q+dquLq(R2)q+ε2uL2(R2)2

that is

R2F(x,u)dxεC1tHr1(R2)2+CδC2tHr1(R2)q.

We use this estimates on functional I+:

I+(u)=12uHr1(R2)2+18πR2R2ln1+|xy||u+(x)|2|u+(y)|2dxdy18πR2R2ln1+1|xy||u+(x)|2|u+(y)|2dxdyR2F+(x,u)dx12uHr1(R2)2+14B1(u+)2,(u+)214B2(u+)2,(u+)2εu+L2(R2)2Bδu+Lq(R2)q

and by the Hardy-Littlewood-Sobolev inequality

I+(u)12uHr1(R2)2+14B1(u+)2,(u+)2C3u+Hr1(R2)4εC1u+Hr1(R2)2BδC2u+Hr1(R2)q=12uHr1(R2)2+14B1(u+)2,(u+)2C3u+Hr1(R2)4εC1u+Hr1(R2)2BδC2u+Hr1(R2)q.

Since u+Hr1(R2)uHr1(R2) we get

I+(u)12εC1uHr1(R2)2C3+BδC2uHr1(R2)q4uHr1(R2)4.

Choosing ε0,12C1 and uHr1(R2)=r, we have

I+(u)C4uHr1(R2)2C3+BδC2uq4uHr1(R2)4=uHr1(R2)2C4C3+BδC2rq4r2,

for some C4 > 0. We take r such that C4 – (C3 + BδC2rq–4) > 0 and so

I+(u)α0,

thus we have the Mountain Pass geometry, and we can apply [25, Theorem 5.40].

Hence u+ satisfies Δu+au12πln1|x||u+|2u+=f(x,u+). Now, multipliyng by u, we get

R2|(Du(x)|2+a|u(x)|2dx=0,

thus u ≡ 0, then u ≥ 0 and it is a nontrivial solution of problem (P). Working with I we find another nontrivial solution of (P) which is non positive in ℝ2. □

Acknowledgement

Dimitri Mugnai is a member of GNAMPA and is supported by the MIUR National Research Project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and by the FFABR “Fondo per il finanziamento delle attività base di ricerca” 2017.

References

  • [1]

    P. Choquard and J. Stubbe, The one-dimensional Schrödinger--Newton equations, Lett. Math. Phys. 81, (2007), 177–184. CrossrefGoogle Scholar

  • [2]

    R. Harrison, I. Moroz and K.P. Tod, A numerical study of the Schrödinger - Newton equation, Nonlinearity 16, (2003), 101–122. CrossrefGoogle Scholar

  • [3]

    P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model - An ODE approach, Differential Integral Equations 21 (2008), 665–679. Google Scholar

  • [4]

    V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topolog. Meth. Nonlin. Analysis 11, (1998), 283–293. CrossrefGoogle Scholar

  • [5]

    T. D’Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A 134A, (2004), 893–906. Google Scholar

  • [6]

    T. D’Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud. 4, (2004), no. 3, 307–322. Google Scholar

  • [7]

    C.O. Alves, G.M. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal. 5 (2016), 331–345. Web of ScienceGoogle Scholar

  • [8]

    A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris, Ser. I 342, (2006), 453–458. CrossrefGoogle Scholar

  • [9]

    A. Ambrosetti and E. Colorado, Standing Waves of Some Coupled Nonlinear Schrödinger Equations, J. Lond. Math. Soc. (2) 75, (2007), no. 1, 67–82. Google Scholar

  • [10]

    M. Du and T. Weth, Ground states and high energy solution of the planar Schrödinger - Poisson system, Nonlinearity 30, (2017), 3492–3515. CrossrefGoogle Scholar

  • [11]

    Y. Li, F. Li and J. Shi, Existence and multiplicity of positive solutions to Schrödinger - Poisson type systems with critical nonlocal term, J. Calc. Var. 56, (2017), 56–134. Google Scholar

  • [12]

    D. Ruiz, The Schrödinger - Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237, (2006), 655–674. CrossrefGoogle Scholar

  • [13]

    J. Seok, Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal. 8, (2019), 1083–1098. Web of ScienceGoogle Scholar

  • [14]

    G. Singh, Nonlocal perturbations of the fractional Choquard equation, Adv. Nonlinear Anal. 8, (2019), 694–706. Web of ScienceGoogle Scholar

  • [15]

    J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, available at arXiv:0807.4959v1, 2008. Google Scholar

  • [16]

    D. Bonheure, S. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal. 272, (2017), 5255–5281. CrossrefGoogle Scholar

  • [17]

    S. Cingolani and T. Weth, On the planar Schrödinger-Poisson system, Ann. H. Poincaré - Anal. Non Linéaire 33, (2016), 169–197. CrossrefGoogle Scholar

  • [18]

    E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 2nd ed., vol. 14, AMS, Providence, Rhode Island, 2001. Google Scholar

  • [19]

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

  • [20]

    M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Academic Press, New York - London, 1978. Google Scholar

  • [21]

    H. Brezis, Functional Analysis, Sobolev spaces and partial differential equations, Springer, New York, 2011. Google Scholar

  • [22]

    D. Mugnai and N.S. Papageorgiu, Wang’s multiplicity result for superlinear (pq)-equations without Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc. 366, (2014), 4919–4937. Google Scholar

  • [23]

    D. Mugnai, Addendum to: Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem, NoDEA. Nonlinear Differential Equations Appl. 11 (2004), no. 3, 379-391, and a comment on the generalized Ambrosetti-Rabinowitz condition, Nonlinear Differ. Equ. Appl. 19, (2012), 299–301. Google Scholar

  • [24]

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

  • [25]

    D. Motreanu, V. V. Motreanu and N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. Google Scholar

  • [26]

    P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1, (1984), no. 4, 223–283. CrossrefGoogle Scholar

  • [27]

    P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1, (1984), no. 2, 109–145. CrossrefGoogle Scholar

About the article

Received: 2018-12-28

Accepted: 2019-03-16

Published Online: 2019-08-29

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 850–865, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0028.

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© 2020 F. Bernini and D. Mugnai, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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