Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane

Luca Battaglia; Jean Van Schaftingen

doi:10.1515/ans-2016-0038

Article Open Access

Existence of Groundstates for a Class of Nonlinear Choquard Equations in the Plane

Luca Battaglia and Jean Van Schaftingen

Published/Copyright: January 11, 2017

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From the journal Advanced Nonlinear Studies Volume 17 Issue 3

Abstract

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equations

- Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) in ⁢ ℝ 2 ,

where Iα is the Riesz potential of order α on the plane ℝ2 under general nontriviality, growth and subcriticality on the nonlinearity F∈C1⁢(ℝ,ℝ).

Keywords: Choquard Equation; Semilinear Elliptic Problem; Variational Method

MSC 2010: 35J91; 35J20

1 Introduction

We are interested in the existence of nontrivial solutions to the class of nonlinear Choquard equations of the form

(P) - Δ ⁢ u + u = ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) in ⁢ ℝ N ,

where N∈ℕ={1,2,…}, Δ is the standard Laplacian operator on the Euclidean space ℝN, Iα:ℝN→ℝ is the Riesz potential of order α∈(0,N) defined for each x∈ℝN∖{0} by

I α ⁢ ( x ) = Γ ⁢ ( N - α 2 ) Γ ⁢ ( α 2 ) ⁢ π N 2 ⁢ 2 α ⁢ | x | N - α ,

and a nonlinearity is described by the function F∈C1⁢(ℝ,ℝ). Solutions of equation (P) are at least formally critical points of the energy functional defined for a function u:ℝN→ℝ by

(1.1) ℐ ⁢ ( u ) = 1 2 ⁢ ∫ ℝ N ( | ∇ ⁡ u | 2 + | u | 2 ) - 1 2 ⁢ ∫ ℝ N ( I α * F ⁢ ( u ) ) ⁢ F ⁢ ( u ) .

In the particular case where F⁢(s)=s2/2 for each s∈ℝ, solutions to the Choquard equation (P) are standing waves solutions of the Hartree equation. In particular, when N=3 and α=2, problem (P) has arisen in various fields of physics: quantum mechanics [20], one-component plasma [11] and self-gravitating matter [15]. In these cases, many existence results have been obtained in literature, with both variational [11, 13, 14] and ordinary differential equations techniques [21, 15, 6] (see also the review [18]). Such methods extend also to the case of homogeneous nonlinearities [16].

When the nonlinearity F is not any more homogeneous, it has been shown that the Choquard equation (P) has a nontrivial solution if the nonlinearity F satisfies the following hypotheses [17]:

There exists s0∈ℝ such that F⁢(s0)≠0.
There exists C>0 such that

| F ′ ⁢ ( s ) | ≤ C ⁢ ( | s | α N + | s | α + 2 N - 2 )

for every s>0.
There holds

lim s → 0 ⁡ F ⁢ ( s ) / | s | 1 + α N = 0 = lim s → 0 ⁡ F ⁢ ( s ) / | s | N + α N - 2 .

The solution u is a groundstate, in the sense that u minimizes the value of the functional ℐ among all nontrivial solutions. Assumptions (F0′), (F1′) and (F2′) are rather mild and reasonable and are “almost necessary” in the sense of Berestycki and Lions [4]: the nontriviality of the nonlinearity condition (F0′) is clearly necessary to have a nontrivial solution; assumption (F1′) secures a proper variational formulation of problem (P) by ensuring that the energy functional ℐ is well-defined on the natural Sobolev space H1⁢(ℝN) through the Hardy–Littlewood–Sobolev and Sobolev inequalities; condition (F2′) is a sort of subcriticality condition with respect to the limiting-case embeddings. The analysis by a Pohožaev identity shows that assumptions (F1′) and (F2′) are necessary in the homogeneous case F⁢(s)=sp/p (see [16]).

The results in [17] can thus be seen as a counterpart for Choquard-type equations of the result of Berestycki and Lions [4] which give similar “almost necessary” conditions for the existence of a groundstate to the equation

(1.2) - Δ ⁢ u + u = G ′ ⁢ ( u ) in ⁢ ℝ N .

The latter equation can be at least formally obtained by (P) by passing to the limit as α→0 and setting G=F2/2.

Whereas the above-mentioned almost necessary conditions for existence of the Choquard equation (P) and for the scalar field equation (1.2) have been obtained in higher dimensions N≥3, the latter result has been extended to the two-dimensional case [3] under the following assumptions:

There exists s0∈ℝ such that G⁢(s0)>|s0|22.
For every θ>0 there exists C=Cθ>0 such that |G′⁢(s)|≤Cθ⁢min⁡{1,s2}⁢eθ⁢|s|2 for every s>0.
lim s → 0 ⁡ G ⁢ ( s ) / | s | 2 < 1 2 .

This raises naturally the question whether there is a similar existence result for the Choquard equation (P) in the planar case.

In the present work, we provide a general existence result for groundstate solutions of problem (P) in the planar case N=2, which is a two-dimensional counterpart of [17] and a counterpart for the Choquard equation of [3]. The counterparts of (F0′), (F1′), (F2′) we need are the following:

There exists s0∈ℝ such that F⁢(s0)≠0.
For every θ>0 there exists C=Cθ>0 such that |F′⁢(s)|≤Cθ⁢min⁡{1,|s|α2}⁢eθ⁢|s|2 for every s>0.
lim s → 0 ⁡ F ⁢ ( s ) / | s | 1 + α 2 = 0 .

Our main result reads as follows.

Theorem 1.1.

If N=2 and F∈C1⁢(R,R) satisfies conditions (F0), (F1) and (F2), then problem (P) has a groundstate solution u∈H1⁢(R2)∖{0}, namely the function u solves (P) and

ℐ ⁢ ( u ) = c := inf ⁡ { ℐ ⁢ ( v ) ∣ v ∈ H 1 ⁢ ( ℝ 2 ) ∖ { 0 } ⁢ is a solution of (P) } .

Let us discuss the assumptions of Theorem 1.1. As above, assumption (F0) is necessary for the existence of a nontrivial solution. As before, condition (F1) ensures the needed well-definedness of the energy functional on the whole space H1⁢(ℝ2). It has a different shape, because in dimension N=2 the critical nonlinearity for Sobolev embeddings is not anymore a power but rather an exponential-type nonlinearity. More precisely, the integral of min⁡{1,u2}⁢eθ⁢|u|2 on ℝ2 is uniformly controlled on H01⁢(B1) if and only if θ⁢∫B1|∇⁡u|2≤4⁢π (see [19, 1]); this is why the parameter θ>0 appears in condition (F1). It will appear that condition (F1) is strong enough at infinity. Indeed, by integrating the function F′, it is possible to observe that

(1.3) lim | s | → ∞ ⁡ | F ⁢ ( s ) | + | F ′ ⁢ ( s ) | ⁢ | s | e θ ⁢ | s | 2 = 0

for every θ>0. A subcriticality condition still needs to be imposed around 0; that is the goal of the subcriticality condition (F2).

Assumptions (F0), (F1) and (F2) are still almost necessary: in the case F⁢(s)=spp, they are satisfied if and only if p>1+α2, and for p≤1+α2 the Choquard equation (P) has no nontrivial solutions (see [16]).

In order to prove Theorem 1.1 the constraint minimization technique used in [4, 3] for the local problem (1.2) does not seem to work, as it introduces a Lagrange multiplier that cannot be absorbed through a suitable dilation because of the presence of three different scalings in the equation and because of the nonhomogeneity of the nonlinearity.

Following [17], we use a mountain pass construction. We start by constructing a Palais–Smale sequence for the mountain pass level

(1.4) b := inf γ ∈ Γ ⁡ sup t ∈ [ 0 , 1 ] ⁡ ℐ ⁢ ( γ ⁢ ( t ) ) ,

where

Γ := { γ ∈ C ⁢ ( [ 0 , 1 ] , H 1 ⁢ ( ℝ 2 ) ) ∣ γ ⁢ ( 0 ) = 0 ⁢ and ⁢ ℐ ⁢ ( γ ⁢ ( 1 ) ) < 0 } .

To avoid relying on an Ambrosetti–Rabinowitz superlinearity condition, we use a scaling trick due to Jeanjean [9], which allows us to construct a Pohožaev–Palais–Smale sequence (Proposition 3.1), namely a Palais–Smale sequence which, in addition, satisfies asymptotically the Pohožaev identity

(1.5) 𝒫 ⁢ ( u ) := ∫ ℝ 2 | u | 2 - ( 1 + α 2 ) ⁢ ∫ ℝ 2 ( I α * F ⁢ ( u ) ) ⁢ F ⁢ ( u ) = 0 .

Such a condition will imply quite directly the boundedness of the sequence in the space H1⁢(ℝ2) and it will be crucial to get the convergence, hence the existence of a solution (Proposition 4.1).

We are left with showing that the solution u is actually a groundstate. To prove this, we first show that the solution u itself satisfies the Pohožaev identity (Proposition 5.2). This will follow by simple calculations once a suitable regularity result is established (Proposition 5.1); this regularity turns out to be easier to prove from assumption (F1) than in the higher-dimensional case [17] where a suitable nonlocal Brezis–Kato regularity had to be proved. The last ingredient that we need is an optimal path γv∈Γ associated to any solution v of (P). The construction of such paths (Proposition 5.3) is inspired by [10, 17] but it is more delicate in our two-dimensional case than in the higher dimensions N≥3, because dilations t↦v(⋅/t)∈H1(ℝN) are not anymore continuous at t=0 when N=2.

The content of the paper is the following: In Section 2 we provide some technical preliminaries. In Section 3 we construct the Pohožaev–Palais–Smale sequence. In Section 4 we show that the sequence converges to a solution of (P). In Section 5 we prove that u is actually a groundstate. In the last section we also state some qualitative result concerning the solutions, which can be proved directly following [17].

2 Preliminaries

In this section, we present some preliminary results which we will need throughout the rest of this paper. We start by reformulating in a more convenient form the Moser–Trudinger inequality of Adachi and Tanaka [1]. This quantitative estimate will play a crucial role throughout the paper.

Proposition 2.1 (Moser–Trudinger inequality).

For any β∈(0,4⁢π) there exists C=Cβ>0 such that for every u∈H1⁢(R2) satisfying

∫ ℝ 2 | ∇ ⁡ u | 2 ≤ 1

one has

∫ ℝ 2 min ⁡ { 1 , | u | 2 } ⁢ e β ⁢ | u | 2 ≤ C β ⁢ ∫ ℝ 2 | u | 2 .

Proof.

The result follows from the fact [1, Theorem 0.1] that under the conditions of the theorem one has

∫ ℝ 2 ( e β ⁢ | u | 2 - 1 ) ≤ C ⁢ ∫ ℝ 2 | u | 2 .

Together with the elementary inequalities valid for every s≥0, this yields

( 1 - 1 e ) ⁢ max ⁡ { 1 , s } ⁢ e s ≤ e s - 1 ≤ max ⁡ { 1 , s } ⁢ e s ,

as desired. ∎

We will also use the Hardy–Littlewood–Sobolev inequality to deal with the nonlocal term (see for example [12, Theorem 4.3]).

Proposition 2.2 (Hardy–Littlewood–Sobolev inequality).

For any p∈[1,2α) and f∈Lp⁢(R2) there exists a constant C=Cα,p such that

∥ I α ∗ f ∥ L 2 ⁢ p 2 - α ⁢ p ⁢ ( ℝ 2 ) ≤ C ⁢ ∥ f ∥ L p ⁢ ( ℝ 2 ) .

Combining the last two results with the assumption on F and (1.3), we deduce that the energy functional is well-defined on H1⁢(ℝ2).

Proposition 2.3.

If F satisfies (F1), then the energy functional I defined by (1.1) is well-defined and continuously differentiable.

Proof.

We first consider the superposition map ℰ defined for each u∈H1⁢(ℝ2) and x∈ℝ2 by ℰ⁢(u)⁢(x)=F′⁢(u⁢(x)). We claim that ℰ is well-defined and continuous as a map from H1⁢(ℝ2) to L4/α⁢(ℝ2). Indeed by assumption (F1), for every θ>0 and s∈ℝ we have

| F ′ ⁢ ( s ) | 4 α ≤ C θ 4 α ⁢ min ⁡ { 1 , s 2 } ⁢ e 4 ⁢ θ α ⁢ | s | 2 .

If u∈H1⁢(ℝ2), we take θ>0 such that ∫ℝ2|∇⁡u|2<α⁢π2⁢θ. We observe that

| F ′ ⁢ ( u ) | 4 α ≤ C θ 4 α ⁢ min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2

on ℝ2, where the right-hand side is integrable in view of the Moser–Trudinger inequality (Proposition 2.1). Therefore, the map ℰ:H1⁢(ℝ2)→L4/α⁢(ℝ2) is well-defined.

If now the sequence (un)n∈ℕ converges to u in H1⁢(ℝ2), then we can assume without loss of generality that

ν := sup n ∈ ℕ ⁡ ∫ ℝ 2 | ∇ ⁡ u | 2 < α ⁢ π 2 ⁢ θ

and that (un)n∈ℕ converges to u almost everywhere. Then, for some constant C≥0, we have

C ⁢ ( min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2 + min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2 ) - | F ′ ⁢ ( u ) - F ′ ⁢ ( u n ) | 4 α ≥ 0

for each n∈ℕ almost everywhere in ℝ2. By Fatou’s lemma we get

lim inf n → ∞ ⁡ ∫ ℝ 2 C ⁢ ( min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2 + min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2 ) - | F ′ ⁢ ( u ) - F ′ ⁢ ( u n ) | 4 α ≥ 2 ⁢ C ⁢ ∫ ℝ 2 min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2 ,

and therefore

lim sup n → ∞ ⁡ ∫ ℝ 2 | F ′ ⁢ ( u ) - F ′ ⁢ ( u n ) | 4 α ≤ C ⁢ lim sup n → ∞ ⁡ ∫ ℝ 2 min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2 - min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2 .

If we consider the set Anλ={x∈ℝ2∣|un⁢(x)|≥λ}, we have by Lebesgue’s dominated convergence theorem

lim sup n → ∞ ⁡ ∫ ℝ 2 ∖ A n λ min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2

≤ lim sup n → ∞ ⁡ ∫ ℝ 2 ∖ A n λ ( min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2 - min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ min ⁡ ( | u | 2 , λ 2 ) ) + ∫ ℝ 2 min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2

≤ ∫ ℝ 2 min ⁡ { 1 , | u | 2 } ⁢ e 4 ⁢ θ α ⁢ | u | 2

for every λ>0. On the other hand, we have by the Cauchy–Schwarz inequality, the Chebyshev inequality and the Moser–Trudinger inequality (Proposition 2.1)

∫ A n λ min ⁡ { 1 , | u n | 2 } ⁢ e 4 ⁢ θ α ⁢ | u n | 2 ≤ | A n λ | 1 2 ⁢ ( ∫ A n λ min ⁡ { 1 , | u n | 2 } ⁢ e 8 ⁢ θ α ⁢ | u n | 2 ) 1 2 ≤ C λ ⁢ ∫ ℝ 2 | u n | 2 .

This allows us to conclude that the map ℰ:H1⁢(ℝ2)→L4/α⁢(ℝ2) is continuous.

We now consider the map ℱ:H1⁢(ℝ2)→L4/(2+α)⁢(ℝ2) defined for each u∈H1⁢(ℝ2) by ℱ⁢(u)=F∘u. We observe that for every s∈ℝ we have

F ⁢ ( s ) = ∫ 0 1 F ′ ⁢ ( τ ⁢ s ) ⁢ s ⁢ d τ ,

and thus for almost every x∈ℝ2 we have

F ⁢ ( u ⁢ ( x ) ) = ∫ 0 1 F ′ ⁢ ( τ ⁢ u ⁢ ( x ) ) ⁢ u ⁢ ( x ) ⁢ d τ .

Thus it follows from the first part of the proof that ℱ is well-defined from H1⁢(ℝ2) to L4/(2+α)⁢(ℝ2).

For the differentiability we consider a sequence (un)n∈ℕ converging strongly to u in H1⁢(ℝ2). We observe that

ℱ ⁢ ( u n ) - ℱ ⁢ ( u ) - ℰ ⁢ ( u ) ⁢ ( u n - u ) = ∫ 0 1 ( ℰ ⁢ ( ( 1 - τ ) ⁢ u + τ ⁢ ( u n ) ) - ℰ ⁢ ( u ) ) ⁢ ( u n - u ) ⁢ d τ

for each n∈ℕ, and thus by Hölder’s inequality

∥ ℱ ⁢ ( u n ) - ℱ ⁢ ( u ) - ℰ ⁢ ( u ) ⁢ ( u n - u ) ∥ L 4 / ( 2 + α ) ≤ ∫ 0 1 ∥ ℰ ⁢ ( ( 1 - τ ) ⁢ u + τ ⁢ ( u n ) ) - ℰ ⁢ ( u ) ∥ L 4 / α ⁢ ∥ u n - u ∥ L 2 .

By the convergence of the sequence (un)n∈ℕ and the continuity of the functional ℰ, it follows that

∥ ℱ ⁢ ( u n ) - ℱ ⁢ ( u ) - ℰ ⁢ ( u n ) ⁢ ( u n - u ) ∥ L 4 / ( 2 + α ) = o ⁢ ( ∥ u n - u ∥ L 2 ) as ⁢ n → ∞ ,

that is, ℰ represents the Fréchet differential of the functional ℱ. Since ℰ is continuous, it follows that ℱ is of class C1.

Finally, we consider the quadratic form 𝒬 defined for f∈L4/(2+α) by

𝒬 ⁢ ( f ) = ∫ ℝ 2 ( I α ∗ f ) ⁢ f .

By the Hardy–Littlewood–Sobolev inequality (Proposition 2.2), the quadratic form 𝒬 is bounded on bounded sets of the space L4/(2+α)⁢(ℝ2). This implies that 𝒬 is continuously differentiable and thus the functional

u ∈ H 1 ⁢ ( ℝ 2 ) ↦ 𝒬 ⁢ ( ℱ ⁢ ( u ) , ℱ ⁢ ( u ) ) = ∫ ℝ 2 ( I α ∗ F ⁢ ( u ) ) ⁢ F ⁢ ( u )

is continuously differentiable. By the smoothness of the norm on a Hilbert space, we conclude that the functional ℐ is continuously differentiable. ∎

Finally, we will use the following improvement of Proposition 2.2 when one has some more Lp integrability.

Proposition 2.4.

For any p∈[1,2α), q∈(2α,+∞) and f∈Lp⁢(R2)∩Lq⁢(R2) there exists C=Cα,p,q such that

∥ I α ∗ f ∥ L ∞ ⁢ ( ℝ 2 ) ≤ C ⁢ ( ∥ f ∥ L p ⁢ ( ℝ 2 ) + ∥ f ∥ L q ⁢ ( ℝ 2 ) ) .

Proof.

The result is classical. We give its short proof for the convenience of the reader. By choosing p, q in those ranges we have (2-α)⁢qq-1<2<(2-α)⁢pp-1. Therefore, through splitting the integral and by the Hölder inequality, we get

| I α * f ⁢ ( x ) | ≤ C ⁢ ∫ ℝ 2 | f ⁢ ( x - y ) | | y | 2 - α ⁢ 𝑑 y

≤ C ⁢ ( ∫ B 1 d ⁢ y | y | ( 2 - α ) ⁢ q q - 1 ) 1 - 1 q ⁢ ∥ f ∥ L q ⁢ ( B 1 ⁢ ( x ) ) + C ⁢ ( ∫ B 1 d ⁢ y | y | ( 2 - α ) ⁢ p p - 1 ) 1 - 1 p ⁢ ∥ f ∥ L p ⁢ ( ℝ 2 ∖ B 1 ⁢ ( x ) )

≤ C ′ ⁢ ( ∥ f ∥ L p ⁢ ( ℝ 2 ) + ∥ f ∥ L q ⁢ ( ℝ 2 ) )

for every x∈ℝ2. ∎

3 Construction of a Pohožaev–Palais–Smale Sequence

In this section, we show the existence of a Pohožaev–Palais–Smale sequence at the level b defined by (1.4). In other words, we construct a sequence of almost critical points which asymptotically satisfies equation (P) and the Pohožaev identity (1.5).

Proposition 3.1.

Suppose the function F∈C1⁢(R,R) satisfies assumptions (F0) and (F1). Then there exists a sequence (un)n∈N in H1⁢(R2) such that the following hold:

ℐ ⁢ ( u n ) → b as n → ∞ .
ℐ ′ ⁢ ( u n ) → 0 strongly in H 1 ⁢ ( ℝ 2 ) ′ as n → ∞ .
𝒫 ⁢ ( u n ) → 0 as n → ∞ .

To prove Proposition 3.1, we first need to show that the energy functional ℐ has the mountain pass geometry, namely that the mountain pass level b is well-defined and nontrivial.

Lemma 3.2.

The critical level b defined by (1.4) satisfies b∈(0,+∞).

Proof.

We start by showing the finiteness of b, which will be done as in [17, Proposition 2.1]. By the definition of the set b, it is sufficient to show that Γ≠∅, which in turn is equivalent to find u0∈H1⁢(ℝ2) such that ℐ⁢(u0)<0. By assumption (F0), we can take s0 such that F⁢(s0)≠0 and we find

∫ ℝ 2 ( I α * F ⁢ ( s 0 ⁢ 𝟏 B 1 ) ) ⁢ F ⁢ ( s 0 ⁢ 𝟏 B 1 ) = F ⁢ ( s 0 ) 2 ⁢ ∫ B 1 ∫ B 1 I α ⁢ ( x - y ) ⁢ 𝑑 x ⁢ 𝑑 y > 0 .

Therefore, by the density of smooth functions in Lq⁢(ℝ2) there will be v0∈H1⁢(ℝ2) with

∫ ℝ 2 ( I α * F ⁢ ( v 0 ) ) ⁢ F ⁢ ( v 0 ) > 0 .

We consider now, for t>0, the function vt:ℝ2→ℝ defined for x∈ℝ2 by vt⁢(x):=v0⁢(xt). This function verifies

ℐ ⁢ ( v t ) = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v 0 | 2 + t 2 2 ⁢ ∫ ℝ 2 | v 0 | 2 - t 2 + α 2 ⁢ ∫ ℝ 2 ( I α * F ⁢ ( v 0 ) ) ⁢ F ⁢ ( v 0 ) .

Therefore, for some t0≫0, the function u0:=vt0 satisfies ℐ⁢(u0)<0.

Let us now show that b>0. By the definition of b, this is equivalent to showing that there exists ε>0 such that for every path γ∈Γ there exists tγ∈[0,1] with ℐ⁢(γ⁢(tγ))≥ε>0. We first assume that u∈H1⁢(ℝ2) and ∫ℝ2(|∇⁡u|2+|u|2)≤δ≪1. In particular, since ∫ℝ2|∇⁡u|2≤1, Proposition 2.1 applies to u with β=2⁢π. Therefore, by Propositions 2.2 and 2.1 and by (1.3), we have

∫ ℝ 2 ( I α * F ⁢ ( u ) ) ⁢ F ⁢ ( u ) ≤ C ⁢ ( ∫ ℝ 2 | F ⁢ ( u ) | 4 2 + α ) 1 + α 2

≤ C ⁢ ( ∫ ℝ 2 min ⁡ { 1 , | u | 2 } ⁢ e 2 ⁢ π ⁢ | u | 2 ) 1 + α 2

≤ C ⁢ ( ∫ ℝ 2 | u | 2 ) 1 + α 2 ,

which is smaller than 14⁢∫ℝ2(|∇⁡u|2+|u|2) if δ is small enough. It follows then that if ∫ℝ2(|∇⁡u|2+|u|2)≤δ, we have

ℐ ⁢ ( u ) ≥ 1 4 ⁢ ∫ ℝ 2 ( | ∇ ⁡ u | 2 + | u | 2 ) .

We now take an arbitrary path γ∈Γ. Since ℐ⁢(γ⁢(1))<0<14⁢∫ℝ2(|∇⁡γ⁢(tγ)|2+|γ⁢(tγ)|2), we have

∫ ℝ 2 ( | ∇ ⁡ γ ⁢ ( 1 ) | 2 + | γ ⁢ ( 1 ) | 2 ) > δ > 0 = ∫ ℝ 2 ( | ∇ ⁡ γ ⁢ ( 0 ) | 2 + | γ ⁢ ( 0 ) | 2 ) .

Therefore, there exists tγ∈(0,1) such that ∫ℝ2(|∇⁡γ⁢(tγ)|2+|γ⁢(tγ)|2)=δ, and hence ℐ⁢(γ⁢(tγ))≥δ4. The lemma follows by taking ε:=δ4. ∎

Proof of Proposition 3.1.

We follow [9, Chapter 2], [8, Chapter 4] and [17, Proposition 2.1]. We consider the map Φ given by

Φ : ℝ × H 1 ⁢ ( ℝ 2 ) → H 1 ⁢ ( ℝ 2 ) , ( σ , v ) ↦ Φ ⁢ ( σ , v ) ⁢ ( x ) := v ⁢ ( e - σ ⁢ x )

and the functional ℐ~=ℐ∘Φ, i.e.,

ℐ ~ ⁢ ( σ , v ) = ℐ ⁢ ( Φ ⁢ ( σ , u ) ) = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + e 2 ⁢ σ 2 ⁢ ∫ ℝ 2 | v | 2 - e ( 2 + α ) ⁢ σ 2 ⁢ ∫ ℝ 2 ( I α * F ⁢ ( v ) ) ⁢ F ⁢ ( v ) ,

which is well-defined and Fréchet-differentiable on the Hilbert space ℝ×H1⁢(ℝ2).

We define now the class of paths

Γ ~ := { γ ~ ∈ C ⁢ ( [ 0 , 1 ] , ℝ × H 1 ⁢ ( ℝ 2 ) ) ∣ γ ~ ⁢ ( 0 ) = ( 0 , 0 ) ⁢ and ⁢ ℐ ~ ⁢ ( γ ~ ⁢ ( 1 ) ) < 0 } .

Since we have Γ={Φ∘γ~∣γ~∈Γ~}, the mountain pass levels of ℐ and ℐ~ coincide, namely

b = inf γ ~ ∈ Γ ~ ⁡ sup t ∈ [ 0 , 1 ] ⁡ ℐ ~ ⁢ ( γ ~ ⁢ ( t ) ) .

Since, by Lemma 3.2, the mountain pass level b is not trivial, we can thus apply the minimax principle (see [24, Theorem 2.9]) and we find a sequence ((σn,vn))n∈ℕ in ℝ×H1⁢(ℝ2) such that

ℐ ~ ⁢ ( σ n , v n ) → n → ∞ b and ℐ ~ ′ ⁢ ( σ n , v n ) → n → ∞ 0 ⁢ strongly in ⁢ ( ℝ × H 1 ⁢ ( ℝ 2 ) ) ′ .

Writing explicitly the derivative of ℐ~ as

ℐ ~ ′ ⁢ ( σ n , v n ) ⁢ [ h , w ] = ℐ ′ ⁢ ( Φ ⁢ ( σ n , v n ) ) ⁢ [ Φ ⁢ ( σ n , w ) ] + 𝒫 ⁢ ( Φ ⁢ ( σ n , v n ) ) ⁢ h ,

we see that the conclusion follows by taking un=Φ⁢(σn,vn). ∎

4 Convergence of the Pohožaev–Palais–Smale Sequence

In this section, we will construct a nontrivial solution of (P) from the sequence given by Proposition 3.1.

Proposition 4.1.

Suppose that the function F∈C1⁢(R,R) satisfies (F1) and (F2) and the sequence (un)n∈N in H1⁢(R2) satisfies the following conditions:

ℐ ⁢ ( u n ) is uniformly bounded.
ℐ ′ ⁢ ( u n ) → 0 strongly in ( H 1 ⁢ ( ℝ 2 ) ) ′ as n → ∞ .
𝒫 ⁢ ( u n ) → 0 as n → ∞ .

Then, up to subsequences, one of the following occurs:

either u n → 0 strongly in H 1 ⁢ ( ℝ 2 ) as n → ∞ ,
or there exist u ∈ H 1 ⁢ ( ℝ 2 ) ∖ { 0 } solving ( (P) ) and a sequence ( x n ) n ∈ ℕ in ℝ 2 such that u n ( ⋅ - x n ) ⇀ u weakly in H 1 ⁢ ( ℝ 2 ) as n → ∞ .

We follow the strategy of [17, Proposition 2.2]. Since the gradient does not appear in the Pohožaev identity (1.5), it will be more delicate to show that the nonlocal term does not vanish.

Proof of Proposition 4.1.

We assume that the first alternative does not hold, namely

lim inf n → ∞ ⁡ ∫ ℝ 2 ( | ∇ ⁡ u n | 2 + | u n | 2 ) > 0 .

Writing

1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u n | 2 + α 2 ⁢ ( 2 + α ) ⁢ ∫ ℝ 2 | u n | 2 = ℐ ⁢ ( u n ) - 𝒫 ⁢ ( u n ) 2 + α

for each n∈ℕ, we deduce that the sequence (un)n∈ℕ is bounded in the space H1⁢(ℝ2). Since ℐ′⁢(un)→0 in H1⁢(ℝ2)′ as n→∞, we have ℐ′⁢(un)⁢[un]→0 as n→∞, therefore

∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ′ ⁢ ( u n ) ⁢ u n = ∫ ℝ 2 ( | ∇ ⁡ u n | 2 + | u n | 2 ) - ℐ ′ ⁢ ( u n ) ⁢ [ u n ] ≥ 1 C .

Taking C0≥supn∈ℕ⁡∫ℝ2(|∇⁡un|2+|un|2), we can apply Proposition 2.1 to 1C0⁢un with β=2⁢π and we obtain

∫ ℝ 2 min ⁡ { 1 , u n 2 } ⁢ e 2 ⁢ π C 0 ⁢ | u n | 2 ≤ C 2 ⁢ π ⁢ ∫ ℝ 2 | u n | 2 C 0 ≤ C 2 ⁢ π

for each n∈ℕ. Moreover, we also have

∫ ℝ 2 | u n | 2 = ( 1 + α 2 ) ⁢ ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ⁢ ( u n ) + 𝒫 ⁢ ( u n )

= ( 1 + α 2 ) ⁢ ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ⁢ ( u n ) + o ⁢ ( 1 )

as n→∞. Therefore, from Proposition 2.2 and by (1.3) we get

1 C ≤ ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ′ ⁢ ( u n ) ⁢ u n ≤ C ⁢ ( ∫ ℝ 2 | F ⁢ ( u n ) | 4 2 + α ⁢ ∫ ℝ 2 ( | F ′ ⁢ ( u n ) | ⁢ | u n | ) 4 2 + α ) 2 + α 4

≤ C ′ ⁢ ( ∫ ℝ 2 min ⁡ { 1 , | u n | 2 } ⁢ e 2 ⁢ π C 0 ⁢ | u n | 2 ) 1 + α 2 ≤ C ′′ ⁢ ( ∫ ℝ 2 | u n | 2 ) 1 + α 2

= C ′′ ⁢ ( ( 1 + α 2 ) ⁢ ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ⁢ ( u n ) + o ⁢ ( 1 ) ) 1 + α 2 ,

namely ∫ℝ2(Iα*F⁢(un))⁢F⁢(un) is bounded from above from zero when n→∞.

We now want to prove that un does not vanish. We will use the following inequality [13, Lemma I.1] (see also [24, Lemma 1.21], [16, Lemma 2.3] and [22, (2.4)]):

∫ ℝ 2 | u n | p ≤ C ⁢ ∫ ℝ 2 ( | ∇ ⁡ u n | 2 + | u n | 2 ) ⁢ ( sup x ∈ ℝ 2 ⁡ ∫ B 1 ⁢ ( x ) | u n | p ) 1 - 2 p .

We will show that the term on the right-hand side is bounded from below by a positive constant for every p>2. By assumption (F2) and (1.3), for every ε>0 there exists Cε,θ>0 such that

| F ⁢ ( s ) | 4 2 + α ≤ ε ⁢ min ⁡ { 1 , | s | 2 } ⁢ e θ ⁢ | s | 2 + C ε , θ ⁢ | s | p .

Therefore,

( sup x ∈ ℝ 2 ⁡ ∫ B 1 ⁢ ( x ) | u n | p ) 1 - 2 p ≥ 1 C ⁢ ∫ ℝ 2 | u n | p ∫ ℝ 2 ( | ∇ ⁡ u n | 2 + | u n | 2 )

≥ 1 C ⁢ C 0 ⁢ C ε ⁢ ( ∫ ℝ 2 | F ⁢ ( u n ) | 4 2 + α - ε ⁢ ∫ ℝ 2 min ⁡ { 1 , | u n | 2 } ⁢ e 2 ⁢ π C 0 ⁢ | u n | 2 )

≥ 1 C ε ′ ⁢ ( ( ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ⁢ ( u n ) ) 2 2 + α - ε ⁢ C ⁢ ∫ ℝ 2 | u n | 2 )

≥ 1 C ε ′ ⁢ ( 1 C ′ - ε ⁢ C ⁢ C 0 ) .

From the arbitrariness of the quantity ε, we get ∫B1⁢(xn)|un|p≥1C for some xn∈ℝ2, for n large enough.

We can now consider the translated sequence (un(⋅-xn))n∈ℕ. Since problem (P) is invariant by translation, this sequence will satisfy the hypotheses of the present proposition, hence we will still denote it as (un)n∈ℕ and we will assume that xn=0 for all n∈ℕ. Since lim infn→∞⁡∫B1|un|p>0, we can assume that this sequence (un)n∈ℕ converges weakly to u∈H1⁢(ℝ2)∖{0}. We just have to show that u solves (P).

The sequence (un)n∈ℕ is bounded in H1⁢(ℝ2). Thus, the sequence (F⁢(un))n∈ℕ is bounded in Lp⁢(ℝ2) for every p≥42+α. Moreover, up to subsequences, un→u almost everywhere as n→∞, so by the continuity of the function F we also have F⁢(un)→F⁢(u) almost everywhere as n→∞. This implies that F⁢(un)⇀F⁢(u) weakly in Lp⁢(ℝ2) for every such p as n→∞. Since 2α>42+α, by Propositions 2.2 and 2.4 we get

I α * F ⁢ ( u n ) ⇀ I α * F ⁢ ( u ) weakly in ⁢ L 4 / ( 2 - α ) ⁢ ( ℝ 2 ) ∩ L ∞ ⁢ ( ℝ 2 ) ⁢ as ⁢ n → ∞ .

By condition (F1) and Proposition 2.1, the sequence (F′⁢(un))n∈ℕ is bounded in Lp⁢(ℝ2) for every p∈[2α,+∞), and by continuity F′⁢(un)→F′⁢(u) almost everywhere as n→∞. Therefore, F′⁢(un)→F′⁢(u) strongly in Llocq⁢(ℝ2) for every q∈[1,+∞) as n→∞, hence

( I α * F ⁢ ( u n ) ) ⁢ F ′ ⁢ ( u n ) ⇀ n → ∞ ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) in ⁢ L loc r ⁢ ( ℝ 2 ) ⁢ for all ⁢ r ∈ [ 1 , + ∞ ) .

Therefore, for every φ∈C01⁢(ℝ2) we have

∫ ℝ 2 ( ∇ ⁡ u ⋅ ∇ ⁡ φ + u ⁢ φ ) = lim n → ∞ ⁡ ∫ ℝ 2 ( ∇ ⁡ u n ⋅ ∇ ⁡ φ + u n ⁢ φ )

= lim n → ∞ ⁡ ∫ ℝ 2 ( I α * F ⁢ ( u n ) ) ⁢ F ′ ⁢ ( u n ) ⁢ φ

= ∫ ℝ 2 ( I α * F ⁢ ( u ) ) ⁢ F ′ ⁢ ( u ) ⁢ φ ,

namely u solves the Choquard equation (P). ∎

Corollary 4.2.

If F satisfies conditions (F0), (F1) and (F2), then problem (P) has a nontrivial solution u∈H1⁢(R2).

Proof.

By Proposition 3.1, the functional ℐ admits a Pohožaev–Palais–Smale sequence (un)n∈ℕ at the level b. We apply Proposition 4.1 to (un)n∈ℕ. If the first alternative occurred, then we would have ℐ⁢(un)→ℐ⁢(0)=0 as n→∞, in contradiction with Lemma 3.2. Therefore, the second alternative must occur, and in particular we get a solution u∈H1⁢(ℝ2)∖{0} of (P). ∎

5 From Solutions to Groundstates

We start by providing a local regularity result for solutions of (P). This result can be obtained quite directly because our growth assumption (F1) gives a good control on Iα*F⁢(u), which, in turn, permits to apply a standard bootstrap method. The equivalent result in higher dimension N≥3 is more delicate to prove (see [17, Theorem 2]) because of the relative weakness of assumption (F1′).

Proposition 5.1.

If F∈C1⁢(R,R) satisfies condition (F1) and if the function u∈H1⁢(R2) solves problem (P), then u∈Wloc2,p⁢(R2) for every p≥1.

Proof.

By (1.3) and Lemma 2.1 we deduce that if v∈H1⁢(ℝ2), then F⁢(v)∈Lp⁢(ℝ2) for every p≥42+α. Since 2α>42+α, by the inequality from Proposition 2.4 we get Iα*F⁢(v)∈L∞⁢(ℝ2).

Therefore, any solution u of (P) verifies

| - Δ ⁢ u + u | ≤ C ⁢ | F ′ ⁢ ( u ) |

with F′⁢(u)∈Llocp⁢(ℝ2) for every p≥1 because of (F1). By standard (interior) regularity theory on bounded domains (see for example [7, Chapter 9]) we deduce that u∈Wloc2,p⁢(ℝ2). ∎

The extra regularity just proved allows us to prove that solutions of (P) satisfy the Pohožaev identity (1.5). The proof of the Pohožaev identity is classical and it is based on testing (P) against a suitable cut-off of x⋅∇⁡u⁢(x), therefore it will be skipped. Details can be found in [17, Theorem 3].

Proposition 5.2 (Pohožaev Identity).

If F∈C1⁢(R,R) satisfies (F1) and u∈H1⁢(R2)∩Wloc2,2⁢(R2) solves (P), then

𝒫 ⁢ ( u ) = ∫ ℝ 2 | u | 2 - ( 1 + α 2 ) ⁢ ∫ ℝ 2 ( I α * F ⁢ ( u ) ) ⁢ F ⁢ ( u ) = 0 .

The Pohožaev identity allows us to show that the mountain pass solution is actually a groundstate. We will argue like [10, Lemma 2.1] and [17, Proposition 2.1], associating to any solution v a path γv∈Γ passing through v. The main difficulty here is that the integral of |∇⁡u|2 is invariant by dilation, therefore we are not allow to join v with 0 by just taking dilations t↦v⁢(⋅t). To overcome this difficulty, we will combine properly dilatations and multiplication by constants [10].

Proposition 5.3.

If F∈C1⁢(R,R) satisfies (F1) and if v∈H1⁢(R2)∖{0} solves (P), then there exists a path γv∈C⁢([0,1],H1⁢(R2)) such that the following hold:

γ v ⁢ ( 0 ) = 0 ;
γ v ⁢ ( 1 2 ) = v ;
ℐ ⁢ ( γ v ⁢ ( t ) ) < ℐ ⁢ ( v ) for every t ∈ [ 0 , 1 ] ∖ { 1 2 } ;
ℐ ⁢ ( γ v ⁢ ( 1 ) ) < 0 .

Proof.

We consider the path γ~:[0,+∞)→H1⁢(ℝ2) given for each τ∈[0,∞) by

( γ ~ ⁢ ( τ ) ) ⁢ ( x ) := { τ τ 0 ⁢ v ⁢ ( x τ 0 ) if ⁢ τ ≤ τ 0 , v ⁢ ( x τ ) if ⁢ τ ≥ τ 0 ,

with τ0≪1 to be chosen later. The function γ~ is clearly continuous on the interval [0,+∞) and in particular at its boundary 0. For τ≥τ0 Proposition 5.2 gives

ℐ ⁢ ( γ ~ ⁢ ( τ ) ) = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + τ 2 2 ⁢ ∫ ℝ 2 | v | 2 - τ 2 + α 2 ⁢ ∫ ℝ 2 ( I α * F ⁢ ( v ) ) ⁢ F ⁢ ( v )

= 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + ( τ 2 2 - τ 2 + α 2 + α ) ⁢ ∫ ℝ 2 | v | 2 ,

which attains its strict maximum in τ=1 and is negative for τ≥τ1, for some τ1≫1. For τ≤τ0 we use (1.3) with θ=(1+α2)⁢π and then apply Proposition 2.1 to the function γ~⁢(τ)/(∫ℝ2|∇⁡γ~⁢(τ)|2)1/2 to obtain

(5.1) ∫ ℝ 2 | F ⁢ ( γ ~ ⁢ ( τ ) ) | 4 2 + α ≤ C ⁢ ∫ ℝ 2 min ⁡ { 1 , | γ ~ ⁢ ( τ ) | 2 } ⁢ e 2 ⁢ π ⁢ | γ ~ ⁢ ( τ ) | 2 ≤ C ⁢ ∫ ℝ 2 | γ ~ ⁢ ( τ ) | 2 ∫ ℝ 2 | ∇ ⁡ γ ~ ⁢ ( τ ) | 2 = C ⁢ τ 0 2 ⁢ ∫ ℝ 2 | v | 2 .

Therefore, because of the Pohožaev identity (Proposition 5.2) and the Hardy–Littlewood–Sobolev inequality (Proposition 2.2), we have

ℐ ⁢ ( γ ~ ⁢ ( τ ) ) = τ 2 2 ⁢ τ 0 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + τ 2 2 ⁢ ∫ ℝ 2 | v | 2 - ∫ ℝ 2 ( I α * F ⁢ ( γ ~ ⁢ ( τ ) ) ) ⁢ F ⁢ ( γ ~ ⁢ ( τ ) )

≤ 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + τ 2 2 ⁢ ∫ ℝ 2 | v | 2 + C ⁢ ( ∫ ℝ 2 | F ⁢ ( γ ~ ⁢ ( τ ) ) | 4 2 + α ) 1 + α 2 .

Therefore, in view of (5.1) and the Pohožaev identity again, we deduce that

ℐ ⁢ ( γ ~ ⁢ ( τ ) ) ≤ 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ v | 2 + τ 0 2 2 ⁢ ∫ ℝ 2 | v | 2 + C ⁢ τ 0 2 + α ⁢ ( ∫ ℝ 2 | v | 2 ) 1 + α 2

= ℐ ⁢ ( v ) + ( τ 0 2 2 - α 2 ⁢ ( 2 + α ) ) ⁢ ∫ ℝ 2 | v | 2 + C ⁢ τ 0 2 + α ⁢ ( ∫ ℝ 2 | v | 2 ) 1 + α 2 ,

which is strictly less than ℐ⁢(v) if τ0=τ0⁢(v) is chosen small enough. Therefore, the function γ~ verifies the following properties:

γ ~ ⁢ ( 0 ) = 0 ;
γ ~ ⁢ ( 1 ) = v ;
ℐ ⁢ ( γ ~ ⁢ ( τ ) ) < ℐ ⁢ ( v ) for every t∈[0,τ1]∖{1};
ℐ ⁢ ( γ ~ ⁢ ( τ 1 ) ) < 0 .

To get the required γv it suffices to take a suitable change of variable γv⁢(t):=γ~⁢(T⁢(τ)) for some function T∈C⁢([0,1],ℝ) satisfying T⁢(0)=0, T⁢(1)=12 and T⁢(τ1)=1. ∎

We are now in a position to prove the main theorem of this work.

Proof of Theorem 1.1.

Let (un)n∈ℕ be the Pohožaev–Palais–Smale sequence given by Proposition 3.1. Then, by Proposition 4.1, it converges weakly to a solution u∈H1⁢(ℝ2)∖{0} of (P). By the definition of groundstates, ℐ⁢(u)≥c and, by Proposition 5.2, we have 𝒫⁢(u)=0 (Proposition 5.2 is applicable in view of Proposition 5.1). Arguing as in [17, Theorem 1], we get successively

ℐ ⁢ ( u ) = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u | 2 + α 2 ⁢ ( 2 + α ) ⁢ ∫ ℝ 2 | u | 2

≤ lim inf n → ∞ ⁡ ( 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u n | 2 + α 2 ⁢ ( 2 + α ) ⁢ ∫ ℝ 2 | u n | 2 )

= lim inf n → ∞ ⁡ ( ℐ ⁢ ( u n ) - 𝒫 ⁢ ( u n ) 2 + α ) = b .

If v∈H1⁢(ℝ2)∖{0} is another solution of the Choquard equation (P), we apply Proposition 5.3 to v and obtain

ℐ ⁢ ( v ) = sup t ∈ [ 0 , 1 ] ⁡ ℐ ⁢ ( γ v ⁢ ( t ) ) ≥ inf γ ∈ Γ ⁡ sup t ∈ [ 0 , 1 ] ⁡ ℐ ⁢ ( γ ⁢ ( t ) ) = b .

By the definition of groundstates and the fact that the solution v is arbitrary, one has b≤c. Putting everything together, we get

c ≤ ℐ ⁢ ( u ) ≤ b ≤ c ,

hence ℐ⁢(u)=b=c. ∎

We point out, as a corollary of the proof of Theorem 1.1, that the convergence in Proposition 4.1 turns out to be actually a strong convergence in H1⁢(ℝ2) and that this gives as a byproduct a compactness property of the set of groundstates of (P).

Corollary 5.4.

Let (un)n∈N be a Pohožaev–Palais–Smale sequence satisfying the assumptions of Proposition 4.1 and in addition

lim n → ∞ ⁡ ℐ ⁢ ( u n ) = c .

Then there exist u∈H1⁢(R2)∖{0} solving (P) and a sequence (xn)n∈N in R2 such that, up to subsequences, un(⋅-xn)→n→∞u strongly in H1⁢(R2). Moreover, the set of groundstates

𝒮 c := { u ∈ H 1 ⁢ ( ℝ 2 ) ∣ u ⁢ solves (P) and ⁢ ℐ ⁢ ( u ) = c }

is compact, up to translations, in H1⁢(R2).

Proof.

We apply Proposition 4.1; the first alternative is excluded by our assumption and the continuity of the functional ℐ at 0. Therefore we get, up to translations, un⇀u as n→∞ in H1⁢(ℝ2) and the function u∈H1⁢(ℝ2)∖{0} solves (P). As in the proof of Theorem 1.1, we get

lim inf n → ∞ ⁡ ( 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u n | 2 + α 2 ⁢ ( 2 + α ) ⁢ ∫ ℝ 2 | u n | 2 ) ≤ c = 1 2 ⁢ ∫ ℝ 2 | ∇ ⁡ u | 2 + α 2 ⁢ ( 2 + α ) ⁢ ∫ ℝ 2 | u | 2 ,

from which it follows that un→u strongly in H1⁢(ℝ2) as n→∞.

To show the compactness of the set of groundstates 𝒮c, we consider an arbitrary sequence (un)n∈ℕ in 𝒮c. Because of Proposition 5.2, it verifies 𝒫⁢(un)=0 for every n∈ℕ, so it satisfies the hypotheses of Proposition 4.1 and of the first part of the present corollary. Therefore, up to subsequences and translations, it will converge to some u which solves (P) and, by the continuity of the functional ℐ in H1⁢(ℝ2), we get u∈𝒮c. ∎

We conclude this paper by the following result on additional qualitative properties of the solution u.

Proposition 5.5.

If F is even and nondecreasing on (0,∞) and u is a groundstate solution of (P), then u has constant sign and is radially symmetric with respect to some point a∈RN.

Proof.

The proof is the same as [17, Propositions 5.2 and 5.3]. We briefly sketch the argument for the convenience of the reader.

To prove the constant-sign property, consider the path γu defined in Proposition 5.3. Since F is an even function, ℐ⁢(|v|)=ℐ⁢(v) for every v∈H1⁢(ℝ2), hence

ℐ ⁢ ( | γ u ⁢ ( t ) | ) < ℐ ⁢ ( | γ u ⁢ ( 1 2 ) | ) = b

for every t∈[0,1]∖{12}. From this one easily deduces that the function |u| is a groundstate solution of (P). Since F′≥0, we can apply the strong maximum principle and get |u|>0, namely u has constant sign. Without loss of generality we assume now that u≥0.

For the symmetry, we follow the strategy of Bartsch, Weth and Willem [2] and its adaptation to the Choquard equation [17, 16]. For any closed half space H⊂ℝ2 we consider the reflection σH with respect to H and define, for every u∈H1⁢(ℝ2), the polarization (see for example [5])

u H ⁢ ( x ) := { max ⁡ { u ⁢ ( x ) , u ⁢ ( σ H ⁢ ( x ) ) } if ⁢ x ∈ H , min ⁡ { u ⁢ ( x ) , u ⁢ ( σ H ⁢ ( x ) ) } if ⁢ x ∉ H .

We first observe that (see [5, Lemma 5.3])

∫ ℝ 2 | ∇ ⁡ u H | 2 + | u H | 2 = ∫ ℝ 2 | ∇ ⁡ u | 2 + | u | 2 .

Moreover, since F is nondecreasing on (0,+∞), we have (F∘u)H=F∘(uH) and thus, in view of the rearrangement inequality for the Riesz potential,

∫ ℝ 2 ( I α ∗ F ⁢ ( u H ) ) ⁢ F ⁢ ( u H ) = ∫ ℝ 2 ( I α ∗ F ⁢ ( u ) H ) ⁢ F ⁢ ( u ) H ≤ ∫ ℝ 2 ( I α ∗ F ⁢ ( u ) ) ⁢ F ⁢ ( u )

with equality if and only if either (F∘u)H=F∘u or (F∘u)H=F∘u∘σH (see [16, Lemma 5.3]). Thus, it follows that ℐ⁢(uH)≤ℐ⁢(u), with equality holding if and only if either F⁢(uH)=F⁢(u) or F⁢(uH)=F⁢(u∘σH) on ℝ2. From this and the definition of the level b, it follows that uH is a groundstate solution of (P), hence either F⁢(uH)=F⁢(u) or F⁢(uH)=F⁢(u∘σH) on ℝ2. In the former case, we easily get f⁢(uH)=f⁢(u), hence uH=u; in the latter, we similarly get uH=u∘σH. Since the hyperplane H is arbitrary, in either case we conclude that the function u is radially symmetric with respect to some point a∈ℝ2 (see [16, Lemma 5.4] and [23, Proposition 3.15]). ∎

Communicated by Zhi-Qiang Wang

Funding source: Fonds De La Recherche Scientifique - FNRS

Award Identifier / Grant number: T.1110.14

Funding statement: This work was supported by the Projet de Recherche (Fonds de la Recherche Scientifique–FNRS) T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”.

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Received: 2016-04-12

Revised: 2016-12-07

Accepted: 2016-12-08

Published Online: 2017-01-11

Published in Print: 2017-07-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/ans-2016-0038

Keywords for this article

Choquard Equation; Semilinear Elliptic Problem; Variational Method

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