Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity

1.1 Main Results

In this paper, we are mainly concerned with positive solutions of (1.1). First, we consider the subcritical case with the Berestycki–Lions conditions. More precisely, we assume the following hypotheses on g.

1. g∈C1⁢(ℝ,ℝ).

2. -∞<lim infτ→0⁡g⁢(τ)τ≤lim supτ→0⁡g⁢(τ)τ=-m<0.

3. lim supτ→∞⁡g⁢(τ)τ2s∗-1≤0, where 2s∗=63-2⁢s.

4. There exists ξ>0 such that G⁢(ξ):=∫0ξg⁢(τ)⁢dτ>0.

Our first result is the following theorem.

In the variational approach to the study of elliptic problems, the Palais–Smale condition ((PS) condition for short) plays a crucial role. To verify the (PS) condition, the so-called Ambrosetti–Rabinowitz condition

has been frequently used in the literature. The main role of ((AR)) is to guarantee the boundedness of the (PS) sequence in some suitable Sobolev space. More recently, Pucci, Xiang and Zhang [32] considered fractional p-Laplacian equations of Schrödinger–Kirchhoff type

With the use of ((AR)), they established the existence of multiple solutions to (1.6) via the Ekeland variational principle and the mountain pass theorem. In fact, ((AR)) is a technical assumption. Many mathematicians have tried to remove or weaken it. In [8], Berestycki and Lions considered the autonomous scalar field equation. Without using ((AR)), they proved the existence of ground state solutions by the constraint variational method. However, it is not easy to use the idea in [8] in order to deal directly with non-autonomous problems. In [26], Jeanjean introduced a monotonicity trick to overcome the difficulty due to the lack of ((AR)) in the non-autonomous case. In [39], without ((AR)), the authors of the present work considered the existence and the concentration of positive solutions to (1.1) in the critical case for s=t=1. It is natural to wonder if similar results can hold for the critical fractional case. This is just our second goal of the present paper. In the critical case, we assume the following hypotheses on g.

1. limτ→0⁡g⁢(τ)τ=-a<0.

2. limτ→∞⁡g⁢(τ)τ2s∗-1=b>0.

3. There exists μ>0 and q<2s∗ such that g⁢(τ)-b⁢τ2s∗-1+a⁢τ≥μ⁢τq-1 for all τ>0.

Our second result is the following theorem.

We conclude by fixing some notation that we will use throughout the paper. We define the norm

∥ u ∥ p := ( ∫ ℝ 3 | u | p ⁢ d x ) 1 / p , p ∈ [ 1 , ∞ ) ,

the value

2 α ∗ := 6 3 - 2 ⁢ α , α ∈ ( 0 , 1 ) ,

and we let u^=ℱ⁢(u) denote the Fourier transform of u.

In the rest of the paper, we use the perturbation approach to prove Theorem 1.1 and Theorem 1.4. Similar arguments can also be found in [39]. The paper is organized as follows. In Section 2, we introduce the functional framework and some preliminary results. In Section 3, we construct the min-max level. In Section 4, we use a perturbation argument to complete the proof of Theorem 1.1 and we give the proof of Theorem 1.4.

2.1 Fractional-Order Sobolev Spaces

The fractional Laplacian (-Δ)α with α∈(0,1) of a function ϕ:ℝ3→ℝ is defined by

ℱ ⁢ ( ( - Δ ) α ⁢ ϕ ) ⁢ ( ξ ) = | ξ | 2 ⁢ α ⁢ ℱ ⁢ ( ϕ ) ⁢ ( ξ ) , ξ ∈ ℝ 3 ,

where ℱ is the Fourier transform, i.e.,

ℱ ⁢ ( ϕ ) ⁢ ( ξ ) = 1 ( 2 ⁢ π ) 3 / 2 ⁢ ∫ ℝ 3 exp ⁡ ( - 2 ⁢ π ⁢ i ⁢ ξ ⋅ x ) ⁢ ϕ ⁢ ( x ) ⁢ d x ,

where i is the imaginary unit. If ϕ is smooth enough, it can be computed by the singular integral

( - Δ ) α ⁢ ϕ ⁢ ( x ) = c α ⁢ P . V . ∫ ℝ 3 ϕ ⁢ ( x ) - ϕ ⁢ ( y ) | x - y | 3 + 2 ⁢ α ⁢ d y , x ∈ ℝ 3 ,

where cα is a normalization constant and P.V. stands for the principal value.

For any α∈(0,1), we consider the fractional-order Sobolev space

H α ⁢ ( ℝ 3 ) = { u ∈ L 2 ⁢ ( ℝ 3 ) : ∫ ℝ 3 | ξ | 2 ⁢ α ⁢ | u ^ | 2 ⁢ d ξ < ∞ }

endowed with the norm

∥ u ∥ α = ( ∫ ℝ 3 ( 1 + | ξ | 2 ⁢ α ) ⁢ | u ^ | 2 ⁢ d ξ ) 1 / 2 , u ∈ H α ⁢ ( ℝ 3 ) ,

and with the inner product

( u , v ) = ∫ ℝ 3 ( 1 + | ξ | 2 ⁢ α ) ⁢ u ^ ⁢ v ^ ¯ ⁢ d ξ , u , v ∈ H α ⁢ ( ℝ 3 ) .

It is easy to see that the inner products

u , v ↦ ∫ ℝ 3 ( 1 + | ξ | 2 ⁢ α ) ⁢ u ^ ⁢ v ^ ¯ ⁢ d ξ   and   u , v ↦ ∫ ℝ 3 ( u ⁢ v + ( - Δ ) α / 2 ⁢ u ⁢ ( - Δ ) α / 2 ⁢ v ) ⁢ d x

on Hs⁢(ℝ3) are equivalent (see [36]). The homogeneous Sobolev space 𝒟α,2⁢(ℝ3) is defined by

𝒟 α , 2 ⁢ ( ℝ 3 ) = { u ∈ L 2 α ∗ ⁢ ( ℝ 3 ) : | ξ | α ⁢ u ^ ∈ L 2 ⁢ ( ℝ 3 ) } ,

which is the completion of C0∞⁢(ℝ3) under the norm

∥ u ∥ 𝒟 α , 2 2 = ∥ ( - Δ ) α / 2 ⁢ u ∥ 2 2 = ∫ ℝ 3 | ξ | 2 ⁢ α ⁢ | u ^ | 2 ⁢ d ξ , u ∈ 𝒟 α , 2 ⁢ ( ℝ 3 ) ,

and the inner product

( u , v ) 𝒟 α , 2 = ∫ ℝ 3 ( - Δ ) α / 2 ⁢ u ⁢ ( - Δ ) α / 2 ⁢ v ⁢ d x , u , v ∈ 𝒟 α , 2 ⁢ ( ℝ 3 ) .

For a further introduction on fractional-order Sobolev spaces, we refer the interested reader to Di Nezza, Palatucci and Valdinoci [17]. Let

H r s ⁢ ( ℝ 3 ) = { u ∈ H 3 ⁢ ( ℝ 3 ) : u ⁢ ( x ) = u ⁢ ( | x | ) } .

Now, we introduce the following Sobolev embedding theorems.

2.2 The Variational Setting

Now, we study the variational setting of (1.1). By Lemma 2.1, for 2⁢t+4⁢s≥3, we have

H s ⁢ ( ℝ 3 ) ↪ L 12 / ( 3 + 2 ⁢ t ) ⁢ ( ℝ 3 ) .

Then, for u∈Hs⁢(ℝ3), by Lemma 2.2, the linear operator P:𝒟t,2⁢(ℝ3)→ℝ defined by

P ⁢ ( v ) = ∫ ℝ 3 u 2 ⁢ v ≤ ∥ u ∥ 12 / ( 3 + 2 ⁢ t ) 2 ⁢ ∥ v ∥ 2 t ∗ ≤ C ⁢ ∥ u ∥ s 2 ⁢ ∥ v ∥ 𝒟 t , 2

is well defined on 𝒟t,2⁢(ℝ3) and is continuous. Thus, it follows from the Lax–Milgram theorem that there exists a unique ϕut∈𝒟t,2⁢(ℝ3) such that (-Δ)t⁢ϕut=λ⁢u2. Moreover, for x∈ℝ3, we have

where we have set

c t = Γ ⁢ ( 3 2 - 2 ⁢ t ) π 3 / 2 ⁢ 2 2 ⁢ t ⁢ Γ ⁢ ( t ) .

Formula (2.1) is called the t-Riesz potential. Substituting (2.1) into (1.1), we can rewrite (1.1) in the equivalent form

We define the energy functional Γλ:Hs⁢(ℝ3)→ℝ by

Γ λ ⁢ ( u ) = 1 2 ⁢ ∫ ℝ 3 | ( - Δ ) s / 2 ⁢ u | 2 ⁢ d x + λ 4 ⁢ ∫ ℝ 3 ϕ u t ⁢ u 2 ⁢ d x - ∫ ℝ 3 G ⁢ ( u ) ⁢ d x

with

G ⁢ ( τ ) = ∫ 0 τ g ⁢ ( ζ ) ⁢ d ζ .

Obviously, the critical points of Γλ are the weak solutions of (2.2).

Setting

T ⁢ ( u ) := 1 4 ⁢ ∫ ℝ 3 ϕ u t ⁢ u 2 ⁢ d x ,

we summarize some properties of ϕut and T⁢(u) which will be used later.

3.1 The Modified Problem

It follows from Lemma 2.4 that Γλ is well defined on Hs⁢(ℝ3) and is of class C1. Since we are concerned with positive solutions of (2.2), similarly to [8] (see also [12]), we modify our problem first. Without loss of generality, we assume that

0 < ξ = inf ⁡ { τ ∈ ( 0 , ∞ ) : G ⁢ ( τ ) > 0 } ,

where ξ is given in (H4). Let

τ 0 = inf ⁡ { τ > ξ : g ⁢ ( τ ) = 0 } ∈ [ ξ , ∞ ]

and define a function g~:ℝ→ℝ by

g ~ ( τ ) = { g ⁢ ( τ ) for ⁢ τ ∈ [ 0 , τ 0 ] , 0 for ⁢ τ ≥ τ 0 ,

and g~⁢(τ)=0 for τ≤0. If u∈Hs⁢(ℝ3) is a solution of (2.2), where g is replaced by g~, then, by the maximum principle (see Cabré and Sire [10]), we get that u is positive and u⁢(x)≤τ0 for any x∈ℝ3, i.e., u is a solution of the original problem (2.2) with g. Thus, from now on, we can replace g by g~, but still use the same notation g. In addition, for τ>0, let

g 1 ⁢ ( τ ) = max ⁡ { g ⁢ ( τ ) + m ⁢ τ , 0 }   and   g 2 ⁢ ( τ ) = g 1 ⁢ ( τ ) - g ⁢ ( τ ) .

Then, we have g2⁢(τ)≥m⁢τ for τ≥0,

and, for any ε>0, there exists Cε>0 such that

Let

G i ⁢ ( u ) = ∫ 0 u g i ⁢ ( τ ) ⁢ d τ , i = 1 , 2 .

Then, by (3.1) and (3.2), for any ε>0, there exists Cε>0 such that

3.2 The Limit Problem

In the following, we will find solutions of (2.2) by seeking critical points of Γλ. If λ=0, (2.2) becomes

which is referred to as the limit problem of (2.2). We define an energy functional for the limit problem (3.4) by

L ⁢ ( u ) = 1 2 ⁢ ∫ ℝ 3 | ( - Δ ) s / 2 ⁢ u | 2 ⁢ d x - ∫ ℝ 3 G ⁢ ( u ) ⁢ d x , u ∈ H s ⁢ ( ℝ 3 ) .

In [12], Chang and Wang proved that, with the same assumptions on g as in Theorem 1.1, there exists a positive ground state solution U∈Hr3⁢(ℝ3) of (3.4). Moreover, each such solution U of (3.4) satisfies the Pohožaev identity

Let S be the set of positive radial ground state solutions U of (3.4). Then, S≠∅ and we have the following compactness result which plays a crucial role in the proof of Theorem 1.1.

As shown in Cho and Ozawa [13], for general s∈(0,1), we do not have a similar radial lemma in Hrs⁢(ℝ3). So the Strauss compactness lemma (see [8]) is not applicable here. Before we prove Proposition 3.1, we begin with the following compactness lemma which is a special case of [12, Lemma 2.4.]

3.3 The Min-Max Level

Take U∈S and let

U τ ⁢ ( x ) = U ⁢ ( x τ ) , τ > 0 .

Then, by the definition of U^=ℱ⁢(U), we know that U^(⋅/τ)=τ3U^(t⋅) and

∫ ℝ 3 | ( - Δ ) s / 2 U τ | 2 d x = ∫ ℝ 3 | ξ | 2 ⁢ s | U ^ ( ξ τ ) | 2 = τ 3 - 2 ⁢ s ∫ ℝ 3 | ( - Δ ) s / 2 U | 2 d x .

By the Pohožaev identity, we have

L ⁢ ( U τ ) = ( τ 3 - 2 ⁢ s 2 - 3 - 2 ⁢ s 6 ⁢ τ 3 ) ⁢ ∫ ℝ 3 | ( - Δ ) s / 2 ⁢ U | 2 .

Thus, there exists τ0>1 such that L⁢(Uτ)<-2 for τ≥τ0. Set

D λ ≡ max τ ∈ [ 0 , τ 0 ] ⁡ Γ λ ⁢ ( U τ ) .

By virtue of Lemma 2.4, we have Γλ⁢(Uτ)=L⁢(Uτ)+O⁢(λ). Note that since maxτ∈[0,τ0]⁡L⁢(Uτ)=E, we get that Dλ→E as λ→0+.

Moreover, similarly to [39], we can prove the following lemma, which is crucial in defining the uniformly bounded set of the mountain paths (see below).

Now, for any λ∈(0,λ1), we define a min-max value Cλ as

C λ = inf γ ∈ Υ λ ⁡ max τ ∈ [ 0 , τ 0 ] ⁡ Γ λ ⁢ ( γ ⁢ ( τ ) ) ,

where

Υ λ = { γ ∈ C ⁢ ( [ 0 , τ 0 ] , H r s ⁢ ( ℝ 3 ) ) : γ ⁢ ( 0 ) = 0 , γ ⁢ ( τ 0 ) = U τ 0 , ∥ γ ⁢ ( τ ) ∥ s ≤ 𝒞 0 + 1 , τ ∈ [ 0 , τ 0 ] } .

Obviously, for τ>0, we have

∥ U τ ∥ s 2 = τ 3 - 2 ⁢ s ⁢ ∥ ( - Δ ) s / 2 ⁢ U ∥ 2 2 + τ 3 ⁢ ∥ U ∥ 2 2 .

Then, we can define U0≡0 so Uτ∈Υλ. Moreover, we have

lim sup λ → 0 + ⁡ C λ ≤ lim λ → 0 + ⁡ D λ = E .

3.4 Proof of Theorem 1.1

Now, for α,d>0, define

Γ λ α := { u ∈ H r s ⁢ ( ℝ 3 ) : Γ λ ⁢ ( u ) ≤ α }

and

S d = { u ∈ H r s ⁢ ( ℝ 3 ) : inf v ∈ S ⁡ ∥ u - v ∥ s ≤ d } .

In the following, we will find a solution u∈Sd of (2.2) for sufficiently small λ>0 and some 0<d<1. The following proposition is crucial for obtaining a suitable (PS) sequence for Γλ and plays a key role in our proof.

By Proposition 3.5, for small d∈(0,1), there exist ω>0, λ0>0 such that