Concentration of Positive Ground State Solutions for Schrödinger–Maxwell Systems with Critical Growth

1 Introduction and Main Results

The nonlinear Schrödinger equation

has been used to study many nonlinear models from different fields of physics. Here ℏ is the Plank constant, m is the mass of the field ψ, W⁢(x) is a given external potential and the nonlinear term describes the interaction effect among many particles. Considering the existence of standing wave type solutions, i.e.,

one is lead to study the semilinear elliptic equation

where ε2=ℏ22⁢m, V⁢(x)=W⁢(x)-ω and g is a real continuous function.

The study of equation (1.3) when the parameter ε goes to zero, known as the semiclassical problems for Schrödinger equation, was used to describe the transition between quantum mechanics and classical mechanics. This kind of problem has been investigated extensively under various hypotheses on the potential and the nonlinearity since the work by Floer and Weinstein [18], see for example [11, 2, 4, 13, 17, 19, 24, 29, 28, 35, 36] and the references therein for recent progress. Among them, Wang and Zeng [36] considered the equation

and showed that the concentration points are located on the middle ground of the competing potential functions and are given explicitly in terms of these functions in some cases. Cingolani and Lazzo [13] obtained a multiplicity result involving the set of global minima of a function which provides some kind of global median value between the minimum of V and the maximum of K and Q. We finally mention the paper of Ambrosetti, Malchiodi and Secchi [4], in which they considered the case Q=0 and proved that the number of solutions of (1.4) is related with the set of minima of a function given explicitly in terms of V,K,r and the dimension N. In a recent paper [17], Ding and Liu considered the equation

( - i ⁢ ε ⁢ ∇ + A ⁢ ( z ) ) 2 ⁢ u + V ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ ( g ⁢ ( | u | ) + | u | 4 ) ⁢ u

for u∈H1⁢(ℝN,ℂ), where the function A:ℝ3→ℝ3 denotes a continuous magnetic potential associated with a magnetic field B (i.e., curl⁡A=B) and g⁢(|u|)⁢u is superlinear and subcritical. Under suitable assumptions on the potentials, they obtained existence and new concentration phenomena of the semiclassical ground states.

If the Schrödinger field (1.2) of equation (1.1) interacts with an unknown electromagnetic field, then one is lead to study the following system of equations in ℝ3:

where K⁢(x) is used to describe the electron charge. When ε=1 and V⁢(x) is a constant, the existence and multiplicity of solutions of equation (1.5) have been widely considered under different assumptions on the nonlinearities, see, for example, [1, 5, 6, 14, 8, 30]. When V⁢(x) is not a constant, we refer readers to [37] for the asymptotically linear case, [25] for the potential well case and [7] for the critical growth case. The case where the electronic potential K⁢(x) is not a constant was considered by Cerami and Varia [12]. In particular, they studied the system

{ - Δ ⁢ u + u + K ⁢ ( x ) ⁢ ϕ ⁢ u = a ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ 3 , - Δ ⁢ ϕ = K ⁢ ( x ) ⁢ u 2 , x ∈ ℝ 3 ,

and, under suitable assumptions on K⁢(x) and a⁢(x), proved the existence of positive solutions. By requiring radial symmetry and asymptotical behavior at infinity, Li, Peng and Yan in [26] studied the existence of infinitely many sign-changing solutions.

The semiclassical Schrödinger–Poisson problem has also attracted a lot of attention. Considering the system

{ - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + ϕ ⁢ u = | u | p - 2 ⁢ u , x ∈ ℝ 3 , - Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 ,

when p∈(1,117) and the potential is a constant, D’Aprile and Wei [15] constructed a family of positive radially symmetric bound states and showed the concentration around a sphere in ℝ3 as ε→0 and, in addition, in [16] proved the existence of multi bump solutions. When V⁢(x) is not a constant, a similar concentration phenomenon was also observed by Ruiz in [32]. Ruiz [31] studied the concentration around a sphere and Ruiz and Varia [33] obtained the existence of multi-bump type solutions and showed that the bumps concentrate around a local minimum of the potential for 3<p<5 by applying Lyapunov–Schmidt reduction methods. Ianni [21], and Ianni and Vaira [22, 23] studied the existence of semiclassical states for Schrödinger–Maxwell systems and constructed a family of radial bound states that concentrate around a sphere or the stationary point of the bounded potential. He and Zou [20] studied the problem

{ - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + ϕ ⁢ u = f ⁢ ( u ) , x ∈ ℝ 3 , - ε 2 ⁢ Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 ,

and obtained the existence of positive solutions concentrating on the minima of V⁢(x). Zhang [39] obtained the existence of solutions concentrating at a local minimum point of the potential V⁢(x).

To study the nonlocal type Schrödinger equation, one of the interesting problems posed in [3, p. 29] is to introduce a suitable auxiliary potential W⁢(x) which can be characterized by V⁢(x) and K⁢(x) in (1.4), and then describe the concentration around the stationary points of W⁢(x). Since the appearance of a nonlocal convolution term in Schrödinger–Maxwell systems brings new difficulties, it would be quite natural to consider how the interaction between the auxiliary potential and the nonlinear potential will affect the existence and concentration of solutions of nonlinear Schrödinger–Maxwell systems. Motivated by the results mentioned for a single Schrödinger equation, the aim of this paper is to study the existence and concentration of solutions of Schrödinger–Maxwell systems in ℝ3 having the form

where ε is a small parameter and V⁢(x), K⁢(x), P⁢(x) and Q⁢(x) are four real continuous function on ℝ3. Notice that the number 2*=6 is the critical exponent for the Sobolev imbedding in ℝ3, so the nonlinearity in this equation is of critical growth at infinity.

Throughout this paper, we denote the standard norm of E:=H1⁢(ℝ3) by

∥ u ∥ = ( ∫ ℝ 3 ( | ∇ ⁡ u | 2 + u 2 ) ) 1 / 2

and the norm of D1,2⁢(ℝ3) by

∥ u ∥ D = ( ∫ ℝ 3 | ∇ u | 2 ) 1 / 2 .

As usual, |⋅|s denotes the norm of the Lebesgue space, i.e.,

| u | s := ( ∫ ℝ 2 | u | s ) 1 / s , 1 ≤ s < ∞ .

Applying the Lax–Milgram theorem, we know that for every u∈H1⁢(ℝ3), there exists a unique ϕu∈D1,2⁢(ℝ3) such that

- Δ ⁢ ϕ u = K ⁢ ( x ) ⁢ u 2

and ϕu can be expressed by

ϕ u ⁢ ( x ) = 1 | x | ∗ ( K ⁢ ( x ) ⁢ u 2 ) = ∫ ℝ 3 K ⁢ ( y ) ⁢ u 2 ⁢ ( y ) | x - y | ⁢ 𝑑 y .

Simple calculations also show that ϕu must satisfy

∥ ϕ u ∥ D ≤ C ⁢ ∥ u ∥ 2   and   ∫ ℝ 3 ϕ u ⁢ u 2 ≤ C ⁢ ∥ u ∥ 4 .

Thus, it can be proved in a standard way that (u,ϕu)∈H1⁢(ℝ3)×D1,2⁢(ℝ3) satisfies system (1.6) if and only if u∈H1⁢(ℝ3) is a solution of

By changing variables, we know that if u is a solution of equation (1.7), then the function v=u⁢(ε⁢x) satisfies

- Δ ⁢ v + V ⁢ ( ε ⁢ x ) ⁢ v + [ 1 | x | ∗ ( K ⁢ ( ε ⁢ x ) ⁢ v 2 ) ] ⁢ K ⁢ ( ε ⁢ x ) ⁢ v = P ⁢ ( ε ⁢ x ) ⁢ g ⁢ ( v ) + Q ⁢ ( ε ⁢ x ) ⁢ | v | 4 ⁢ v .

This suggests some convergence, as ε→0, of the family of solutions to a solution v0 of the limit problem

- Δ ⁢ v + V ⁢ ( 0 ) ⁢ v + K 2 ⁢ ( 0 ) ⁢ [ 1 | x | ∗ v 2 ] ⁢ v = P ⁢ ( 0 ) ⁢ g ⁢ ( v ) + Q ⁢ ( 0 ) ⁢ | v | 4 ⁢ v .

In order to state the main results, we suppose that the nonlinearity g:ℝ+→ℝ satisfies the following hypotheses:

1. We have

   g ⁢ ( 0 ) = 0   and   lim s → 0 ⁡ g ⁢ ( s ) s = 0 .

2. There exist 2<p<6 and C0>0 such that

   | g ⁢ ( s ) | ≤ C 0 ⁢ ( 1 + s p - 1 )   for all ⁢ s ≥ 0 .

3. There exist 4<q<6 and θ>4 such that

   c 0 ⁢ s q ≤ θ ⁢ G ⁢ ( s ) ≤ g ⁢ ( s ) ⁢ s   for all ⁢ s > 0 ,

   where G⁢(t)=∫0tg⁢(s)⁢𝑑s. This is the well-known Ambrosetti–Rabinowitz condition.

4. s→g⁢(s)/s3 is strictly increasing on (0,+∞).

Since we are interested in the existence of positive solutions, we may assume that

g ⁢ ( s ) = 0   for all ⁢ s ≤ 0 .

In the following, we denote by S the best Sobolev constant:

S | u | 6 2 ≤ ∫ ℝ 3 | ∇ u | 2 .

Moreover, we define

S q := inf z ∈ E , | z | q = 1 ∥ z ∥ 2 .

Let V,K,P,Q:ℝ3→ℝ be four continuous and bounded functions with

We consider the case where the following two assumptions hold:

1. ℳ:=𝒫∩𝒬={x∈ℝ3:P⁢(x)=τmax,Q⁢(x)=νmax}≠∅.

2. We have τmax>τ∞ and there exist R>0 and x*∈ℳ such that

   V ⁢ ( x * ) ≤ V ⁢ ( x ) , K ⁢ ( x * ) ≤ K ⁢ ( x )   for all ⁢ | x | ≥ R .

In this case, we can define

𝒜 V := { x ∈ ℳ : V ⁢ ( x ) ≤ V ⁢ ( x * ) } ∪ { x ∉ ℳ : V ⁢ ( x ) < V ⁢ ( x * ) }

and

𝒜 K := { x ∈ ℳ : K ⁢ ( x ) ≤ K ⁢ ( x * ) } ∪ { x ∉ ℳ : K ⁢ ( x ) < K ⁢ ( x * ) } .

The first result of this paper is the following theorem.

We have another two cases to consider for assumptions (PQ1) and (PQ2) introduced above. Each case is characterized by two assumptions. The first one is as follows:

1. 𝒳 := 𝒦 ∩ 𝒬 = { x ∈ ℝ 3 : K ⁢ ( x ) = γ min , Q ⁢ ( x ) = ν max } ≠ ∅ .

2. We have γ∞>γmin and there exist R>0 and x*∈𝒳 such that

   V ⁢ ( x * ) ≤ V ⁢ ( x ) , P ⁢ ( x * ) ≥ P ⁢ ( x )   for all ⁢ | x | ≥ R .

Set

𝒜 V := { x ∈ 𝒳 : V ⁢ ( x ) ≤ V ⁢ ( x * ) } ∪ { x ∉ 𝒳 : V ⁢ ( x ) < V ⁢ ( x * ) }

and

𝒜 P := { x ∈ 𝒳 : P ⁢ ( x ) ≥ P ⁢ ( x * ) } ∪ { x ∉ 𝒳 : P ⁢ ( x ) > P ⁢ ( x * ) } .

The second case is as follows:

1. 𝒲:=𝒱∩𝒬={x∈ℝ3:V⁢(x)=κmin,Q⁢(x)=νmax}≠∅.

2. We have κ∞>κmin and there exist R>0 and x*∈𝒲 such that

   K ⁢ ( x * ) ≤ K ⁢ ( x ) , P ⁢ ( x * ) ≥ P ⁢ ( x )   for all ⁢ | x | ≥ R .

For this case, we set

𝒜 K := { x ∈ 𝒲 : K ⁢ ( x ) ≤ K ⁢ ( x * ) } ∪ { x ∉ 𝒲 : K ⁢ ( x ) < K ⁢ ( x * ) }

and

𝒜 P := { x ∈ 𝒲 : P ⁢ ( x ) ≥ P ⁢ ( x * ) } ∪ { x ∉ 𝒲 : P ⁢ ( x ) > P ⁢ ( x * ) } .

In this paper, we will use the following notations:

1. C, Ci denote positive constants.

2. BR denote the open ball centered at the origin with radius R>0.

3. C0∞⁢(ℝ3) denotes the space of the functions infinitely differentiable with compact support in ℝ3.

4. Let E be a real Hilbert space and I:E→ℝ a functional of class 𝒞1. We say that (un)⊂E is a Palais–Smale ((PS) for short) sequence at c for I if (un) satisfies

   I ⁢ ( u n ) → c   and   I ′ ⁢ ( u n ) → 0   as ⁢ n → ∞ .

5. We say that I satisfies the (PS) condition at c, if any (PS) sequence at c possesses a convergent subsequence.

The paper is organized as follows. In Section 2 we outline some preliminary results related to elliptic equations with critical exponent. In Section 3 we study an auxiliary problem which will play an important role in proving the main results. In Section 4 we prove the existence and concentration results for Schrödinger–Maxwell systems with critical exponent. We will give the details of the proof for the case where the concentration phenomenon is dominated by the nonlinear potential P⁢(x). Also, we will sketch the proof for the case where the concentration phenomenon is dominated by the electronic potential K⁢(x).

2 Variational Framework

Changing variable by u⁢(x)=v⁢(ε⁢x), it is possible to see that equation (1.7) is equivalent to

Then the energy functional can be defined by

I ε ⁢ ( u ) = 1 2 ⁢ ∥ u ∥ ε 2 + 1 4 ⁢ ∫ ℝ 3 [ 1 | x | ∗ ( K ⁢ ( ε ⁢ x ) ⁢ u 2 ) ] ⁢ K ⁢ ( ε ⁢ x ) ⁢ u 2 - ∫ ℝ 3 P ⁢ ( ε ⁢ x ) ⁢ G ⁢ ( u ) - 1 6 ⁢ ∫ ℝ 3 Q ⁢ ( ε ⁢ x ) ⁢ | u | 6 ,

where G⁢(u)=∫0ug⁢(s)⁢𝑑s and

∥ u ∥ ε := ( ∫ ℝ 3 ( | ∇ ⁡ u | 2 + V ⁢ ( ε ⁢ x ) ⁢ | u | 2 ) ) 1 / 2

is an equivalent norm on E. From the growth assumption on g, we know that Iε is well defined on E and belongs to 𝒞1. In the following, the associated Nehari manifold for Iε will be denoted by 𝒩ε, that is,

𝒩 ε = { u ∈ E : u ≠ 0 , 〈 I ε ′ ⁢ ( u ) , u 〉 = 0 } .

Using conditions (g1)–(g3), it is easy to see that there exists α>0, independent of ε, such that

∥ u ∥ ε ≥ α   for all ⁢ u ∈ 𝒩 ε .

One can easily check that the functional Iε satisfies the mountain-pass geometry, that is, the following lemma holds.

Applying the mountain-pass theorem without the (PS) condition [38], we know that there exists a (PS)cε sequence (un)⊂E, i.e.,

I ε ′ ⁢ ( u n ) → 0 , I ε ⁢ ( u n ) → c ε ,

where cε is defined by

Moreover, there exists a constant c>0, independent of ε, such that cε>c. Using the monotonicity condition (g4), for each u∈E\{0}, there exists a unique t=t⁢(u) such that t⁢(u)⁢u∈𝒩ε and

I ε ⁢ ( t ⁢ ( u ) ⁢ u ) = max s ≥ 0 ⁡ I ε ⁢ ( s ⁢ u ) .

Thus, cε can also be characterized by

c ε = inf u ∈ 𝒩 ε ⁡ I ε ⁢ ( u ) .

In order to study the concentration of the ground states of critical Schrödinger–Maxwell systems, we would like to recall some basic results for systems with constant coefficients. Consider the system

where κ∈[κmin,κmax], γ∈[γmin,γmax], τ∈[τmin,τmax], ν∈[νmin,νmax] are four constants. The associated energy functional is denoted by

Φ κ , γ , τ , ν * ( u ) := 1 2 ∥ u ∥ κ 2 + γ 2 4 ∫ ℝ 3 [ 1 | x | ∗ u 2 ] u 2 - τ ∫ ℝ 3 G ( u ) - ν 6 ∫ ℝ 3 | u | 6 ,

where

∥ u ∥ κ := ( ∫ ℝ 3 ( | ∇ ⁡ u | 2 + κ ⁢ | u | 2 ) ) 1 / 2 .

It is not difficult to verify that the functional Φκ,γ,τ,ν* also has a mountain pass level mκ,γ,τ,ν* characterized by

m κ , γ , τ , ν * := inf u ∈ E \ { 0 } ⁡ max t ≥ 0 ⁡ Φ κ , γ , τ , ν * ⁢ ( t ⁢ u ) .

In fact, we can estimate the mountain pass level mκ,γ,τ,ν*, as seen in the following lemma.