In this paper, we study the existence of normalized ground state solution of the following Schrödinger-Maxwell equations

( − Δ u + u + ϕ u + λ u = f ( u )     in   ℝ N , − Δ ϕ = u 2     in   ℝ N , (1.1)

where ϕ : ℝ N → ℝ and 2 < N < 6 , the parameter λ ∈ ℝ appears as a Lagrange multiplier. The unknowns of the equations are the field u associated to the particle and the electric potential ϕ , and satisfying the normalization condition

∫ ℝ N | u | 2 d x = a , (1.2)

we prescribe a > 0 . Hence, we have

( − Δ u + u + ϕ u + λ u = f ( u )     in   ℝ N , − Δ ϕ = u 2     in   ℝ N , ∫ ℝ N | u | 2 d x = a . (1.3)

where u belongs to the Hilbert space

H = { u ∈ H r 1 ( ℝ N ) : ∫ ℝ N | ∇ u | 2 + u 2 d x < ∞ } ,

and

H r 1 ( ℝ N ) = { u ∈ H 1 ( ℝ N ) : u ( x ) = u ( | x | ) } .

The space H is endowed with the norm

‖ u ‖ H 2 = ∫ ℝ N ( | ∇ u | 2 + u 2 ) d x .

Let D 1,2 ≡ D 1,2 ( ℝ N ) = { u ∈ L 2 * ( ℝ N ) : ∇ u ∈ L 2 ( ℝ N ) } with respect to the norm

‖ u ‖ D 1,2 2 = ∫ ℝ N | ∇ u | 2 d x .

For any 2 < s < 2 * , L s ( ℝ N ) is endowed with the norm

| u | s s = ∫ ℝ N | u | s d x .

Obviously, the embedding H ↪ L s ( ℝ N ) is compact (see [1] ).

By the variational nature, the weak solutions of (1.1) are critical points of the functional J : H × D 1,2 → ℝ defined by

J ( u , ϕ ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + V ( x ) u 2 ) d x − 1 4 ∫ ℝ N | ∇ ϕ | 2 d x + 1 2 ∫ ℝ N   ϕ u 2 d x − ∫ ℝ N   F ( u ) d x ,

where F ( t ) = ∫ 0 t   f ( s ) d s is a rather general nonlinearity. Then, it is clear that the function J is C 1 on H × D 1,2 and has the strong indefiniteness. We can know that the weak solutions of (1.1) ( u , ϕ ) ∈ H × D 1,2 are critical points of the functional J. By standard arguments, the function J is C 1 on H × D 1,2 .

In recent years, normalized solutions of Schrödinger equations have been widely studied. When searching for the existence of normalized solutions of Schrödinger equations in ℝ N , appears a new mass-critical exponent

l = 2 + 4 N .

Now, let us review the involved works. In the mass-subcritical case, Zuo Yang and Shijie Qi [2] proved that for all a > 0 , the following Schrödinger equations with potentials and non-autonomous nonlinearities

( − Δ u + V ( x ) u + λ u = f ( x , u )     in     ℝ N , ∫ ℝ N | u | 2 d x = a , u ∈ H 1 ( ℝ N ) ,

have a normalized solutions. Nicola Soave [3] in the mass-subcritical proved the nonlinear Schrödinger equation with combined power nonlinearities mass- critical and mass-supercritical cases studied of:

( − Δ u = λ u + μ | u | p − 2 u + | u | 2 * − 2 , u   in   ℝ N , N ≥ 3, ∫ ℝ N | u | 2 d x = a , u ∈ H 1 ( ℝ N ) .

have several stability/instability and existence/non-existence results of normalized ground state solutions. For g ( u ) is a superlinear, subcritical, Thomas Bartsch [4] studied the existence of infinitely many normalized solutions for the problem

− Δ u − g ( u ) = λ u , u ∈ H 1 ( ℝ N ) ,

By establishing the compactness of the minimizing sequences, Tianxiang Gou and Louis Jeanjean [5] in the mass-subcritical studied the existence of multiple positive solutions to the nonlinear Schrödinger systems:

( − Δ u = λ 1 u + μ 1 | u 1 | p 1 − 2 u 1 + β r 1 | u | r 1 − 2 u 1 | u 2 | r 2 , − Δ u = λ 2 u + μ 2 | u 2 | p 2 − 2 u 2 + β r 2 | u | r 1 | u 2 | r 2 − 2 u 2 .

In the mass-subcritical case, Masataka Shibata [6] studied for the nonlinear Schrödinger equations with the minimizing problem:

E a = inf { I ( u ) = 1 2 ∫ ℝ N | ∇ u | 2 d x − ∫ ℝ N   F ( | u | ) d x | u ∈ H 1 ( ℝ N ) , ∫ ℝ N | u | 2 d x = a }

where F ( t ) = ∫ 0 t   f ( s ) d s is a general nonlinear term. They proved E a is attained. That is to say, the Schrödinger equations have normalized solutions.

Moreover, for the I ( u ) = 1 2 ∫ ℝ N | ∇ u | 2 + V ( x ) | u | 2 d x − ∫ ℝ N   F ( | u | ) d x case, Norihisa

Ikoma and Yasuhito Miyamoto [7] showed the existence of the minimizer of the minimization problem E a , where V ( x ) → 0 as | x | → ∞ . They also obtained the conclusions that the normalized solutions of Schrödinger equations exist. In the mass-subcritical condition, Zhen Chen and Wenming Zou [8] basing on the refined energy estimates proved the existence of normalized solutions to the Schrödinger equations.

Other related normalized solutions problems of Schrödinger can be seen in [9] [10] [11] [12] [13] . Thus, the main purpose of this paper is to study the solution of Schrödinger-Maxwell equations satisfying normalization condition by using above results. In particular, the situation we consider will involve the presence of potential ϕ . In addition, the nonlinear term f ( u ) is mass-sub- critical and satisfies the following appropriate assumptions. In this case, the functional I is bounded from below and coercive on S ( a ) , which will be proved in Lemma 2.5.

We assume the following conditions throughout the paper:

(f1) f : ℝ N → ℝ is continuous.

(f2) l i m s → 0 f ( s ) s = 0 and l i m | s | → + ∞ f ( s ) | s | l − 1 = 0 with l = 2 + 4 N .

Moreover, c and c i are positive constants which may change from line to line.

Our main result is the following theorem:

Theorem 1.1 Suppose (f1) and (f2) hold. Then, for any a > 0 , problem (1.3) has a normalized ground state solution.

Since the functional J exhibits a strong indefiniteness. To avoid the difficulty we use the reduction method. Thus, we shall introduce the method.

For any u ∈ H , us consider the linear operator T ( u ) : D 1,2 → ℝ defined as

T ( u ) = ∫ ℝ N   u 2 v d x . (2.1)

Then, there exists a positive constant c 1 such that

∫ ℝ N   u 2 v d x ≤ ‖ u 2 ‖ L 2 N N + 2 ‖ v ‖ L 2 * ≤ ‖ u ‖ L 4 N N + 2 2 ‖ v ‖ L 2 * ≤ c 1 ‖ u ‖ H 2 ‖ v ‖ D 1,2 ,

because the following embeddings are continuous:

H ↪ L s ( ℝ N ) , ∀ s ∈ [ 2,2 * ] and D 1,2 ( ℝ N ) ↪ L 2 * ( ℝ N ) .

We set

g ( φ , v ) = ∫ ℝ N   ∇ φ ⋅ ∇ v d x ,   φ , v ∈ D 1 , 2 .

Obviously, g ( φ , v ) is linear in φ and v respectively.

Moreover, there exists a positive constant c 2 and c 3 such that for any φ , v ∈ D 1,2 ,

| g ( φ , v ) | ≤ c 2 ‖ φ ‖ D 1,2 ‖ v ‖ D 1,2 , (2.2)

g ( φ , v ) ≥ c 3 ‖ φ ‖ D 1,2 2 . (2.3)

Combining (2.2) and (2.3) we know that g ( φ , v ) is bounded and coercive. Hence, by the Lax-Milgram theorem we have that for every u ∈ H , for any v ∈ D 1,2 , there exists a unique ϕ u ∈ D 1,2 such that

T ( u ) v = g ( ϕ u , v ) .

Then, for any v ∈ D 1,2 , we obtain

∫ ℝ N   u 2 v d x = ∫ ℝ N   ∇ ϕ u ⋅ ∇ v d x , (2.4)

and using integration by parts, we have

∫ ℝ N   ∇ ϕ u ⋅ ∇ v d x = − ∫ ℝ N   v Δ ϕ u d x .

Therefore,

− Δ ϕ u = u 2 (2.5)

in a weak sense, and ϕ u has the following integral expression:

ϕ u = 1 4 π ∫ ℝ N u 2 ( y ) | x − y | d y , (2.6)

The functions ϕ u possess the following properties:

Lemma 2.1 For any u ∈ H , we have:

1) ‖ ϕ u ‖ D 1,2 ≤ c 4 ‖ u ‖ L 4 N N + 2 2 , where c 4 > 0 is independent of u. As a consequence there exists c 5 > 0 such that

∫ ℝ N   ϕ u u 2 d x ≤ c 5 ‖ u ‖ H 4 ;

2) ϕ u ≥ 0 .

Proof. 1) For any u ∈ H , using (2.5) we have

‖ ϕ u ‖ D 1,2 2 = ∫ ℝ N | ∇ ϕ u | 2 d x = − ∫ ℝ N   ϕ u Δ ϕ u d x = ∫ ℝ N   ϕ u u 2 d x ≤ ‖ ϕ u ‖ L 2 * ‖ u 2 ‖ L 2 N N + 2 ≤ c 4 ‖ ϕ u ‖ D 1,2 ‖ u ‖ L 4 N N + 2 2 ,

where c 4 is a positive constant. Hence, we obtain that

‖ ϕ u ‖ D 1,2 ≤ c 4 ‖ u ‖ L 4 N N + 2 2 ,

therefore there exists a positive constant c 5 such that

∫ ℝ N   ϕ u u 2 d x ≤ c 4 ‖ ϕ u ‖ D 1,2 ‖ u ‖ L 4 N N + 2 2 ≤ c 4 2 ‖ u ‖ L 4 N N + 2 4 ≤ c 5 ‖ u ‖ H 4 , (2.7)

because we know for any s ∈ [ 2,2 * ] , H ↪ L s ( ℝ N ) .

2) Obviously, by the expression (2.6) the conclusion holds. □

Now let us consider the functional I : H → ℝ N ,

I ( u ) : = J ( u , ϕ u ) .

Then I is C 1 .

By the definition of J, we have

I ( u ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + V ( x ) u 2 ) d x − 1 4 ∫ ℝ N | ∇ ϕ u | 2 d x + 1 2 ∫ ℝ N   ϕ u u 2 d x − ∫ ℝ N   F ( u ) d x .

Multiplying both members of (2.5) by ϕ u and integrating by parts, we obtain

∫ ℝ N | ∇ ϕ u | 2 d x = ∫ ℝ N   ϕ u u 2 d x .

Therefore, the functional I may be written as

I ( u ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + V ( x ) u 2 ) d x + 1 4 ∫ ℝ N   ϕ u u 2 d x − ∫ ℝ N   F ( u ) d x . (2.8)

The following lemma is Proposition 2.3 in [5] .

Lemma 2.2 The following statements are equivalent:

1) ( u , ϕ ) ∈ H × D 1,2 ( ℝ N ) is a critical point of J.

2) u is a critical point of I and ϕ = ϕ u .

Hence u is a solution to (1.3) if and only if u is the critical point of the functional (2.8). The critical point can be obtained as the minimizer under the constraint of L 2 -sphere

S ( a ) = { u ∈ H : ∫ ℝ N   u 2 d x = a } . (2.9)

We shall study the constraint problem as follows:

E a = inf u ∈ S ( a ) I ( u ) . (2.10)

The solution of (13) u = u ˜ is called a normalized ground state solution satisfying problem (3) if it has minimal energy among all solutions:

d I | S ( a ) ( u ˜ ) = 0   and   I ( u ˜ ) = inf { I ( u ) : d I | S ( a ) ( u ˜ ) = 0 , u ˜ ∈ S ( a ) } .

In this paper, we will be especially interested in the existence of normalized ground state solutions.

Lemma 2.3 We define Φ : H → D r 1,2 , Φ ( u ) = ϕ u , which is also the solution of the Equation (2.5) in D 1,2 . Let { u n } ⊂ S ( a ) be a minimizing sequence of I with satisfying u n ⇀ u in H . Then, Φ ( u n ) → Φ ( u ) in D 1,2 and we obtain

∫ ℝ N   Φ ( u n ) u n 2 d x → ∫ ℝ N   Φ ( u ) u 2 d x as n → ∞ . (2.11)

Proof. By (2.1), the following expressions hold

T ( u n ) v = ∫ ℝ N   u n 2 v d x ,   T ( u ) v = ∫ ℝ N   u 2 v d x .

Since u ∈ H and the embedding H r 1 ↪ L s is compact for any s ∈ ( 2,2 * ) , clearly we have

u 2 ∈ L 1 ( ℝ N ) ∩ L N ( ℝ N ) , (2.12)

then, by interpolation we have

u 2 ∈ L N 2 ( ℝ N ) .

Using again (2.12), we get

u 2 ∈ L 2 N N + 2 ( ℝ N ) .

Moreover, { u n } be a minimizing sequence and u n ⇀ u in H , we obtain

u n 2 → u 2     in   L 2 N N + 2 . (2.13)

Therefore, we get

| T ( u n ) v − T ( u ) v | = | ∫ ℝ N   u n 2 v d x − ∫ ℝ N   u 2 v d x | ≤ | u n 2 − u 2 | L 2 N N + 2 | v 6 | L 2 * ,

which implies that T ( u n ) converges strongly to T ( u ) .

Hence, we obtain

Φ ( u n ) → Φ ( u )     in   D 1,2 ,

Φ ( u n ) → Φ ( u )     in   L 2 * . (2.14)

By (2.13) and (2.14), we know that conclusion (2.11) holds. □

Lemma 2.4 (Gagliardo-Nirenberg inequality). For all u ∈ H , we have

‖ u ‖ p p ≤ C ( N ) ‖ ∇ u ‖ p ′ 2 ‖ u ‖ 2 p − p ′ , 2 < p < 2 *

where C ( N ) is a positive constant depending on N and p ′ = N ( p − 2 ) 2 p .

Lemma 2.5 Suppose (f1) and (f2) hold, than for any a > 0 , the functional I is bounded from below and coercive on S(a).

Proof. Assumptions (f1) and (f2) imply that for any ε > 0 , there exist C ε > 0 such that

F ( s ) ≤ C ε | s | 2 + ε | s | l , ∀ s ∈ ℝ .

Hence, according to Lemma 2.4 with p = l = 2 + 4 N , we obtain that

| ∫ ℝ N   F ( u ) d s | ≤ C ε ‖ u ‖ 2 2 + ε ‖ u ‖ l l ≤ C ε ‖ u ‖ 2 2 + ε C ( N ) ‖ ∇ u ‖ 2 2 ‖ u ‖ 2 4 N

Choose ε such that ε C ( N ) a 2 N = 1 4 , than

I ( u ) = 1 2 ∫ ℝ N ( | ∇ u | 2 + u 2 ) d x + 1 4 ∫ ℝ N   ϕ u u 2 d x − ∫ ℝ N   F ( u ) d x ≥ 1 2 ‖ ∇ u ‖ 2 2 + 1 2 ∫ ℝ N   u 2 d x − ∫ ℝ N   F ( u ) d x ≥ 1 4 ‖ ∇ u ‖ 2 2 − C a > − ∞

Therefore, I is bounded from below and coercive on S ( a ) . □

The following lemma is Lemma 2.2 in [6] .

Lemma 2.6 Suppose (f1) and (f2) hold and { u n } n ∈ N is a bounded sequence in

H . If l i m n → ∞ | u n | 2 2 = 0 holds, then it is true that

l i m n → ∞ ∫ ℝ N   F ( u n ) d x = 0.

Next, we collect a variant of Lemma 2.2 in [14] . The proof is similar, so we omit it.

Lemma 2.7 Suppose (f1) and (f2) hold and { u n } n ∈ N is a bounded sequence in H , then we have u n ⇀ u in H , thus

l i m n → ∞ ∫ ℝ N [ F ( u n ) − F ( u ) − F ( u n − u ) ] d x = 0.

Proof of Theorem 1.1. Let { u n } ⊂ S ( a ) be a minimizing sequence of I with concerning E a . Then, by (9) we obtain

I ( u n ) = 1 2 ∫ ℝ N ( | ∇ u n | 2 + u n 2 ) d x + 1 4 ∫ ℝ n   ϕ u n u n 2 d x − ∫ ℝ N   F ( u n ) d x .

According to Lemma 2.5, the sequence { u n } is bounded in H . Letting u 0 be in H . Moreover, we know that the embedding H ↪ L s ( ℝ N ) is compact. Hence, we conclude

u n ⇀ u 0     in   H , (2.15)

u n → u 0     in   L s ( ℝ N ) ,   2 < s < 2 * , (2.16)

u n → u 0     a .e .   in   ℝ N .

We also have

I ( u 0 ) = 1 2 ∫ ℝ N ( | ∇ u 0 | 2 + u 0 2 ) d x + 1 4 ∫ ℝ n   ϕ u 0 u 0 2 d x − ∫ ℝ N   F ( u 0 ) d x .

Since (19) holds, we have l i m n → ∞ | u n − u 0 | 2 2 = 0 . Then, by Lesmma 2.6 we obtain

l i m n → ∞ ∫ ℝ N   F ( u n − u 0 ) d x = 0.

Moreover, by Lemma 2.7 we have

l i m n → ∞ ∫ ℝ N [ F ( u n ) − F ( u 0 ) ] d x = 0.

which implies

∫ ℝ N   F ( u n ) d x → ∫ ℝ N   F ( u 0 ) d x     as     n → ∞ . (2.17)

Hence, combining weak lower semicontinuity of the norm ‖ ⋅ ‖ H , Lemma 2.3 and (2.17), we have

E a ≤ I ( u 0 ) ≤ lim inf n → ∞ I ( u n ) = E a ,

which implies I ( u 0 ) = E a . Then, u 0 satisfies

( − Δ u 0 + u 0 + ϕ u 0 + λ u 0 = f ( u 0 )     in   ℝ N , − Δ ϕ = u 0 2     in   ℝ N ,

and ∫ ℝ N | u 0 | 2 d x = a . Therefore, problem (1.3) has a normalized ground state solution. □

The author declares no conflicts of interest.

Wang, Z.Y. (2023) Normalized Solutions of Mass-Subcritical Schrödinger-Maxwell Equations. Open Access Library Journal, 10: e10956. https://doi.org/10.4236/oalib.1110956