Consider the following nonlinear Schrödinger-Kirchhoff type problem:

− ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) in R N (1.1)

where constants a > 0 , b ≥ 0 ， V ∈ ( R N , R ) and f ∈ ( R N × R , R ) satisfy some assumptions.

In (1.1), if a = 1 , b = 0 , then Equation (1.1) is the following well-known Schrödinger equation:

− Δ u + V ( x ) u = f ( x , u ) , in R N (1.2)

The Schrödinger equation has been studied by Brezis and Lieb [1] for a general class of autonomous, and by many authors [2] [3] [4] for periodic data.

If V ( x ) ≡ 0 and R N are replaced by a smooth bounded domain Ω ∈ R N , then Equation (1.1) is a Dirichlet problem of Kirchhoff type [5] :

{ − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u = f ( x , u ) ,     on   Ω u = 0 ,                                                                                   in   ∂ Ω (1.3)

Many interesting studies by variational methods can be found in [6] - [11]. It is related to the following stationary analogue of the equation:

u t t − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u = f ( x , u ) (1.4)

which is an extension of classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early studies of Kirchhoff equation can be refer to in [12] - [19] and the references therein.

In order to reduce some extra statement we need to describe the eigenvalue of the Schrodinger operator − a Δ + V . We are considered the sequence { λ i } , λ i < 0 < λ i + 1 < λ ∞ = lim i → ∞ λ i of minimax values, it is known that λ ∞ , if finite, is lower bound of the essential spectrum σ ( − a Δ + V ) of Schrodinger operator − a Δ + V (many details can see [20]).

Our aim is to study ground state solutions to (1.1) with a class of nonlinearities. The energy functional I : X → R given by:

I ( u ) = a 2 ∫ R N | ∇ u | 2 d x + b 4 ( ∫ R N | ∇ u | 2 d x ) 2 + 1 2 ∫ R N V ( x ) u 2 d x − ∫ R N F ( x , u ) d x (1.5)

Set F ( x , u ) = ∫ 0 u f ( x , s ) d s . We make the following assumptions:

(V) V ∈ C ( R N ) bounded from below, there is M > 0 , such that

meas { x ∈ R N | V ( x ) < M } < ∞

(F) f : R N × R → R is measurable in x ∈ R N and continuous in u ∈ R for a.e. x ∈ R N .

F : R N × R → R is differentiable with respect to the second variable u ∈ R and F ( x , 0 ) = 0 for a.e. x ∈ R N .

(F1) There are A > 0 and 2 < p < 2 * = 2 N N − 2 , such that

| f ( x , u ) | ≤ A ( 1 + | u | p − 1 ) for all u ∈ R and a.e. x ∈ R N .

(F2) f ( x , u ) = o ( | u | ) as | u | → 0 uniformly in x ∈ R N .

Our goal is to find a ground state solution of Equation (1.1) i.e., a critical point being a minimizer of I on the Nehari-Pankov manifold defined as follows:

N = { u ∈ X \ X − | 〈 I ′ ( u ) , u 〉 = 0     and     〈 I ′ ( u ) , v 〉 = 0     for     any     v ∈ X − } .

Since N contains all critical points of I, then a ground state is a energy solution. To conquer the linking geometry of I and to structure N , we give the following assumptions:

(F3) f ( x , u ) u ≥ 4 F ( x , u ) ≥ 0 for all u ∈ R and a.e. x ∈ R N .

(F4) F ( x , u ) | u | 4 → ∞ as | u | → ∞ uniformly in x ∈ R N .

We have our main result as follow.

Theorem 1.1. Suppose that (V), (F), (F1)-(F4) are satisfied. Then (1.1) has a ground state solution, there is a nontrivial critical point of I, such that I ( u ) = inf N I .

Our approach is the same a new linking-type result of [21] involving the Nehari-Pankov manifold. In the next section, we present a critical point theory and variational setting. In Section 3, we state some relevant lemmas and prove Theorem 1.1.

Set Hilbertspace

H 1 ( R N ) = { u ∈ L 2 ( R N ) , ∇ u ∈ L 2 ( R N ) }

with the norm

‖ u ‖ H 1 = ( ∫ R N ( | ∇ u | 2 + u 2 ) d x ) 1 2

To prove our theorems, let

X = { u ∈ H 1 ( R N ) | ∫ R N ( a | ∇ u | 2 + V ( x ) u 2 ) d x < ∞ }

and define the inner product of X by the following formula:

〈 u , v 〉 = ∫ R N a 〈 ∇ u + , ∇ u + 〉 + V ( x ) 〈 u + , u + 〉 d x − ∫ R N a 〈 ∇ u − , ∇ u − 〉 + V ( x ) 〈 u − , u − 〉 d x

and norm given by ‖ u ‖ = 〈 u , u 〉 1 2 , where u = u + + u − , v = v + + v − ∈ X = X + ⊕ X − ,

therefore X is a Hilbert space with the norm ‖ ‖ . It is easy to see that

∫ R N ( a | ∇ u | 2 + V ( x ) u 2 ) d x = ‖ u + ‖ 2 − ‖ u − ‖ 2 .

Let X − is the finite dimentional space spanned by the negative eigenfunctions with λ i < 0 , ( i = 1 , ⋯ , n ) and let X + = ( X − ) ⊥ . In view of (V), we may find continuous projections u + and u − of X onto X + and X − , respectively, such that X = X + + X − and X + is the positive eigenspace and X − is the negative eigenspace of the operator − a Δ + V .

For any t ∈ [ 2 , 2 * ] , the embedding x → L t ( R N ) is continuous. Consequently, there is a constant γ t > 0 , such that

‖ u ‖ t ≤ γ t ‖ u ‖ , ∀ u ∈ X (2.1)

Moreover, we know that under assumption (V), the embedding x → L t ( R N ) is compact for any t ∈ [ 2 , 2 * ) by lemma 3.4 in [22].

Then the relative functional of (1.1) can be writed by

I ( u ) = 1 2 ‖ u + ‖ 2 − 1 2 ‖ u − ‖ 2 + 1 4 ‖ u ‖ 2 4 − ∫ R N F ( x , u ) d x

and under assumptions (F1) and (F2), I ∈ C 1 ( X , R ) , for any u , v ∈ X ,

〈 I ′ ( u ) , v 〉 = ( a + b ∫ R N | ∇ u | 2 d x ) ∫ R N ∇ u ∇ v d x + ∫ R N V ( x ) u v d x − ∫ R N f ( x , u ) v d x .

From the assumptions that (F1) and (F2), we know a functional I ∈ C 1 ( X , R ) , then the following conditions hold:

(A1) I is lower semicontinuous.

(A2) I ′ is weak continuous.

Let P ⊂ X \ X − and P ≠ ∅ . The linking geometry of I is described by the following conditions:

(A3) There exists r > 0 , such that inf u ∈ X + , ‖ u ‖ = r I ( u ) > 0 .

(A4) For every u ∈ P , there exists R ( u ) > r , such that sup ∂ M ( u ) I ≤ I ( 0 ) = 0 where M ( u ) : = { t u + v ∈ X | t ≥ 0 , v ∈ X −   and   ‖ t u + v ‖ ≤ R ( u ) } .

(A5) If u ∈ N , then I ( u ) ≥ I ( t u + v ) for any t ≥ 0 and v ∈ X − where N = { u ∈ P | 〈 I ′ ( u ) , u 〉 = 0   and   〈 I ′ ( u ) , v 〉 = 0   for   any   v ∈ X − } .

If P = X \ X − , then N has been introduced by Pankov [23].

For any A ⊂ X , I ⊂ [ 0 , ∞ ) , such that 0 ∈ I , and h : A × I → X , we collect the following assumptions [24] :

(h1) h is a continuous.

(h2) h ( 0 , u ) = u for all u ∈ A .

(h3) I ( u ) ≥ I ( h ( u , t ) ) for all ( u , t ) ∈ A × I .

(h4) each ( u , t ) ∈ A × I has an open neighborhood W in the product topology of ( X , T ) and I such that the set { v − h ( v , s ) : ( v , s ) ∈ W ∩ ( A × I ) } is contained in a finite dimentional subspace of X.

Theorem 2.1. (linking theory [21] [24]) Suppose that I ∈ C 1 ( X , R ) satisfies conditions A1-A4).

Then there exists a Cerami sequence { u n } (i.e. I ( u n ) → c , ( 1 + ‖ u n ‖ ) I ′ ( u n ) → ), where

c = inf u ∈ P inf h ∈ Γ ( u ) sup u ′ ∈ M ( u ) I ( h ( u ′ , 1 ) ) > 0

Γ ( u ) = { h ∈ C ( M ( u ) × [ 0 , 1 ] ) | h   satisfies   ( h1 )   -   ( h4 ) } .

Suppose that in addition (A5) hold. Then c ≤ inf N I , and if c ≥ I ( u ) , for some critical point u ∈ P . then

c = inf N I

where N = { u ∈ P | 〈 I ′ ( u ) , u 〉 = 0   and   〈 I ′ ( u ) , v 〉 = 0     for   any   v ∈ X − } .

We need the following lemmas.

Lemma 3.1. Assume that I ∈ C 1 ( X , R ) and assumptions (V), (F), (F1) and (F2) are satisfied, then conditions (A1) and (A2) are hold.

Proof. Set J ( u ) = ∫ R N F ( x , u ) d x . According to (1.5), it suffices to show that J ( u ) is weakly continuous on X.

For any ε > 0 , by (F1) and (F2), there is C ε > 0 , such that

| f ( x , u ) | = ε | u | + C ε | u | p − 1 for u ∈ R (3.1)

let { u n } ⊂ X and u n ⇀ u in X, then { u n } is bounded in X and converges to u in L t ( R N ) , where 2 ≤ t < 2 * , by (3.1) we have

| J ( u n ) − J ( u ) | = | ∫ R N F ( x , u n ( x ) ) − F ( x , u ( x ) ) d x | = | ∫ R N ∫ 0 1 f ( x , u + s ( u n − u ) ) ( u n − u ) d s d x | ≤ ∫ R N [ ε | u + s ( u n − u ) | + C ε | u + s ( u n − u ) | p − 1 ] | u n − u | d x ≤ ∫ R N [ ε ( | u | + 1 2 | u n − u | ) + C ε , p ( | u | p − 1 + | u n − u | p − 1 ) ] | u n − u | d x ≤ c ( ε + C ε , p ‖ u n − u ‖ p ) ≤ C ε , n

Therefore, J ( u ) is weakly continuous on X. This shows (A1) and (A2) hold.□

Lemma 3.2. Assume that assumptions (F), (F1) and (F2) are satisfied, then condition (A3) are hold.

Proof. For any 0 < ε < 1 4 γ 2 2 ( γ 2 is depended on (2.1)), by (3.1) we know that there is C ε > 0 such that

| F ( x , u ) | ≤ ε 2 | u | 2 + C ε p | u | p for u ∈ R . (3.2)

Note that (F4) implies p > 4 , then there is r > 0 , set S = { u ∈ X + , ‖ u ‖ = r } , for every u ∈ S , from (3.2) we have

I ( u ) = 1 2 ‖ u + ‖ + 1 4 ‖ u ‖ 2 4 − ∫ R N F ( x , u ) d x ≥ 1 2 ‖ u + ‖ 2 − ε 2 ‖ u ‖ 2 2 − C ε p ‖ u ‖ p p ≥ 1 4 ( 1 − γ 2 2 ε ) ‖ u + ‖ 2

Therefore

inf u ∈ s I ( u ) ≥ 1 4 ( 1 − γ 2 2 ε ) r 2 > 0

This shows (A3) hold. □

Hence similarly as in [17] [19], we obtain that conditions (A1)-(A3) are hold. Moreover, obviously, I ( 0 ) = 0 , I has the linking geometry.

Lemma 3.3. Assume that assumptions (V), (F), (F1)-(F4) are satisfied, then condition (A4) is hold.

Proof. Choose a fixed u ∈ P and there are t n > 0 and v n ∈ X − such that I ( t n u + v n ) > 0 and ‖ t n u + v n ‖ → ∞ as n → ∞ , let z n = t n u + v n ‖ t n u + v n ‖ and we let z n ⇀ z in X and z n ( x ) → z ( x ) a.e. R N for some z ∈ X . Since

0 < I ( t n u + v n ) ≤ 1 2 ‖ t n u + ‖ 2 − 1 2 ‖ t n u − + v n ‖ 2 + 1 4 ‖ t n u + ‖ 4

then

0 < t n 4 ‖ t n u + v n ‖ 4 ‖ u + ‖ 4 = ‖ z n + ‖ 4 ≤ ‖ z n ‖ 4 = 1

and we may assume that z + ≠ 0 . Hence z ≠ 0 and

| t n u ( x ) + v n ( x ) | = | z n ( x ) | ‖ t n u + v n ‖ → ∞ as n → ∞ and z ( x ) ≠ 0 .

Then by (F4) and Fatou’s lemma, we have

0 < I ( t n u + v n ) ‖ t n u + v n ‖ 4 ≤ γ 2 4 4 − ∫ R N F ( x , t n u + v n ) | t n u + v n | 4 | z n | 4 d x → − ∞ as t n → ∞

and is a contradiction. This shows (A4) hold. □

Remark 3.4. The inspection of proof of lemma 3.3, then due to conclusion of that lemma 3.3, for any t ≥ 0 , v ∈ X − such that (A5) holds.

Lemma 3.5. Assume that (V), (F), (F1) and (F2) are satisfied. If { u n } ∈ X is a bounded Cerami sequence in X, then { u n } has a convergent subsequence.

Proof: Let { u n } is a Cerami sequence in X, that is I ( u n ) → c and ( 1 + ‖ u n ‖ ) I ′ ( u n ) → 0 .

From I ( u n ) → ∞ as ‖ u n ‖ → ∞ , we get that I is coercive.

Hence it is easy to get that { u n } is bounded. Since

o ( ‖ u n ‖ ) = 〈 I ′ ( u n ) , u n 〉 = ‖ u n + ‖ 2 − ‖ u n − ‖ 2 + ‖ u n ‖ 2 4 − ∫ R N f ( x , u n ) u n d x

One has

o ( 1 ) + ‖ u n − ‖ 2 = ‖ u n + ‖ 2 + ‖ u n ‖ 2 4 − ∫ R N f ( x , u n ) u n d x (3.3)

Passing to a subsequence, we may assume that u n ⇀ u in X. Therefore

0 = 〈 I ′ ( u ) , u 〉 = ‖ u + ‖ 2 − ‖ u − ‖ 2 + ‖ u ‖ 2 4 − ∫ R N f ( x , u ) u d x

and we have

‖ u − ‖ 2 = ‖ u + ‖ 2 + ‖ u ‖ 2 4 − ∫ R N f ( x , u ) u d x (3.4)

Moreover, since dim X − < ∞ , we have that u n − → u − . Then combine (3.3) with (3.4) push out

lim n → ∞ ( ‖ u n + ‖ 2 + ‖ u n ‖ 2 4 − ∫ R N f ( x , u n ) u n d x ) = ‖ u + ‖ 2 + ‖ u ‖ 2 4 − ∫ R N f ( x , u ) u d x (3.5)

It follows from (A2) and Fatou’s lemma that

lim _ n → ∞ ( ‖ u n ‖ 2 4 − ∫ R N f ( x , u n ) u n d x ) ≥ ‖ u ‖ 2 4 − ∫ R N f ( x , u ) u d x (3.6)

Combining (3.5) and (3.6), we can get

lim _ n → ∞ ‖ u n + ‖ 2 ≤ ‖ u + ‖ 2

and then ‖ u n + ‖ → ‖ u + ‖ . Consequently, u n → u in X.

This shows lemma’s conclusion hold. □

Proof of Theorem 1.1. Observe that (F3) implies that P = X \ X − . In view of theorem 2.1, there is a bounded Cerami sequence { u n } and by lemma 3.5, { u n } has a convergent subsequence. Then passing to a subsequence, let u n ⇀ u in X, then u n ( x ) → u ( x ) a.e. in R N . From lemma 3.1, we can obtain that I is lower semicontinuous. One has that

∫ R N | ∇ u | 2 d x ≤ lim _ n → ∞ ∫ R N | ∇ u n | 2 d x

Observe that by (F3) and in view of Fatou’s lemma that

c = lim _ n → ∞ I ( u n ) = lim _ n → ∞ ( I ( u n ) − 1 4 〈 I ′ ( u n ) , u n 〉 ) ≥ I (u)

Since u ∈ N , then by theorem 2.1 we have

c = inf N I = I (u)

This shows the main result hold. □

To sum up the above arguments, we through the linking theory to prove the existence of ground state solution of Schrödinger-Kirchhoff type equation.

The author declares no conflicts of interest regarding the publication of this paper.

Li, W.D. (2020) Ground States of Nonlinear Schrödinger-Kirchhoff Type Equation. Journal of Applied Mathematics and Physics, 8, 307-314. https://doi.org/10.4236/jamp.2020.82025