Consider the following nonlinear Schrödinger-Kirchhoff type problem:
where constants ， and satisfy some assumptions.
In (1.1), if , then Equation (1.1) is the following well-known Schrödinger equation:
, in (1.2)
The Schrödinger equation has been studied by Brezis and Lieb  for a general class of autonomous, and by many authors    for periodic data.
If and are replaced by a smooth bounded domain , then Equation (1.1) is a Dirichlet problem of Kirchhoff type  :
Many interesting studies by variational methods can be found in  - . It is related to the following stationary analogue of the equation:
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  -  and the references therein.
In order to reduce some extra statement we need to describe the eigenvalue of the Schrodinger operator . We are considered the sequence , of minimax values, it is known that , if finite, is lower bound of the essential spectrum of Schrodinger operator (many details can see ).
Our aim is to study ground state solutions to (1.1) with a class of nonlinearities. The energy functional given by:
Set . We make the following assumptions:
(V) bounded from below, there is , such that
(F) is measurable in and continuous in for a.e. .
is differentiable with respect to the second variable and for a.e. .
(F1) There are and , such that
for all and a.e. .
(F2) as uniformly in .
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:
Since contains all critical points of I, then a ground state is a energy solution. To conquer the linking geometry of I and to structure , we give the following assumptions:
(F3) for all and a.e. .
(F4) as uniformly in .
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 .
Our approach is the same a new linking-type result of  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.
2. Preliminaries and Variational Setting
with the norm
To prove our theorems, let
and define the inner product of X by the following formula:
and norm given by , where , ,
therefore X is a Hilbert space with the norm . It is easy to see that
Let is the finite dimentional space spanned by the negative eigenfunctions with and let . In view of (V), we may find continuous projections and of X onto and , respectively, such that and is the positive eigenspace and is the negative eigenspace of the operator .
For any , the embedding is continuous. Consequently, there is a constant , such that
Moreover, we know that under assumption (V), the embedding is compact for any by lemma 3.4 in .
Then the relative functional of (1.1) can be writed by
and under assumptions (F1) and (F2), , for any ,
From the assumptions that (F1) and (F2), we know a functional , then the following conditions hold:
(A1) I is lower semicontinuous.
(A2) is weak continuous.
Let and . The linking geometry of I is described by the following conditions:
(A3) There exists , such that .
(A4) For every , there exists , such that where .
(A5) If , then for any and where .
If , then N has been introduced by Pankov .
For any , such that , and , we collect the following assumptions  :
(h1) h is a continuous.
(h2) for all .
(h3) for all .
(h4) each has an open neighborhood W in the product topology of and I such that the set is contained in a finite dimentional subspace of X.
Theorem 2.1. (linking theory  ) Suppose that satisfies conditions A1-A4).
Then there exists a Cerami sequence (i.e. , ), where
Suppose that in addition (A5) hold. Then , and if , for some critical point . then
3. Proof of Theorem
We need the following lemmas.
Lemma 3.1. Assume that and assumptions (V), (F), (F1) and (F2) are satisfied, then conditions (A1) and (A2) are hold.
Proof. Set . According to (1.5), it suffices to show that is weakly continuous on X.
For any , by (F1) and (F2), there is , such that
let and in X, then is bounded in X and converges to u in , where , by (3.1) we have
Therefore, 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 ( is depended on (2.1)), by (3.1) we know that there is such that
for . (3.2)
Note that (F4) implies , then there is , set , for every , from (3.2) we have
This shows (A3) hold. □
Hence similarly as in  , we obtain that conditions (A1)-(A3) are hold. Moreover, obviously, , 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 and there are and such that and as , let and we let in X and a.e. for some . Since
and we may assume that . Hence and
as and .
Then by (F4) and Fatou’s lemma, we have
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 , such that (A5) holds.
Lemma 3.5. Assume that (V), (F), (F1) and (F2) are satisfied. If is a bounded Cerami sequence in X, then has a convergent subsequence.
Proof: Let is a Cerami sequence in X, that is and .
From as , we get that I is coercive.
Hence it is easy to get that is bounded. Since
Passing to a subsequence, we may assume that in X. Therefore
and we have
Moreover, since , we have that . Then combine (3.3) with (3.4) push out
It follows from (A2) and Fatou’s lemma that
Combining (3.5) and (3.6), we can get
and then . Consequently, in X.
This shows lemma’s conclusion hold. □
Proof of Theorem 1.1. Observe that (F3) implies that . In view of theorem 2.1, there is a bounded Cerami sequence and by lemma 3.5, has a convergent subsequence. Then passing to a subsequence, let in X, then a.e. in . From lemma 3.1, we can obtain that I is lower semicontinuous. One has that
Observe that by (F3) and in view of Fatou’s lemma that
Since , then by theorem 2.1 we have
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.
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