Research Article | Open Access

# Multiplicity of Solutions for Nonlocal Elliptic System of -Kirchhoff Type

**Academic Editor:**Josip E. Peฤariฤ

#### Abstract

This paper is concerned with the following nonlocal elliptic system of (*p,q*)-Kirchhoff
type , in ฮฉ, , in ฮฉ, , on Under bounded condition on *M* and some novel and periodic condition on *F*, some new
results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by
means of Bonanno's multiple critical points theorems without the Palais-Smale condition
and Ricceri's three critical points theorem, respectively.

#### 1. Introduction and Preliminaries

We are concerned with the following nonlocal elliptic system of -Kirchhoff type: where is a bounded smooth domain, , , is the Laplacian operator and , are continuous functions with bounded conditions.There are two positive constants such that Furthermore, is a function such that is measurable in for all and is in for a.e. , and denotes the partial derivative of with respect to . Moreover, satisfies the following. for a.e. . There exist two positive constants and a positive real function such that

The system (1.1) is related to a model given by the equation of elastic strings which was proposed by Kirchhoff [1] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings, where the parameters in (1.5) have the following meanings: is the mass density, is the initial tension, is the area of the cross-section, is the Young modulus of the material, and is the length of the string. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations.

Later, (1.5) was developed to the form where is a given function. After that, many people studied the nonlocal elliptic boundary value problem which is the stationary counterpart of (1.6). It is pointed out in [2] that (1.7) models several physical and biological systems, where describes a process which depends on the average of itself (e.g., population density). By using the methods of sub and supersolutions, variational methods, and other techniques, many results of (1.7) were obtained, we can refer to [2โ12] and the references therein. In particular, Alves et al. [2, Theoremโโ4] supposes that satisfies bounded condition and satisfies condition , that is, for some and such that where one positive solutions for (1.7) was obtained. It is well known that condition plays an important role for showing the boundedness of Palais-Smale sequences. More recently, Corrรชa and Nascimento in [13] studied a nonlocal elliptic system of Kirchhoff type where is the unit exterior vector on , and satisfy suitable assumptions, where a special and important condition: periodic condition on nonlinearity was assumed. They obtained the existence of a weak solution for the nonlocal elliptic system of Kirchhoff type under Neumann boundary condition via Ekeland's Variational Principle.

In the present paper, our objective is to consider the nonlocal elliptic system of Kirchhoff-type (1.1), instead of the nonlocal elliptic system of Kirchhoff type and single Kirchhoff type equation. Under bounded condition on and some novel conditions without condition and periodic condition on , we will prove the existence of two solutions and three solutions of system (1.1) by means of one multiple critical points theorem without the Palais-Smale condition of Bonanno in [14] and an equivalent formulation [15, Theoremโโ2.3] of Ricceri's three critical points theorem [16, Theoremโโ1], respectively.

In order to state our main results, we need the following preliminaries.

Let be the Cartesian product of two Sobolev spaces, which is a reflexive real Banach space endowed with the norm where and denote the norms of and , respectively. That is, for all and .

Since and , and are compactly embedded in . Let then we have . Furthermore, it is known from [17] that where denotes the Gamma function and is the Lebesgue measure of . Additionally, (1.11) is an equality when is a ball.

Recall that is called a weak solution of system (1.1) if for all . Define the functional given by for all , and where By the conditions and , it is easy to see that and a critical point of corresponds to a weak solution of the system (1.1).

Now, giving and choosing such that , where . Next we give some notations. Moreover, let be positive constants, denote

Now we are ready to state our main results for the system (1.1)

Theorem 1.1. *Assume that hold and there are three positive constants with such that
**
Then, for each **
there exists a positive real number such that the system (1.1) has at least two weak solutions whose norms in are less than some positive constant .*

Theorem 1.2. *Assume that hold and there are two positive constants , with such that ** for a.e. and all ; **. **Then there exist an open interval and a positive real number such that, for each , the system (1.1) has at least three weak solutions whose norms are less than .*

#### 2. Proofs of Main Results

Before proving the results, we state one multiple critical points theorem without the Palais-Smale condition of Bonanno in [13] and an equivalent formulation [14, Theoremโโ2.3] of Ricceri's three critical points theorem [15, Theoremโโ1], which are our main tools.

Theorem 2.1 (see [14, Theoremโโ2.1]). *Let be a reflexive real Banach space, and let be two sequentially weakly lower semicontinuous functions. Assume that is (strongly) continuous and satisfies . Assume also that there exist two constants and such that **;
**;**; **where
**
Then, for each
**
the functional has two local minima which lie in and , respectively.*

Theorem 2.2 (see [15, Theoremโโ2.3]). *Let be a separable and reflexive real Banach space. is a continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functional whose Gรขteaux derivative admits a continuous inverse on ; is a continuously Gรขteaux differentiable functional whose Gรขteaux derivative is compact. Suppose that ** for each ;**There are a real number , and such that ;**.**Then there exist an open interval and a positive real number such that, for each , the equation has at least three weak solutions whose norms in are less than .*

First, we give one basic lemma.

Lemma 2.3. *Assume that and hold; let
**
for all . Then and are continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the Gรขteaux derivative of admits a continuous inverse on and the Gรขteaux derivative of is compact.*

*Proof. *By condition , it is easy to see that is continuously Gรขteaux differentiable. Moreover, the Gรขteaux derivative of admits a continuous inverse on . Thanks to , and , is continuously Gรขteaux differentiable and sequentially weakly lower semicontinuous functional whose Gรขteaux derivative is compact. Next We will prove that is a sequentially weakly lower semicontinuous functional. Indeed, for any with in , then in and in . Therefore,
due to the weakly lower semicontinuity of norm. Hence by virtue of the continuity and monotonicity of and , we conclude that
Consequently, is a sequentially weakly lower semicontinuous functional.

*Proof of Theorem 1.1. *Let
for all . Under condition , by a simple computation, we have

Therefore, (2.7) implies that

Put

Denote
and is the closure of in the weak topology.

Set
Then , where and
Consequently, (2.7) and (2.12) imply that
Furthermore, (2.13) implies that
On the other hand, by , (1.17), and, (2.11), one has
For each with , and , by (2.7), we conclude
Therefore, the combination of (2.15) and (2.16) implies
By (2.14) and (2.17), we have
Similarly, for every such that , where is a positive real number, and , one has
By virtue of being sequentially weakly lower semicontinuous, then . Consequently,
It implies that
By (2.18)โ(2.22), we conclude
Therefore, the conditions , , and in Theorem 2.1 are satisfied. Consequently, by Lemma 2.3 and above facts, the functional has two local minima , which lie in and , respectively. Since , are the solutions of the equation
Then are the weak solutions of system (1.1).

Since , by (1.10) and (2.7),
which implies there exists a positive real number such that the norms of in are less than some positive constant . This completes the proof of Theorem 1.1.

*Proof of Theorem 1.2. *Let
for all . By and (2.7), we have
where are positive constants. Since , (2.27) implies that
The same as in (2.11), defining a function , and letting , then (2.12) is also satisfied. Choosing , by (2.7), (2.12), and , we conclude
By and the definitions of and , one has
For every such that , and , one has
By the combination of (2.30) and (2.31), we have
Therefore,
Note that , we conclude that
Hence, by Lemma 2.3 and above facts, and satisfy all conditions of Theorem 2.2; then the conclusion directly follows from Theorem 2.2.

#### Acknowledgments

The authors would like to thank the referee for the useful suggestions. This work is supported in part by the National Natural Science Foundation of China (10961028), Yunnan NSF Grant no. 2010CD086, and the Foundation of young teachers of Qujing Normal University (2009QN018).

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Copyright ยฉ 2011 Bitao Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.