Research Article | Open Access
Multiplicity of Solutions for Nonlocal Elliptic System of -Kirchhoff Type
This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type , in Ω, , in Ω, , 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  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  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 [2–12] and the references therein. In particular, Alves et al. [2, Theorem 4] 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, Corrêa and Nascimento in  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  and an equivalent formulation [15, Theorem 2.3] of Ricceri's three critical points theorem [16, Theorem 1], 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  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  and an equivalent formulation [14, Theorem 2.3] of Ricceri's three critical points theorem [15, Theorem 1], which are our main tools.
Theorem 2.1 (see [14, Theorem 2.1]). 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, Theorem 2.3]). Let be a separable and reflexive real Banach space. is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on ; is a continuously Gâteaux differentiable functional whose Gâteaux 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 Gâteaux differentiable and sequentially weakly lower semicontinuous functionals. Moreover, the Gâteaux derivative of admits a continuous inverse on and the Gâteaux derivative of is compact.
Proof. By condition , it is easy to see that is continuously Gâteaux differentiable. Moreover, the Gâteaux derivative of admits a continuous inverse on . Thanks to , and , is continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux 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
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)–(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.
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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