Abstract
In this paper, we study the solutions of a nonlocal elliptic system of Kirchhoff type on a bounded domain based on the three critical points theorem introduced by Ricceri. Firstly, we establish the existence of three weak solutions under appropriate hypotheses; then, we prove the existence of at least three weak solutions for the nonlocal elliptic system of Kirchhoff type.
Keywords:
Kirchhoff type system; multiple solutions; three critical points theory1 Introduction and main results
We consider the boundary problem involving Kirchhoff
where () is a bounded smooth domain, , , , is the pLaplacian operator . are functions such that , are measurable in Ω for all and , are continuously differentiable in for a.e. . is the partial derivative of F with respect to i, , so is . , , are continuous functions which satisfy the following bounded conditions.
(M) There exist two positive constants , such that
Here and in the sequel, X denotes the Cartesian product of two Sobolev spaces and , i.e., . The reflexive real Banach space X is endowed with the norm
Since and , and are compactly embedded in . Let
then one has . Furthermore, it is known from [1] that
where Γ is the gamma function and is the Lebesgue measure of Ω. As usual, by a weak solution of system (1.1), we mean any such that
System (1.1) is related to the stationary version of a model established by Kirchhoff [2]. More precisely, Kirchhoff proposed the following model:
which extends D’Alembert’s wave equation with free vibrations of elastic strings, where ρ denotes the mass density, denotes the initial tension, h denotes the area of the crosssection, E denotes the Young modulus of the material, and L denotes the length of the string. Kirchhoff’s model considers the changes in length of the string produced during the vibrations.
Later, (1.1) was developed into the following form:
where is a given function. After that, many authors studied the following problem:
which is the stationary counterpart of (1.6). By applying variational methods and other techniques, many results of (1.7) were obtained, the reader is referred to [313] and the references therein. In particular, Alves et al. [[3], Theorem 4] supposed that M satisfies bounded condition (M) and satisfies the condition
where ; one positive solution for (1.7) was given.
In [14], using Ekeland’s variational principle, Corrêa and Nascimento proved the existence of a weak solution for the boundary problem associated with the nonlocal elliptic system of pKirchhoff type
where η is the unit exterior vector on ∂Ω, and , (), f, g satisfy suitable assumptions.
In [15], when in (1.1), Bitao Cheng et al. studied the existence of two solutions and three solutions of the following nonlocal elliptic system:
In this paper, our objective is to prove the existence of three solutions of problem (1.1) by applying the three critical points theorem established by Ricceri [16]. Our result, under appropriate assumptions, ensures the existence of an open interval and a positive real number ρ such that, for each , problem (1.1) admits at least three weak solutions whose norms in X are less than ρ. The purpose of the present paper is to generalize the main result of [15].
Now, for every and choosing , with , such that , where , put
Moreover, let a, c be positive constants and define
Our main result is stated as follows.
Theorem 1.1Assume thatsuch that, and suppose that there exist four positive constantsa, b, γandβwith, , , and a functionsuch that
Then there exist an open intervaland a positive real numberρwith the following property: for eachand for two Carathéodory functionssatisfying
there existssuch that, for each, problem (1.1) has at least three weak solutions () whose normsare less thanρ.
2 Proof of the main result
First we recall the modified form of Ricceri’s three critical points theorem (Theorem 1 in [16]) and Proposition 3.1 of [17], which is our primary tool in proving our main result.
Theorem 2.1 ([16], Theorem 1)
Suppose that X is a reflexive real Banach space and thatis a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on, and that Φ is bounded on each bounded subset ofX; is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; is an interval. Suppose that
for all, and that there existssuch that
Then there exist an open intervaland a positive real numberρwith the following property: for everyand everyfunctionalwith compact derivative, there existssuch that, for each, the equation
has at least three solutions inXwhose norms are less thanρ.
Proposition 2.1 ([17], Proposition 3.1)
Assume thatXis a nonempty set and Φ, Ψ are two real functions onX. Suppose that there areandsuch that
Then, for eachhsatisfying
one has
Before proving Theorem 1.1, we define a functional and give a lemma.
By conditions (M) and (j3), it is clear that and a critical point of H corresponds to a weak solution of system (1.1).
Lemma 2.2Assume that there exist two positive constantsa, bwithsuch that
Then there existand, such that
and
Proof We put
and . Then we can verify easily and, in particular, we have
and
Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that
Under condition (M), by a simple computation, we have
Setting and applying the assumption of Lemma 2.2
from (2.6) and (2.7), we obtain
Since, , for each , from condition (j1) of Lemma 2.2, we have
Hence, based on condition (j2), we get
□
Now, we can prove our main result.
Proof of Theorem 1.1 For each , let
From the assumption of Theorem 1.1, we know that Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Additionally, the Gâteaux derivative of Φ has a continuous inverse on . Since , , Ψ and J are continuously Gâteaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X. In particular, for each ,
Hence, the weak solutions of problem (1.1) are exactly the solutions of the following equation:
and so the first condition of Theorem 2.1 is satisfied. By Lemma 2.2, there exists such that
and
From (1.3), we have
for each . Let for each such that
From (2.11), we get
Then, from (2.10) and (2.12), we find
Hence, we have
Fix h such that
by (2.9), (2.13) and Proposition 2.1, with and , we obtain
and so assumption (2.1) of Theorem 2.1 is satisfied.
Now, with , from (2.8) and (2.14), all the assumptions of Theorem 2.1 hold. Hence, our conclusion follows from Theorem 2.1. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (no. 201204LX672).
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