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 theory
1 Introduction and main results
where () is a bounded smooth domain, , , , is the p-Laplacian 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.
then one has . Furthermore, it is known from  that
System (1.1) is related to the stationary version of a model established by Kirchhoff . 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 cross-section, 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:
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 [3-13] and the references therein. In particular, Alves et al. [, Theorem 4] supposed that M satisfies bounded condition (M) and satisfies the condition
In , 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 p-Kirchhoff type
In , 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 . 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 .
Moreover, let a, c be positive constants and define
Our main result is stated as follows.
2 Proof of the main result
Theorem 2.1 (, 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
has at least three solutions inXwhose norms are less thanρ.
Proposition 2.1 (, Proposition 3.1)
Then, for eachhsatisfying
Before proving Theorem 1.1, we define a functional and give a lemma.
Proof We put
Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that
Under condition (M), by a simple computation, we have
from (2.6) and (2.7), we obtain
Hence, based on condition (j2), we get
Now, we can prove our main result.
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:
From (1.3), we have
From (2.11), we get
Then, from (2.10) and (2.12), we find
Hence, we have
Fix h such that
and so assumption (2.1) of Theorem 2.1 is satisfied.
The authors declare that they have no competing interests.
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.
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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