Existence of three solutions for a nonlocal elliptic system of -Kirchhoff type
© Chen et al.; licensee Springer 2013
Received: 1 May 2013
Accepted: 10 July 2013
Published: 25 July 2013
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.
1 Introduction and main results
- (M)There exist two positive constants , such that(1.2)
for all .
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.
where ; one positive solution for (1.7) was given.
where η is the unit exterior vector on ∂ Ω, and , (), f, g satisfy suitable assumptions.
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 .
Our main result is stated as follows.
Theorem 1.1 Assume that such that , and suppose that there exist four positive constants a, b, γ and β with , , , and a function such that
for a.e. and all ;
for a.e. and all ;
for a.e. .
Then there exist an open interval and a positive real number ρ with the following property: for each and for two Carathéodory functions satisfying
for all ,
there exists such that, for each , problem (1.1) has at least three weak solutions () whose norms are less than ρ.
2 Proof of the main result
Theorem 2.1 (, Theorem 1)
has at least three solutions in X whose norms are less than ρ.
Proposition 2.1 (, Proposition 3.1)
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 2.2 Assume that there exist two positive constants a, b with such that
, for a.e. and all ;
Now, we can prove our main result.
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. □
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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