- Open Access
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).
- Talenti G: Some inequalities of Sobolev type on two-dimensional spheres. Internat. Schriftenreihe Numer. Math. 5. In General Inequalities. Edited by: Walter W. Birkhäuser, Basel; 1987:401-408.View ArticleGoogle Scholar
- Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.MATHGoogle Scholar
- Alves CO, Corrêa FJSA, Ma TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 2005, 49(1):85-93. 10.1016/j.camwa.2005.01.008MATHMathSciNetView ArticleGoogle Scholar
- Cheng B, Wu X: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal., Theory Methods Appl. 2009, 71(10):4883-4892. 10.1016/j.na.2009.03.065MATHMathSciNetView ArticleGoogle Scholar
- Cheng B, Wu X, Liu J: Multiplicity of nontrivial solutions for Kirchhoff type problems. Bound. Value Probl. 2010., 2010: Article ID 268946Google Scholar
- Chipot M, Lovat B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal., Theory Methods Appl. 1997, 30(7):4619-4627. 10.1016/S0362-546X(97)00169-7MATHMathSciNetView ArticleGoogle Scholar
- D’Ancona P, Spagnolo S: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 1992, 108(2):247-262.MATHMathSciNetView ArticleGoogle Scholar
- He X, Zou W: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal., Theory Methods Appl. 2009, 70(3):1407-1414. 10.1016/j.na.2008.02.021MATHMathSciNetView ArticleGoogle Scholar
- Ma T, Muñoz Rivera JE: Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 2003, 16(2):243-248. 10.1016/S0893-9659(03)80038-1MATHMathSciNetView ArticleGoogle Scholar
- Ma T: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal., Theory Methods Appl. 2005, 63: 1967-1977. 10.1016/j.na.2005.03.021View ArticleGoogle Scholar
- Mao A, Zhang Z: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal., Theory Methods Appl. 2009, 70(3):1275-1287. 10.1016/j.na.2008.02.011MATHMathSciNetView ArticleGoogle Scholar
- Perera K, Zhang Z: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 2006, 221(1):246-255. 10.1016/j.jde.2005.03.006MATHMathSciNetView ArticleGoogle Scholar
- Zhang Z, Perera K: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 2006, 317(2):456-463. 10.1016/j.jmaa.2005.06.102MATHMathSciNetView ArticleGoogle Scholar
- Corrêa FJSA, Nascimento RG: On a nonlocal elliptic system of p -Kirchhoff-type under Neumann boundary condition. Math. Comput. Model. 2009, 49(3-4):598-604. 10.1016/j.mcm.2008.03.013MATHView ArticleMathSciNetGoogle Scholar
- Cheng B, Wu X, Liu J:Multiplicity of solutions for nonlocal elliptic system of -Kirchhoff type. Abstr. Appl. Anal. 2011., 2011: Article ID 526026 10.1155/2011/526026Google Scholar
- Ricceri B: A three critical points theorem revisited. Nonlinear Anal. 2009, 70(9):3084-3089. 10.1016/j.na.2008.04.010MATHMathSciNetView ArticleGoogle Scholar
- Ricceri B: Existence of three solutions for a class of elliptic eigenvalue problems. Math. Comput. Model. 2000, 32(11-13):1485-1494. 10.1016/S0895-7177(00)00220-XMATHMathSciNetView ArticleGoogle Scholar
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