Multiplicity of positive solutions for Kirchhoff type problem involving critical exponent and sign-changing weight functions
© Chu; licensee Springer. 2014
Received: 29 August 2013
Accepted: 25 December 2013
Published: 16 January 2014
This paper is devoted to the study of a class of Kirchhoff type problems with critical exponent, concave nonlinearity, and sign-changing weight functions. By means of variational methods, the multiplicity of the positive solutions to this problem is obtained.
MSC:35J20, 35J60, 47J30, 58E50.
1 Introduction and main results
where Ω is a smooth bounded domain in with and the parameters . The weight functions , satisfy the following conditions:
() and , where ;
() there exist positive constants and such that and in ;
() and ;
() and for all ;
() there exists such that as .
proposed by Kirchhoff in  as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in the length of the string produced by transverse vibrations. It received great attention only after Lions  proposed an abstract framework for the problem. The solvability of the Kirchhoff type problem (2) has been paid much attention to by various authors. The positive solutions of such a problem are obtained by using variational methods [3–5]. Perera and Zhang  obtained a nontrivial solution of problem (2) via the Yang index and the critical group. He and Zou  obtained infinitely many solutions by using the local minimum methods and the fountain theorems. Recently, when is a continuous superlinear nonlinearity with critical growth, the existence of positive solutions of the Kirchhoff type problem has been studied [8–13]. Moreover, the paper  considered problem (2) with concave and convex nonlinearities by using a Nehari manifold and fibering map methods, and one obtained the existence of multiple positive solutions. In addition, the corresponding results of the Kirchhoff type problem can be found in [15–25], and the references therein.
In the present paper, we deal with problem (1) and consider the existence and multiplicity of positive solutions of problem (1). About the critical growth situation, the aforementioned papers only showed the existence of positive solutions of the Kirchhoff type problem. Moreover, involving the concave and convex nonlinearities,  only considered the subcritical growth case. Therefore, our purpose is to extend the result of  to critical growth. The main results of this paper extend the corresponding results in  and .
Before stating our results, we give some notations and assumptions. Let , (), . In addition, we denote positive constants by . The main results of this paper are as follows.
Theorem 1 Let , and . Suppose that () and () hold, then there exists such that problem (1) for all has at least one positive solution.
Theorem 2 Let , and . Suppose that (), (), (), () and () hold, then there exists such that problem (1) for all has at least two positive solutions.
Remark 1 Our Theorem 2 extends the results for the critical case of Theorem 1.1 in . Our Theorem 2 shows that we have at least two positive solutions of problem (1), but the authors of the reference only obtain at least one positive solution of problem (1). In addition, the results of Theorem 2.1 in  are extended to critical growth.
This paper is organized as follows. In Section 2, we give the local Palais-Smale condition. The proof of Theorems 1 and 2 is provided in Section 3.
2 The local Palais-Smale condition
for any .
Definition A sequence is called a sequence of I if and as . We say that I satisfies the condition if any sequence of I has a convergent subsequence.
Lemma 1 Let , and . Assume that () and () hold. If is a sequence of I, then is bounded in .
Set , we see that is bounded in . □
Set , and we have . □
Lemma 3 Let , and . Assume that () and () hold, then I satisfies the condition with = + + + − , where A is the positive constant given in Lemma 2.
which contradicts the fact that . Therefore, we have , which implies that in . Hence I satisfies the condition with . □
3 The proof of the main results
In this section, we show the proofs of our Theorems 1 and 2. Before we come to the proof of Theorem 1, we first recall the following lemma in .
Lemma 4 Let , and . Then there exists such that .
By applying the Ekeland’s variational principle in , we obtain the result that there exists a sequence of I.
which implies that is a solution of problem (1). After a direct calculation, we derive , which implies . Since , we have . Applying the Harnack inequality , we see that is a positive solution of problem (1). The proof of Theorem 1 is completed. □
Lemma 5 Let , and . Assume that (), (), (), (), and () hold, then there exists , such that for any , we can find such that .
which implies that . Hence we have . Since , we have . By the Harnack inequality, we obtain the result that is the second positive solution of problem (1). The proof of Theorem 2 is completed. □
The author would like to thank the referees for valuable comments and suggestions on improving this paper. This paper was supported by Science and Education Youth culture project in Guizhou Province (Contract Number: Guizhou Provincial Institute of Zi (2012) No. 157). This paper also was supported by Science and Technology Foundation of Guizhou Province (No. J2141; No. LKM31).
- Kirchhoff G: Mechanik. Teubner, Leipzig; 1883.Google Scholar
- Lions JL: On some questions in boundary value problems of mathematical physics. North-Holland Math. Stud. 30. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations. North-Holland, Amsterdam; 1978:284-346. (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977)Google Scholar
- Ma TF, Munoz Rivera JE: Positive solutions for a nonlinear elliptic transmission problem. Appl. Math. Lett. 2003, 16(2):243-248. 10.1016/S0893-9659(03)80038-1MathSciNetView ArticleGoogle Scholar
- Alves CO, Corra 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.2005.01.008MathSciNetView ArticleGoogle Scholar
- Bensedki A, Bouchekif M: On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity. Math. Comput. Model. 2009, 49: 1089-1096. 10.1016/j.mcm.2008.07.032View ArticleGoogle Scholar
- Perera K, Zhang ZT: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 2006, 221: 246-255. 10.1016/j.jde.2005.03.006MathSciNetView ArticleGoogle Scholar
- He XM, Zou WM: Infinitely many positive solutions for Kirchhoff-type problems. Nonlinear Anal. 2009, 70(3):1407-1414. 10.1016/j.na.2008.02.021MathSciNetView ArticleGoogle Scholar
- Figueiredo GM: Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 2013, 401: 706-713. 10.1016/j.jmaa.2012.12.053MathSciNetView ArticleGoogle Scholar
- Liang SH, Shi SY:Soliton solutions to Kirchhoff type problems involving the critical growth in . Nonlinear Anal. 2013, 81: 31-41.MathSciNetView ArticleGoogle Scholar
- Wang J, Tian LX, Xu JX, Zhang FB: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 2012, 253(7):2314-2351. 10.1016/j.jde.2012.05.023MathSciNetView ArticleGoogle Scholar
- Sun YJ, Liu X: Existence of positive solutions for Kirchhoff type problems with critical exponent. J. Partial Differ. Equ. 2012, 25(2):187-198.MathSciNetGoogle Scholar
- Alves CO, Correa FJSA, Figueiredo GM: On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2010, 2(3):409-417.MathSciNetGoogle Scholar
- Ahmed H, Mohammed M, Najib T: Existence of solutions for p -Kirchhoff type problems with critical exponent. Electron. J. Differ. Equ. 2011., 2011: Article ID 105Google Scholar
- Chen CY, Kuo YC, Wu TF: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 2011, 250: 1876-1908. 10.1016/j.jde.2010.11.017MathSciNetView ArticleGoogle Scholar
- Li YH, Li FY, Shi JP: Existence of a positive solution to Kirchhoff type problems without compactness conditions. J. Differ. Equ. 2012, 253(7):2285-2294. 10.1016/j.jde.2012.05.017View ArticleGoogle Scholar
- He XM, Zou WM:Existence and concentration behavior of positive solutions for a Kirchhoff equation in . J. Differ. Equ. 2012, 252(2):1813-1834. 10.1016/j.jde.2011.08.035MathSciNetView ArticleGoogle Scholar
- Chen CS, Huang JC, Liu LH: Multiple solutions to the nonhomogeneous p -Kirchhoff elliptic equation with concave-convex nonlinearities. Appl. Math. Lett. 2013, 26(7):754-759. 10.1016/j.aml.2013.02.011MathSciNetView ArticleGoogle Scholar
- Cheng BT, Wu X, Liu J: Multiple solutions for a class of Kirchhoff type problems with concave nonlinearity. Nonlinear Differ. Equ. Appl. 2012, 19(5):521-537. 10.1007/s00030-011-0141-2MathSciNetView ArticleGoogle Scholar
- Azzouz N, Bensedik A: Existence results for an elliptic equation of Kirchhoff type with changing sign data. Funkc. Ekvacioj 2012, 55(1):55-66. 10.1619/fesi.55.55MathSciNetView ArticleGoogle Scholar
- Sun J, Liu SB: Nontrivial solutions of Kirchhoff type problems. Appl. Math. Lett. 2012, 25(3):500-504.MathSciNetGoogle Scholar
- Aouaoui S: Existence of three solutions for some equation of Kirchhoff type involving variable exponents. Appl. Math. Comput. 2012, 218(13):7184-7192. 10.1016/j.amc.2011.12.087MathSciNetView ArticleGoogle Scholar
- Liu W, He XM: Multiplicity of high energy solutions for superlinear Kirchhoff equations. J. Appl. Math. Comput. 2012, 39(1-2):473-487. 10.1007/s12190-012-0536-1MathSciNetView ArticleGoogle Scholar
- Liu DC, Zhao PH: Multiple nontrivial solutions to a p -Kirchhoff equation. Nonlinear Anal. 2012, 75(13):5032-5038. 10.1016/j.na.2012.04.018MathSciNetView ArticleGoogle Scholar
- Sun JJ, Tang CL: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 2011, 74(4):1212-1222. 10.1016/j.na.2010.09.061MathSciNetView ArticleGoogle Scholar
- Yang Y, Zhang JH: Positive and negative solutions of a class of nonlocal problems. Nonlinear Anal. 2010, 73(1):25-30. 10.1016/j.na.2010.02.008MathSciNetView ArticleGoogle Scholar
- Talenti G: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 1976, 110(4):353-372.MathSciNetView ArticleGoogle Scholar
- Brézis H, Lieb E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc. 1983, 88: 486-490.View ArticleGoogle Scholar
- Chu CM, Tang CL: Multiple results for critical quasilinear elliptic systems involving concave-convex nonlinearities and sign-changing weight functions. Bull. Malays. Math. Soc. 2013, 36(3):789-805.MathSciNetGoogle Scholar
- Ekeland I: On the variational principle. J. Math. Anal. Appl. 1974, 47: 324-353. 10.1016/0022-247X(74)90025-0MathSciNetView ArticleGoogle Scholar
- Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 1967, 20: 721-747. 10.1002/cpa.3160200406MathSciNetView ArticleGoogle Scholar
- Hsu TS, Lin HL: Multiple positive solutions for singular elliptic equations with concave-convex nonlinearities and sign-changing weights. Bound. Value Probl. 2009., 2009: Article ID 584203Google Scholar
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 347-381.MathSciNetView ArticleGoogle Scholar
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