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Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type

  • Ghasem A. Afrouzi EMAIL logo and Armin Hadjian
Published/Copyright: February 17, 2016
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Abstract

Existence results of positive solutions for a nonlocal elliptic problem of Kirchhoff-type are established. The approach is based on variational methods.

MSC 2010: 34B15; 47J10

Funding statement: This research work has been supported by a research grant from the University of Mazandaran.

References

[1] Afrouzi G. A., Hadjian A. and Heidarkhani S., Non-trivial solutions for a two-point boundary value problem, Ann. Polon. Math. 108 (2013), no. 1, 75–84. 10.4064/ap108-1-6Search in Google Scholar

[2] Alves C. O., Corrêa F. S. J. A. and Ma T. F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), no. 1, 85–93. 10.1016/j.camwa.2005.01.008Search in Google Scholar

[3] Bonanno G., A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), no. 5, 2992–3007. 10.1016/j.na.2011.12.003Search in Google Scholar

[4] Bonanno G., Di Bella B. and O’Regan D., Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62 (2011), no. 4, 1862–1869. 10.1016/j.camwa.2011.06.029Search in Google Scholar

[5] Bonanno G., Heidarkhani S. and O’Regan D., Nontrivial solutions for Sturm-Liouville systems via a local minimum theorem for functionals, Bull. Aust. Math. Soc. 89 (2014), no. 1, 8–18. 10.1017/S000497271300035XSearch in Google Scholar

[6] Bonanno G., Molica Bisci G. and Rădulescu V. D., Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations, Manuscripta Math. 142 (2013), no. 1–2, 157–185. 10.1007/s00229-012-0596-4Search in Google Scholar

[7] Bonanno G., Molica Bisci G. and Rădulescu V. D., Weak solutions and energy estimates for a class of nonlinear elliptic Neumann problems, Adv. Nonlinear Stud. 13 (2013), no. 2, 373–389. 10.1515/ans-2013-0207Search in Google Scholar

[8] Bonanno G. and Pizzimenti P. F., Neumann boundary value problems with not coercive potential, Mediterr. J. Math. 9 (2012), no. 4, 601–609. 10.1007/s00009-011-0136-6Search in Google Scholar

[9] Bonanno G. and Pizzimenti P. F., Existence results for nonlinear elliptic problems, Appl. Anal. 92 (2013), no. 2, 411–423. 10.1080/00036811.2011.625013Search in Google Scholar

[10] Bonanno G. and Sciammetta A., An existence result of one nontrivial solution for two point boundary value problems, Bull. Aust. Math. Soc. 84 (2011), no. 2, 288–299. 10.1017/S0004972711002255Search in Google Scholar

[11] Bonanno G. and Sciammetta A., Existence and multiplicity results to Neumann problems for elliptic equations involving the p-Laplacian, J. Math. Anal. Appl. 390 (2012), no. 1, 59–67. 10.1016/j.jmaa.2012.01.012Search in Google Scholar

[12] Cammaroto F. and Vilasi L., Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), no. 5, 1841–1852. 10.1016/j.na.2010.10.057Search in Google Scholar

[13] Cheng B. and Wu X., Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 (2009), no. 10, 4883–4892. 10.1016/j.na.2009.03.065Search in Google Scholar

[14] Chipot M. and Lovat B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), no. 7, 4619–4627. 10.1016/S0362-546X(97)00169-7Search in Google Scholar

[15] Chung N. T., Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities, Electron. J. Qual. Theory Differ. Equ. 2012 (2012), 42. Search in Google Scholar

[16] Chung N. T., Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Var. Elliptic Equ. 58 (2013), no. 12, 1637–1646. 10.1080/17476933.2012.701289Search in Google Scholar

[17] Chung N. T. and Toan H. Q., On a class of degenerate nonlocal problems with sign-changing nonlinearities, Bull. Malays. Math. Sci. Soc. (2) 37 (2014), no. 4, 1157–1167. Search in Google Scholar

[18] Corrêa F. J. S. A. and Figueiredo G. M., On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 (2006), no. 2, 263–277. 10.1017/S000497270003570XSearch in Google Scholar

[19] Dai G. and Hao R., Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), no. 1, 275–284. 10.1016/j.jmaa.2009.05.031Search in Google Scholar

[20] Graef J. R., Heidarkhani S. and Kong L., A variational approach to a Kirchhoff-type problem involving two parameters, Results Math. 63 (2013), no. 3–4, 877–889. 10.1007/s00025-012-0238-xSearch in Google Scholar

[21] Heidarkhani S., Non-trivial solutions for a class of (p1,,pn)-biharmonic systems with Navier boundary conditions, Ann. Polon. Math. 105 (2012), no. 1, 65–76. 10.4064/ap105-1-6Search in Google Scholar

[22] Heidarkhani S., Non-trivial solutions for two-point boundary-value problems of fourth-order Sturm–Liouville type equations, Electron. J. Differential Equations 2012 (2012), 27. Search in Google Scholar

[23] Heidarkhani S., Infinitely many solutions for systems of n two-point Kirchhoff-type boundary value problems, Ann. Polon. Math. 107 (2013), no. 2, 133–152. 10.4064/ap107-2-3Search in Google Scholar

[24] Heidarkhani S., Afrouzi G. A. and O’Regan D., Existence of three solutions for a Kirchhoff-type boundary-value problem, Electron. J. Differential Equations 2011 (2011), 91. Search in Google Scholar

[25] Heidarkhani S. and Henderson J., Infinitely many solutions for nonlocal elliptic systems of (p1,,pn)-Kirchhoff type, Electron. J. Differential Equations 2012 (2012), 69. Search in Google Scholar

[26] Heidarkhani S. and Tian Y., Three solutions for a class of gradient Kirchhoff-type systems depending on two parameters, Dynam. Systems Appl. 20 (2011), no. 4, 551–562. Search in Google Scholar

[27] Kirchhoff G., Vorlesungen über Mathematische Physik. Erster Band: Mechanik, B. G. Teubner, Leipzig, 1897. Search in Google Scholar

[28] Perera K. and Zhang Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), no. 1, 246–255. 10.1016/j.jde.2005.03.006Search in Google Scholar

[29] Pucci P. and Serrin J., The strong maximum principle revisited, J. Differential Equations 196 (2004), no. 1, 1–66. Erratum in: J. Differential Equations 207 (2004), no. 1, 226–227. 10.1016/j.jde.2003.05.001Search in Google Scholar

[30] Ricceri B., A general variational principle and some of its applications. Fixed point theory with applications in nonlinear analysis, J. Comput. Appl. Math. 113 (2000), no. 1–2, 401–410. 10.1016/S0377-0427(99)00269-1Search in Google Scholar

[31] Ricceri B., On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46 (2010), no. 4, 543–549. 10.1007/s10898-009-9438-7Search in Google Scholar

[32] Talenti G., Some inequalities of Sobolev type on two-dimensional spheres, General Inequalities 5 (Oberwolfach 1986), Internat. Schriftenreihe Numer. Math. 80, Birkhäuser, Basel (1987), 401–408. 10.1007/978-3-0348-7192-1_32Search in Google Scholar

Received: 2014-3-5
Accepted: 2015-4-7
Published Online: 2016-2-17
Published in Print: 2016-9-1

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