EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR PERTURBED KIRCHHOFF TYPE ELLIPTIC PROBLEMS WITH HARDY POTENTIAL

Abstract

In this paper, we prove the existence of at least three weak solutions for a doubly eigenvalue elliptic systems involving the p-biharmonic equation with Hardy potential of Kirchho type with Navier boundary condition. More precisely, by using variational methods and three critical points theorem due to B. Ricceri, we establish multiplicity results on the existence of weak solutions for such problems where the reaction term is a nonlinearity function f which satises in the some convenient growth conditions. Indeed, using a consequence of the critical point theorem due to Ricceri, which in it the coercivity of the energy Euler functional was required and is important, we attempt the existence of multiplicity solutions for our problem under algebraic conditions on the nonlinear parts. We also give an explicit example to illustrate the obtained result.

Keywords

• Multiplicity of weak solutions,perturbed Kirchho type elliptic problems,Hardy potential,Critical points.

References

1. Alves, C. O., Corrˆea, F. J. S. A., and Ma, T. F., (2005), Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49, pp. 85-93.
2. Alves, C.O., and Figueiredo, G., (2016), Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal. 5, no. 1, pp. 1-26.
3. Aouaoui, S., (2012), Existence of three solutions for some equation of Kirchhoff type involving variable exponents, Appl. Math. Comput. 218, pp. 7184-7192.
4. Arosio, A., and Panizzi, S., (1996), On the well-posedness of the Kirchhoff string, Trans. Am. Math. Soc. 348, pp. 305-330.
5. Autuori, G., Colasuonno, F., and Pucci, P., (2014), On the existence of stationary solutions forhigher-order p-Kirchhoff problems, Commun. Contemp. Math. 16, 1450002, (43 pages).
6. Baraket, S., and Molica Bisci, G., (2017), Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal. 6, no. 1, pp. 85-93.
7. Berchio, E., Cassani, D., and Gazzola, F., (2010), Hardy-Rellich inequalities with boundary remainderterms and applications, Manusc. Math. 131, pp. 427-458.
8. Bonanno, G., (2003), Some remarks on a three critical points theorem, Nonlinear Anal. 54, pp. 651- 665.

9. Bonanno, G., and Chinn`ı, A., (2014), Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418, pp. 812-827.
10. Bonanno, G., and B. Di Bella, B., (2008), A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343, pp. 1166-1176.
11. Bonanno, G., Di Bella, B., and O’Regan, D., (2011), Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62, pp. 1862-1869.
12. Bouali, T., and Guefaifia, R., (2015), Existence and uniqueness of weak solution for a nonlocal problem involving the p-laplacian, Int. J. Pure Appl. Math. Vol. 98, No. 1, pp. 11-21.
13. Boureanu, M., Radulescu, V., and Repovs, D., (2016), On a p(·)-biharmonic problem with no-flux boundary condition, Comput. Math. Appl. 72, no. 9, pp. 2505-2515.
14. Cabada, A., Cid, J. A., and Sanchez, L., (2007), Positivity and lower and upper solutions for fourth- order boundary value problems, Nonlinear Anal. TMA 67, pp. 1599-1612.
15. Chhetri, M., Shivaji, R., (2005), Existence of a positive solution for a p-Laplacian semipositone prob- lem, Boundary Value Problems, 2005, (3), pp. 323-327.
16. Chipot, M., and Lovat, B., (1997), Some remarks on non local elliptic and parabolic problems, Non- linear Anal. TMA 30, pp. 4619-4627.
17. Chung, N. T., (2011), Multiple solutions for a fourth order elliptic equation with Hardy type potential, Acta Univ. Apulen. Math. Inform. 28, pp. 115-124.
18. Colasuonno, F., and Pucci, P., (2011), Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. TMA 74, pp. 5962-5974.
19. Dai, G., (2013), Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)- Laplacian, Applicable Analysis, 92, (1), pp. 191-210.
20. Dai, G., and Ma, R., (2011), Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data, Nonlinear Anal. RWA, 12, pp. 2666-2680.
21. Davies, E. B., and Hinz, A. M., (1998), Explicit constants for Rellich inequalities in Lp(Ω), Math. Z. 227, pp. 511-523.
22. Ferrara, M., Khademloo, S., and Heidarkhani, S., (2014), Multiplicity results for perturbed fourth- order Kirchhoff type elliptic problems, Appl. Math. Comput. 234, pp. 316-325.
23. Figueiredo, G. M., (2013), Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401, pp. 706-713.
24. Ghergu, M., and R˘adulescu, V., (2012), Nonlinear PDEs. Mathematical models in biology, chemistry and population genetics, Springer Monographs in Mathematics. Springer, Heidelberg.
25. Ghergu, M., and R˘adulescu, V., (2008) Singular Elliptic Problems. Bifurcation and Asymptotic Anal- ysis, Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford Univ. Press.
26. Ghergu, M., and R˘adulescu, V., (2003), Sublinear singular elliptic problems with two parameters, J. Differ. Equ. 195, pp. 520-536.
27. Graef, J. R., Heidarkhani, S., and Kong, L., (2013), A variational approach to a Kirchhoff-type problem involving two parameters, Results Math. 63, pp. 877-889.
28. Graef, J. R., Heidarkhani, S., and Kong, L., (2013), Multiple solutions for a class of (p1, ..., pn)- biharmonic systems, Commu. Pure Appl. Anal. (CPAA) 12, (3), pp. 1393-1406.
29. Hamydy, A., Massar, M., and Tsouli, N., (2011), Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Differential Equations, Vol. 2011, No. 105, pp. 1-8.
30. Han, X., and Dai, G., (2012), On the sub-supersolution method for p(x)-Kirchhoff type equations, J. Inequa. Appl. 2012, 283.
31. He, X., and Zou, W., (2009), Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. TMA 70, pp. 1407-1414.
32. Heidarkhani, S., (2014), Existence of non-trivial solutions for systems of n fourth order partial differ- ential equations, Math. Slov. 64, No. 5, pp. 1249-1266.
33. Heidarkhani, S., (2013), Infinitely many solutions for systems of n two-point boundary value Kirchhoff- type problems, Ann. Polon. Math. 107, No.2, pp. 133-152.
34. Heidarkhani, S., Afrouzi, G. A., and O’Regan, D., (2011), Existence of three solutions for a Kirchhoff- type boundary-value problem, Electron. J. Differential Equations, Vol. 2011, No. 91, pp. 1-11.
35. Heidarkhani, S., Caristi, G., and Ferrara, M., (2016), Perturbed Kirchhoff-type Neumann problems in Orlicz-Sobolev spaces, Comput. Math. Appl. (CAMWA) 71, pp. 2008-2019.
36. Heidarkhani, S., Ferrara, M., and Khademloo, S., (2016), Nontrivial solutions for one-dimensional fourthorder Kirchhoff-type equations, Mediterr. J. Math. 13, pp. 217-236.
37. Heidarkhani, S., and Henderson, J., (2012), Infinitely many solutions for a class of nonlocal elliptic systems of (p, ..., pn)-Kirchhoff type, Electron. J. Differential Equations, Vol. 2012, No. 69, pp. 1-15.
38. Heidarkhani, S., and Henderson, J., (2012), Multiple solutions for a nonlocal perturbed elliptic problem of p-Kirchhoff type, Commu. Appl. Non. Anal. 19, (3), pp. 25-39.
39. Heidarkhani, S. , Khademloo, S., and Solimaninia, A., (2014), Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem, Portugal. Math. (N.S.) Vol. 71, Fasc. 1, pp. 39-61.
40. Heidarkhani, S., Tian, Y., and Tang, C. L., (2012), Existence of three solutions for a class of (p1, ..., pn)- biharmonic systems with Navier boundary conditions, Ann. Polon. Math. 104, No.3, 261-277.
41. Huang, Y., and Liu, X., (2014), Sign-changing solutions for p-biharmonic equations with Hardy po- tential, J. Math. Anal. Appl. 412, pp. 142-154.
42. Kirchhoff, G., (1883), Vorlesungen uber mathematische Physik, Mechanik. Teubner, Leipzig.
43. Lazer, A. C., and Mckenna, P. J., (1990), Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev. 32, pp. 537- 578.
44. Li, L., (2016), Two weak solutions for some singular fourth order elliptic problem, Electron. J. Qua. Theo. Diff. Equ. 2016, No.1, pp. 1-9.
45. Li, Y., Li, F., and Shi, J., (2012), Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ. 253, pp. 2285-2294.
46. Mao, A., and Zhang, Z., (2009), Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. TMA 70, pp. 1275-1287.
47. Massar, M., (2013), Existence and multiplicity solutions for nonlocal elliptic problems, Electron. J. Differential Equations, Vol. 2013, No. 75, pp. 1-14.
48. Massar, M., Hssini, E. M., Tsouli, N., and Talbi, M., (2014), Infinitely many solutions for a fourth- order Kirchhoff type elliptic problem, J. Math. Comput. Sci. 8, pp. 33-51.
49. Molica Bisci, G., R˘adulescu, V., and Servadei, R., (2016), Variational methods for nonlocal frac- tional problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge.
50. Molica Bisci, G., and Repovs, D., (2014), Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 59, no. 2, pp. 271-284.
51. Pei, R., and Zhang, J., (2013), Sign-changing solutions for a fourth-order elliptic equation with Hardy singular terms, J. Appl. Math. 2013, pp. 1-6.
52. Perera, K., and Zhang, Z. T., (2006), Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equ. 221, pp. 246-255.
53. P´erez-Llanos, M., and Primo, A., (2014), Semilinear biharmonic problems with a singular term, J. Differ. Equ. 257, pp. 3200-3225.
54. Pucci, P., Xiang, M., and Zhang, B., (2016), Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal. 5, no. 1, pp. 27-55.
55. R˘adulescu, V., (2008), Combined effects in nonlinear singular elliptic problems with convenction, Rev. Roum. Math. Pures Appl. 53, pp. 543-553.
56. R˘adulescu, V., and Repovs, D., (2012), Combined effects in nonlinear problems arising in the study of anisotropic continuous media, Nonlinear Anal. 75, no. 3, pp. 1524-1530.
57. Ricceri, B., (2011), A further refinement of a three critical points theorem, Nonlinear Anal. 74, pp. 7446-7454.
58. Ricceri, B., (2010), On an elliptic Kirchhoff-type problem depending on two parameters, J. Global Optim. 46, pp. 543-549.
59. Shahruz, S. M., and Parasurama, S. A., (1998), Suppression of vibration in the axially moving Kirch- hoff string by boundary control, J. Sound Vib. 214, pp. 567-575.
60. Simon, J., (1978), R´egularit´e de la solution d’une ´equation non lin´ear dans RN, in: Journ´ees d’Analyse Non Lin´ear (Proc. Conf., Besancon, 1977),(P. B´enilan, J. Robert, eds.), Lecture Notes in Math., 665, pp. 205-227, Springer, Berlin-Heidelberg-New York.
61. Talenti, G., (1976), Elliptic equations and rearrangements, Ann. Sc. Norm. Super Pisa Cl. Sci. 3, pp. 697-718.
62. Wang, F., and An, Y., (2012), Existence and multiplicity of solutions for a fourth-order elliptic equation, Bound. Value Probl. 2012, No. 6, pp. 1-9.
63. Wang, F., Avci, M., and An, Y., (2014), Existence of solutions for fourth-order elliptic equations of Kirchhoff-type, J. Math. Anal. Appl. 409, pp. 140-146.
64. Wang, Y., and Shen, Y., (2009), Nonlinear biharmonic equations with Hardy potential and critical parameter, J. Math. Anal. Appl. 355, pp. 649-660.
65. Xu, M., and Bai, C., (2015), Existence of infinitely many solutions for perturbed Kirchhoff type elliptic problems with Hardy potential, Electron. J. Differential Equations, Vol. 2015, No. 268, pp. 1-9.
66. Zeidler, E., (1985), Nonlinear Functional Analysis and its Applications, vol. II, Springer, Berlin- Heidelberg-New York.