Existence results for perturbed fourth-order Kirchhoff type elliptic problems with singular term
DOI:
https://doi.org/10.5269/bspm.44841Abstract
Under appropriate growth conditions on the nonlinearity, the existence of multiple solutions for a perturbed
nonlocal fourth-order Kirchhoff-type problem involving the Hardy term:$$\Delta_p ^2 u-\big[M(\int_{\Omega}|\nabla u|^{p}dx)\big]^{p-1}\Delta_{p}u-\mu\frac{|u|^{p-2}u}{|x|^{2p}}= \lambda f(x,u),$$is established. Our main tools are based on variational methods and some critical points
theorems. We give some examples to illustrate the obtained results.
References
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36. Z. Xiu, L. Zhao, J. Chen and S. Li, Multiple solutions on a p-biharmonic equation with nonlocal term. Bound. Value Probl, 1, 154-164, (2016). https://doi.org/10.1186/s13661-016-0662-z
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2. A. Amrouss, F Moradi, M. Moussaoudi, Existence of solutions for fourth-order pdes with variable exponents. E. J. D. E, No. 153, pp. 1-13, (2009).
3. G. Autuori, F. Colasuonno and P. Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems. Commun. Contemp. Math, 16, 1450002 [43 pages] (2014). https://doi.org/10.1142/S0219199714500023
4. G. Bonanno, Relations between the mountain pass theorem and local minima. Adv. Nonlinear Anal, 1, 205-220, (2012). https://doi.org/10.1515/anona-2012-0003
5. G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031-3059, (2008). https://doi.org/10.1016/j.jde.2008.02.025
6. G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functionals with a weak compactness condition. Appl. Anal. 89, 1-10, (2010). https://doi.org/10.1080/00036810903397438
7. A. Cabada, J. A. Cid, L. Sanchez, Positivity and lower and upper solutions for fourth-order boundary value problems. Nonlinear Anal. TMA, 67, 1599-1612, (2007). https://doi.org/10.1016/j.na.2006.08.002
8. P. Candito and R. Livrea, Infinitely many solutions for a nonlinear Navier boundary value problem involving the p-biharmonic. Studia Univ. "Babe¸s-Bolyai", Mathematica, Volume LV, Number 4, December 2010.
9. P. Candito and G. Molica Bisci, Multiple solutions for a Navier boundary value problem involv- ing the p-biharmonic operator. Discrete and Continuous Dynamical Systems Series, (4), 741-751, (2012). https://doi.org/10.3934/dcdss.2012.5.741
10. P. Candito, L. Li and R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. TMA, 75, 6360-6369, (2012). https://doi.org/10.1016/j.na.2012.07.015
11. E. B. Davies, A. M. Hinz, Explicit constants for Rellich inequalities in Lp(Ω). Math. Z, 227, 511-523, (1998). https://doi.org/10.1007/PL00004389
12. X. Fan, Q. Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem. Nonlinear Anal, 52, 1843-1852, (2003). https://doi.org/10.1016/S0362-546X(02)00150-5
13. M. Ferrara, S. Khademloo and S. Heidarkhani, Multiplicity results for perturbed fourth-order Kirchhoff type elliptic problems. Appl. Math. Comput, 234, 316-325, (2014). https://doi.org/10.1016/j.amc.2014.02.041
14. M. Ghergu and V. Radulescu, Singular elliptic problems. Bifurcation and Asymptotic Analysis Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford Univ. Press, (2008).
15. M. Ghergu and V. Radulescu, Sublinear Singular elliptic problems with tow parameters. J. Differ. Equ, 195, 520-536, (2003). https://doi.org/10.1016/S0022-0396(03)00105-0
16. J. R. Graef, S. Heidarkhani, L. Kong, Multiple solutions for a class of (p1, . . . , pn)-biharmonic systems. Commun. Pure Appl. Anal. (CPAA) 12, 1393-1406, (2013). https://doi.org/10.3934/cpaa.2013.12.1393
17. S. Heidarkhani, M. Ferrara, A. Salari and G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions. Complex Var. Elliptic Equ, 61, 1494-1516, (2016). https://doi.org/10.1080/17476933.2016.1182520
18. S. Heidarkhani, S. Khademloo and A. Solimaninia, Multiple solutions for a perturbed fourth-order Kirchhoff type elliptic problem. Portugal. Math. (N.S.), 71, Fasc. 1, 39-61, (2014). https://doi.org/10.4171/PM/1940
19. S. Heidarkhani, Y. Tian, C. L. Tang, Existence of three solutions for a class of (p1, . . . , pn)-biharmonic systems with Navier boundary conditions. Ann. Polon. Math, 104, 261-277, (2012). https://doi.org/10.4064/ap104-3-4
20. L. Kong, On a fourth order elliptic problem with p(x)-biharmonic operator. Appl. Math. Lett, 27, 21-25, (2014). https://doi.org/10.1016/j.aml.2013.08.007
21. A. C. Lazer and P.J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Rev, 32, 537-578, (1990). https://doi.org/10.1137/1032120
22. L. Li, Two weak solutions for some singular fourth order elliptic problems, Electron. J. Qual. Theory Differ. Equ, 1, 1-9, (2016). https://doi.org/10.14232/ejqtde.2016.1.1
23. L. Li and C. L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Nonlinear Anal. TMA, 73, 796-805, (2010). https://doi.org/10.1016/j.na.2010.04.018
24. C. Li, C.L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic. Nonlinear Anal. TMA, 72, 1339-1347, (2010). https://doi.org/10.1016/j.na.2009.08.011
25. L. Liu and C. Chen, Infinitely many solutions for p-biharmonic equation with general potential and concave-convex nonlinearity in RN . Bound. Value Probl, 1, 1-9, (2016). https://doi.org/10.1186/s13661-015-0510-6
26. H. Liu and N. Su, Existence of three solutions for a p-biharmonic problem. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal, 15(3), 445-452, (2008).
27. X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-laplacian. Nonlinearity, 29, 357-374, (2016). https://doi.org/10.1088/0951-7715/29/2/357
28. G. Molica Bisci and V. Radulescu, Applications of local linking to nonlocal Neumann problems. Commun. Contemp. Math, 17, 1450001 [17 pages], (2014). https://doi.org/10.1142/S0219199714500011
29. G. Molica Bisci and V. Radulescu, Mountain pass solutions for nonlocal equations. Annales AcademiæScientiarum FennicæMathematica, 39, 579-592, (2014). https://doi.org/10.5186/aasfm.2014.3921
30. G. Molica Bisci and D. Repovs, Multiple solutions of p-biharmonic equations with Navier boundary conditions. Complex Vari. Elliptic Equ, 59, 271-284, (2014). https://doi.org/10.1080/17476933.2012.734301
31. V. Radulescu, Combined effects in nonlinear Singular elliptic problems with convenction. Rev. Roum. Math. Pures Appl, 53, 543-553, (2008).
32. B. Ricceri, On an elliptic Kirchhoff-type problem depending an two parameters. J. Global Optim, 46, 543-549, (2010). https://doi.org/10.1007/s10898-009-9438-7
33. G. Talenti, Elliptic equations and rearrangements. Ann. Sc. Norm. Super Pisa Cl. Sci, 3, 697-718, (1976).
34. Y. Wang and Y. Shen, Nonlinear biharmonic equations with Hardy potential and critical parameter. J. Math. Anal. Appl, 355, 649-660, (2009). https://doi.org/10.1016/j.jmaa.2009.01.076
35. H. Xie and J. Wang, Infinitely many solutions for p-harmonic equation with singular term. J. Inequal. Appl, 2013, No9, 13pp, (2013). https://doi.org/10.1186/1029-242X-2013-9
36. Z. Xiu, L. Zhao, J. Chen and S. Li, Multiple solutions on a p-biharmonic equation with nonlocal term. Bound. Value Probl, 1, 154-164, (2016). https://doi.org/10.1186/s13661-016-0662-z
37. M. Xu and C. Bai, Existence of infinitely many solutions for perturbed Kirchhoff type elliptic problems with Hardy potential. Electron. J. Diff. Equ, Vol. 2015, No. 268, pp. 1-9, (2015).
38. E. Zeidler, Nonlinear functional analysis and its applications. Vol. II, III. Berlin-Heidelberg- New York 1985. https://doi.org/10.1007/978-1-4612-5020-3
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