Existence of weak solutions for second-order boundary-value problems of Kirchhoff-type with variable exponents
DOI:
https://doi.org/10.5269/bspm.44096Palabras clave:
Three solutions, Kirchhoff-type problem, Neumann problem, Variable exponent Sobolev spacesResumen
In this paper, we investigate the existence of multiple solutions
for a second-order boundary value problems of Kirchhoff-type
equation involving a $p(x)$-Laplacian.
Referencias
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18. L. Li, C- L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Nonlinear Anal. TMA 73 (2010), 796-805. https://doi.org/10.1016/j.na.2010.04.018
19. L. Li, C- L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. TMA 72 (2010), 1339-1347. https://doi.org/10.1016/j.na.2009.08.011
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21. G. Molica Bisci, D. Repovs, Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 59 (2014), 271-284. https://doi.org/10.1080/17476933.2012.734301
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23. C. Qian, Z. Shen, M. Yang, Existence of solutions for p(x)-Laplacian nonhomogeneous Neumann problems with indefinite weight, Nonlinear Anal. 11 (2010), 446-458. https://doi.org/10.1016/j.nonrwa.2008.11.019
24. B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (2009), 4151-4157. https://doi.org/10.1016/j.na.2009.02.074
25. X. J. Wang, R. Yuan, Existence of periodic solutions for p(t)-Laplacian systems, Nonlinear Anal. 70 (2009), 866-880. https://doi.org/10.1016/j.na.2008.01.017
26. V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33-66. https://doi.org/10.1070/IM1987v029n01ABEH000958
2. G. Bonanno, A. Chinnı, Discontinuous elliptic problems involving the p(x)−Laplacian, Math. Nachr. 284 (2011), 639-652 https://doi.org/10.1002/mana.200810232
3. F. Cammaroto, A. Chinnı, B. Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, Nonlinear Anal. 71 (2009), 4486-4492. https://doi.org/10.1016/j.na.2009.03.009
4. G. Bonanno, B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343(2008), 1166-1176. https://doi.org/10.1016/j.jmaa.2008.01.049
5. G. Bonanno, B. Di Bella and D. O. Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62 (2011), 1862- 1869. https://doi.org/10.1016/j.camwa.2011.06.029
6. G. Bonanno, G. D'Aguı and A. Sciammetta, One-dimensional non- linear boundary value problems with variable exponent, Discrete and continuous Dynamical Systems, Series S, 11(2) (2018), 179-191. https://doi.org/10.3934/dcdss.2018011
7. G. Bonanno, B. Di Bella and D. O. Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl. 62 (2011), 1862-1869. https://doi.org/10.1016/j.camwa.2011.06.029
8. A. Cabada, J.A. Cid and L. Sanchez, Positivity and lower and upper solutions for for fourth-order boundary value problems, Nonlinear Anal. 67 (2007), 1599-1612. https://doi.org/10.1016/j.na.2006.08.002
9. G. D'Aguı, Second-order boundary-value problems with variable exponent, Electron. J. Differential Equations, Vol. 2014 (2014), No. 68, 1-10.
10. L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017 Springer-Verlag, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8
11. X. L. Fan, X. Y. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , Nonlinear Anal. 59 (2004), 173-188. https://doi.org/10.1016/S0362-546X(04)00254-8
12. X. L. Fan, D. Zhao, On the spaces L p(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424-446. https://doi.org/10.1006/jmaa.2000.7617
13. S. Heidarkhani, S. Moradi and D. Barill, Existence results for second- order boundary-value problems with variable exponents, Nonlinear Anal. (RWA), 44 (2018), 40-53. https://doi.org/10.1016/j.nonrwa.2018.04.003
14. S. Heidarkhani, S. Moradi and S. A. Tersian, Three solutions for second-order boundary-value problems with variable exponents Electronic Journal of Qualitative Theory of Differential Equations 13 2018, 1-19. https://doi.org/10.14232/ejqtde.2018.1.33
15. S. Heidarkhani, A. Salari, G. Caristi, D. Barilla, Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions, Bound. Value Probl. 2017 (2017), 86. https://doi.org/10.1186/s13661-017-0817-6
16. G. Kirchhoff, Vorlesungen uber mathematische Physik, Mechanik. Teubner, Leipzig (1883).
17. O. Kovacik, J. Rakosnık, On the spaces Lp(x) and W1,p(x), Czechoslovak Math. 41 (1991), 592-618. https://doi.org/10.21136/CMJ.1991.102493
18. L. Li, C- L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Nonlinear Anal. TMA 73 (2010), 796-805. https://doi.org/10.1016/j.na.2010.04.018
19. L. Li, C- L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. TMA 72 (2010), 1339-1347. https://doi.org/10.1016/j.na.2009.08.011
20. M. Mihailescu, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplacian operator, Nonlinear Anal. 67 (2007), 1419-1425. https://doi.org/10.1016/j.na.2006.07.027
21. G. Molica Bisci, D. Repovs, Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equ. 59 (2014), 271-284. https://doi.org/10.1080/17476933.2012.734301
22. J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, Springer, Berlin, 1983. https://doi.org/10.1007/BFb0072210
23. C. Qian, Z. Shen, M. Yang, Existence of solutions for p(x)-Laplacian nonhomogeneous Neumann problems with indefinite weight, Nonlinear Anal. 11 (2010), 446-458. https://doi.org/10.1016/j.nonrwa.2008.11.019
24. B. Ricceri, A further three critical points theorem, Nonlinear Anal. TMA 71 (2009), 4151-4157. https://doi.org/10.1016/j.na.2009.02.074
25. X. J. Wang, R. Yuan, Existence of periodic solutions for p(t)-Laplacian systems, Nonlinear Anal. 70 (2009), 866-880. https://doi.org/10.1016/j.na.2008.01.017
26. V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv. 29 (1987), 33-66. https://doi.org/10.1070/IM1987v029n01ABEH000958
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2022-01-24
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