Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions
DOI:
https://doi.org/10.5269/bspm.44144Abstract
In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.
References
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21. Yucedag, Z., Avci, M., Mashiyev, R. On an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition, Mathematical Modelling and Analysis. 17(2), 161-170, 2012. https://doi.org/10.3846/13926292.2012.655788
2. Avci, M., Cekic, B., Mashiyev, R. A., Existence and multiplicity of the solutions of the p(x)-Kirchhoff type equation via genus theory, Math. Methods Appl. Sci. 34 (14), 1751-1759, (2011). https://doi.org/10.1002/mma.1485
3. Bonanno, G., Candito, P., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, J. Differential Equations. 244, 3031-3059, (2008). https://doi.org/10.1016/j.jde.2008.02.025
4. Bonanno, G., Bisci, G. M., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Boundary Value Problems. 2009, 1-12, (2009). https://doi.org/10.1155/2009/670675
5. Brezis, H., Analyse fonctionnelle: theorie et applications, Masson, Paris, 1992.
6. Chung, N. T., Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Variables and Elliptic Equations. 58 (12), 1637-1646, (2013). https://doi.org/10.1080/17476933.2012.701289
7. Chung, N. T., Toan, H.Q., On a class of degenerate nonlocal problems with sign-changing nonlinearities, Bull. Malays. Math. Sci. Soc. 37 (4), 1157-1167, (2014).
8. Colasuonno, F., Pucci, P., Multiplicity of solutions for p(x)-polyharmonic Kirchhoff equations, Nonlinear Anal. (TMA). 74, 5962-5974, (2011). https://doi.org/10.1016/j.na.2011.05.073
9. Correa, F. J. S. A., Costa, A. C. dos R., On a bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus, Math. Methods Appl. Sci. 38 (1), 87-93, (2015). https://doi.org/10.1002/mma.3051
10. Dai, G., Three solutions for a nonlocal Dirichlet boundary value problem involving the p(x)-Laplacian. Applicable Analysis. 92 (1), 191-210, (2013). https://doi.org/10.1080/00036811.2011.602633
11. Fan, X., On nonlocal p(x)- Laplacian Dirichlet problems , Nonlinear Anal. 72, 3314-3323, (2010). https://doi.org/10.1016/j.na.2009.12.012
12. Guo, E., Zhao, P., Existence and multiplicity of solutions for nonlocal p(x)-Laplacian equations with nonlinear Neumann boundary conditions, Boundary Value Problems. 2012:1, 1-11, (2012). https://doi.org/10.1186/1687-2770-2012-1
13. Heidarkhani, S., Afrouzi, G.A., Ferrara, M., Moradi, S., Variational approaches to impulsive elastic beam equations of Kirchhoff type, Complex Variables and Elliptic Equations. 61 (7), 931-968, (2016). https://doi.org/10.1080/17476933.2015.1131681
14. Heidarkhani, S., De Araujo, A. L. A., Afrouzi, G. A., Moradi, S., Multiple solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions, Math. Nach. 291 (2-3), 326-342, (2018). https://doi.org/10.1002/mana.201600425
15. Kirchhoff, G., Mechanik, Teubner, Leipzig, Germany, 1883.
16. Li, Q., Young, Z., Existence of multiple solutions for a p-Kirchhoff problem with the non-linear boundary condition, Appl. Anal., to appear, https://doi.org/10.1080/00036811.2017.1395859
17. Ni, W. M., Serrin, J., Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo (suppl.) 8. 171-185, (1985).
18. Ni, W. M., Serrin, J., Existence and non-existence theorems for ground states for quasilinear partial differential equations, Att. Conveg. Lincei. 77, 231-257, (1985).
19. Struwe, M., Variational Methods, Second edition, Springer- Verlag, 2008.
20. Wang, W. B., Tang, W., Infinitely many solutions for Kirchhoff type problems with nonlinear Neumann boundary conditions, Electron. J. Diff. Equ. 2016 (188), 1-9, (2016).
21. Yucedag, Z., Avci, M., Mashiyev, R. On an elliptic system of p(x)-Kirchhoff-type under Neumann boundary condition, Mathematical Modelling and Analysis. 17(2), 161-170, 2012. https://doi.org/10.3846/13926292.2012.655788
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2021-12-18
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