# CONSTANT-SIGN SOLUTIONS OF p-LAPLACIAN TYPE OPERATORS ON TIME SCALES VIA VARIATIONAL METHODS

• Zhang, Li (Fundamental Teaching Department Beijing Union University) ;
• Ge, Weigao (Department of Mathematics Beijing Institute of Technology)
• Published : 2012.11.30

#### Abstract

The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with p-Laplacian type operators $$\{({\alpha}(t,x^{\Delta}(t)))^{\Delta}-a(t){\phi}_p(x^{\sigma}(t))+f({\sigma}(t),x^{\sigma}(t))=0,\;{\Delta}-a.e.\;t{\in}I\\x^{\sigma}(0)=0,\\{\beta}_1x^{\sigma}(1)+{\beta}_2x^{\Delta}({\sigma}(1))=0,$$ where ${\beta}_1$, ${\beta}_2$ > 0, $I=[0,1]^{k^2}$, ${\alpha}({\cdot},x({\cdot}))$ is an operator of $p$-Laplacian type, $\mathbb{T}$ is a time scale. Some sufficient conditions for the existence of constant-sign solutions are obtained.

#### References

1. R. P. Agarwal, M. Bohner, and P. Rehak, Half Linear Dynamic Equations, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2, 157, Kluwer Acad. Publ., Dordrecht, 2003.
2. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. Rodriguez-Vivero, Existence of multiple positive solutions for second order nonlinear dynamic BVPs by variational methods, J. Math. Anal. Appl. 331 (2007), no. 2, 1263-1274. https://doi.org/10.1016/j.jmaa.2006.09.051
3. R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. Rodriguez-Vivero, Multiple positive solutions of singular Dirichlet problems on time scales via variational methods, Nonlinear Anal. 67 (2007), no. 2, 368-381. https://doi.org/10.1016/j.na.2006.05.014
4. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser, Boston, 2001.
5. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
6. A. Cabada and D. R. Vivero, Criterions for absolute continuity on time scales, J. Difference Equ. Appl. 11 (2005), no. 11, 1013-1028. https://doi.org/10.1080/10236190500272830
7. G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1979), no. 2, 332-336.
8. L. Jiang and Z. Zhou, Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales, Nonlinear Anal. 69 (2008), no. 4, 1376-1388. https://doi.org/10.1016/j.na.2007.06.034
9. J. Mawhin and M. Willen, Critical Point Theorem and Hamiltonian Systems, Appl. Math. Sci., Vol. 74, Springer, New York, 1989.
10. J. Simon, Regularite de la solution dune equation non lineaire dans $R^n$, Journees d'Analyse Non Lineaire (Proc. Conf., Besanon, 1977), pp. 205227, Lecture Notes in Math., 665, Springer, Berlin, 1978.