Constant-sign solutions of $p$-Laplacian type operators on time scales via variational methods
Bull. Korean Math. Soc. 2012 Vol. 49, No. 6, 1131-1145
https://doi.org/10.4134/BKMS.2012.49.6.1131
Published online November 30, 2012
Li Zhang and Weigao Ge
Beijing Union University, Beijing Institute of Technology
Abstract : The purpose of this paper is to use an appropriate variational framework to discuss the boundary value problem with $p$-Laplacian type operators $$ \left\{\begin{array} {llcc} (\alpha(t,x^{\Delta}(t)))^{\Delta}-a(t)\phi_p(x^{\sigma}(t))+f(\sigma(t),x^{\sigma}(t))=0,~~\Delta{\text -\rm a.e.}~t\in I\nonumber\\ x^{\sigma}(0)=0,\nonumber\\\beta_1x^{\sigma}(1)+\beta_2x^{\Delta}(\sigma(1))=0, \end{array}\nonumber \right. $$ 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.
Keywords : $p$-Laplacian, time scale, variational, constant-sign
MSC numbers : 34B24, 35A15
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