Bulletin of the
Korean Mathematical Society BKMS

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