The existence and uniqueness of weak solutions obstacle Problems with variable growth and weak monotonicity

Abstract

In this article, we investigate the presence of weak solutions for obstacle problems $\displaystyle\int_{\Omega}\mathcal{A}(z,u,Du):D(\upsilon-u)+\phi(u):D(\upsilon-u)\mathrm{~d}z  \geq 0$, for $\upsilon$ belonging to the following convex set  $\mathcal{K}_{\psi, \theta}$, applying the Young measure theory and a theorem by Kinderlehrer and Stampacchia, the desired outcome is achieved.