Existence and uniqueness results for a class of obstacle problem via Young’s measure theory

The purpose of this article is to prove the existence and uniqueness of weak solutions to the following obstacle problem of p-Laplace-type: \(\begin{aligned} \displaystyle \int _{\Omega }\sigma _1(z,Du-\mathcal {F}(u)):D(v-u)+\sigma _2(z,Du):(v-u)+ \left\langle u\vert u\vert ^{p-2}, v- u\right\rangle \mathrm {~d}z\ge 0, \end{aligned}\) $$\begin{array}{c}\hfill {\displaystyle {\int }_{\mathrm{\Omega }}{\sigma }_1(z,Du-\mathcal{F}(u)):D(v-u)+{\sigma }_2(z,Du):(v-u)+〈{u|u|}^{p-2},,,v,-,u〉\mathrm{d}z\ge 0,}\end{array}$$ with data belonging to the dual of Sobolev spaces. The main result is demonstrated by means of Kinderlehrer and Stampacchia’s Theorem and Young’s measure theory.