Weak solutions to quasilinear elliptic obstacle problems

Abstract

We study a class of obstacle problems in Sobolev spaces of the form\begin{gather*}\begin{cases}\displaystyle\int_{\Omega}\Big(a(\vert D \varpi\vert) D \varpi ):D(\mathcal{U}-\varpi)+ \left\langle \varpi\vert \varpi\vert^{r-2}, \mathcal{U}-\varpi\right\rangle\Big) \mathrm{dy} \geq 0, \\\\\mathcal{U}\in \wp_{\ell,g}. \end{cases}\end{gather*}We prove the existence of a weak solution via Young measure theory and a theorem of Kinderlehrer and Stampacchia. Since our operator does not satisfy the property of monotonicity which is necessary in the proof, we suppose another condition to overcome this situation.