Existence and uniqueness of solution for a unilateral problem in Sobolev spaces with variable exponent

Benali Aharrouch ¹   ; Jaouad Bennouna ¹

¹ Sidi Mohamed Ben Abdellah University Faculty of Sciences Dhar El Mahraz Laboratory LAMA, Department of Mathematics P.O. Box 1796 Atlas Fez, Morocco

Applicationes Mathematicae, Tome 46 (2019) no. 2, pp. 175-189

Résumé

We study the existence and uniqueness of the obstacle problem associated to the equation $$ -\operatorname{div}(a(x,u,\nabla u)+\phi(u))+g(x,u)=f-\operatorname{div}F $$ in the framework of Sobolev spaces with variable exponent, where $F\in (L^{r(\cdot)}(\varOmega))^N$ and $ f\in L^{q(\cdot)}(\varOmega)$ with $$ \begin{cases} r(x) \gt \frac{N}{p(x)-1},\quad r(x)\geq p’(x) \forall x \in \varOmega,\\ q(x) \gt \max\biggl(\frac{N}{p(x)},1\biggr),\quad q(x)\geq p’(x) \forall x \in\varOmega, \end{cases} $$ for a log-Lipschitz function $p:\overline \varOmega \to [1,+\infty)$.

Détail

DOI : 10.4064/am2372-2-2019

Keywords: study existence uniqueness obstacle problem associated equation operatorname div nabla phi f operatorname div framework sobolev spaces variable exponent where cdot varomega cdot varomega begin cases frac quad geq forall varomega max biggl frac biggr quad geq forall varomega end cases log lipschitz function overline varomega infty