I would like to know it the following sequence $\psi_{\epsilon}$ is bounded in a Lebesgue Space/Sobolev Space with variable expoent: $\psi_{\epsilon}=\int_{\Omega'} |\nabla u_{\epsilon}|^{-p(x)} dx$
Notation:
- Here $\Omega' \subset\subset \Omega$, where $\Omega$ is a open bounded set in $\mathbb{R}^{n}$.
- $p:\Omega \to (1,\infty)$ is a lipschitz continuous function with $\inf p >1$
- Finally $u \in W^{1,p(.)}(\Omega)$, and $u_{\epsilon}$ is the inf-convolution of $u$ in $\Omega$, i.e, $$u_{\epsilon}=\inf_{y \in\Omega}\Big\{u(y)+\frac{1}{q\epsilon^{q-1}}|x-y|^{q}\Big\}$$, and $q$ is a real number s.t, $q\geq 2$ ,