$\mathrm{\Gamma }$-convergence for power-law functionals with variable exponents

We study the $\mathrm{\Gamma }$-convergence of the functionals $F_n(u):= || f(\cdot,u(\cdot),Du(\cdot))||{p_n(\cdot)}$and$$\mathcal{F}$n(u):= \int{\Omega} \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$definedon$X\in {L^1(\Omega,$\mathbb{R}$^d), L^\infty(\Omega,$\mathbb{R}$^d), C(\Omega,$\mathbb{R}$^d)}$ (endowed with their usual norms) with effective domain the Sobolev space $W^{1,p_n(\cdot )}(\mathrm{\Omega },{\mathbb{R}}^d)$. Here $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a bounded open set, $N,d\ge 1$ and the measurable functions $p_n:\bar{\mathrm{\Omega }}\to (1,+\mathrm{\infty })$ satisfy the conditions ${\mathop{\rm ess: sup }}{\ \overline \Omega} p_n \le , \beta , {\mathop{\rm ess: inf }}{\ \overline \Omega} p_n $ for a fixed constant $\beta >1$ and $ {\mathop{\rm ess: inf }}{\ \overline \Omega} p_n \rightarrow + \infty$as$n \rightarrow + \infty$\mathrm{.}Weshowthatwhen$f(x,u,\cdot)$ is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as $n\to \mathrm{\infty }$, the sequences $(F_n{)}_n$ $\mathrm{\Gamma }$-converges in $X$ to the functional $F$ represented as $F(u)= || f(\cdot,u(\cdot),Du(\cdot))||{\infty}$ontheeffectivedomain$W^{1,\infty}(\Omega, $\mathbb{R}$^d )$\mathrm{.}Moreoverweshowthatthe$\Gamma$-$\lim_n \mathcal F_n$isgivenbythefunctional$ $\mathcal{F}$(u):=\left{\begin {array}{lll} !!!!!! & 0 & \hbox{if } || f(\cdot,u(\cdot),Du(\cdot)) ||{\infty}\leq 1,\ !!!!!! & +\infty & \hbox{otherwise in } X.\ \end{array}\right. $