Let $Omega$ be an open bounded subset of $mathbb{R}^n$ and $p > 2$. Let $uin W_0^{1, p}(Omega)cap L^{infty}(Omega)$ and $(u_n)_nsubset W_0^{1, p}(Omega)cap L^{infty}(Omega)$ be a sequence such that $u_nlongrightarrow u$ in $W_0^{1, p}(Omega)cap L^{infty}(Omega)$. Moreover, suppose {that a} fixed $Ok>0$ exists such that $Vert u_nVert_{infty}leq Ok$. I need to present that $$ int_{Omega} leftvertnabla u_n -nabla u rightvert left(vertnabla u_nvert +vertnabla uvertright)^{p- 2} dxlongrightarrow 0.$$

I proceed on this means. By utilizing Holder inequality, we’ve

start{align}

&int_{Omega} leftvertnabla u_n -nabla u rightvert left(vertnabla u_nvert +vertnabla uvertright)^{p- 2} dx

&leq left(int_{Omega}vertnabla u_n -nabla uvert^{p-1}proper)^{frac{1}{p-1}} left(int_{Omega} left(vertnabla u_nvert +vertnabla uvertright)^{p-1} dxright)^{frac{p-2}{p-1}}

&leq CVert u_n – uVert_{W_{0}^{1, p-1}}left(int_{Omega}left(vertnabla u_nvert^{p-1} +vertnabla uvert^{p-1}proper) dxright)^{frac{p-2}{p-1}}

&leq CVert u_n -uVert_{W_{0}^{1, p-1}}left(Vert u_nVert_{W_{0}^{1, p-1}}^{p-2} + Vert uVert_{W_{0}^{1, p-1}}^{p- 2} dxright)longrightarrow 0

finish{align}

since $u_nlongrightarrow u$ in $W_0^{1, p}(Omega)$ and the opposite hypotheses.

I want to know whether it is everithing okay with my proof. May anybody assist me?

Thanks prematurely!