Define $A=\{ u \in W^{1,p}(]0,r[) \mbox{s.t} u(0)=0, u(r)=q \},$ where $r,q>0$.
Now for $a,b>0$, $g(s)=\int_0^s \beta(u)du$, where $\beta: \mathbb{R} \rightarrow \mathbb{R}$ is a non-decreasing continuous function, with $\beta(0)=0,$
let $J:A \rightarrow \mathbb{R}$ be such that
$$J(w)=\frac{1}{p}\int_0^r |w'(t)|^p e^{-at}dt +b\int_0^r g(w(t))e^{-at}dt, \quad \mbox{for all} \quad w \in A.$$
Can you help me in proving that $J$ is continuous on $A$?
I tried to solve it using sequential continuity:
Let $(w_n) \subset W^{1,p}(]0,r[)$ s.t $w_n \rightarrow w$ as $n \rightarrow \infty$.
$0 \leq |J(w_n)-J(w)| \leq cst\bigl| ||w_n'||_{L^p}^p- ||w'||_{L^p}^p\bigr|+b\int_0^r |g(w_n(t))-g(w(t))|dt.$
1) How can I relate $||w_n'||_{L^p}^p- ||w'||_{L^p}^p$ with $||w_n'-w'||_{L^p}^p$ where the latter is simply less than or equal $||w_n-w||_{W^{1,p}}^p.$
2) I know that $g$ is continuous and even differentiable, but I don't have that $w_n(t) \rightarrow w(t)$ in $\mathbb{R}!!$
Can I please have some help? Or another method if mine is not appropriate?