In the book Critical Point Theory and Hamiltonian Systems by Jean Mawhin and Michel Willem one considers a linear functional of the form
$$\langle \phi(u),v\rangle = \int_{0}^{T}\Big[\left\langle D_xL(t,u(t),\dot{u}(t)),v(t)\right\rangle+\left\langle D_yL(t,u(t),\dot{u}(t)),\dot{v}(t)\right\rangle\Big]dt$$ where $u$ and $v$ are members of the space of $T$-period Sobolev functions: $W_T^{1,p}$ and $L$ possesses some integrability conditions stated at the end of the question. The map $v\mapsto \langle\phi(u),v\rangle$ is the directional (Gateaux) derivative of a non-linear functional. One wants to show that the non-linear functional is Frechet differentiable and so one must show that $u\mapsto \phi(u)$ which is a map from $W_{T}^{1,p}$ to $(W_T^{1,p})^\ast$ is continuous.
However in the proof of continuity they only refer to a result by Krasnosel'skii.
Properties of $L$.
$L:[0,T]\times \mathbb{R}^N\times \mathbb{R}^N\rightarrow \mathbb{R}$ is such that $(x,y)\mapsto L(t,x,y)$ is $C^1$ for a.e. $t$. There are functions $a\in C(\mathbb{R}^+,\mathbb{R}^+)$, $b\in L^1(0,T;\mathbb{R}^N)$ and $c\in L^q(0,T;\mathbb{R}^+)$ where $p^{-1}+q^{-1} = 1$ such that \begin{align*} |L(t,x,y)|&\leq a(|x|)(b(t)+|y|^p)\\ |D_xL(t,x,y)|&\leq a(|x|)(b(t)+|y|^p)\\ |D_yL(t,x,y)| &\leq a(|x|)(c(t)+|y|^{p-1}). \end{align*}
My attempt
I want to apply something like Lebesgue's Dominated Convergence Theorem however I don't think this is possible in this case:
Suppose $u_k\rightarrow u$ in $W_T^{1,p}$ then $\|u_k-u\|_\infty\rightarrow 0$. Cauchy Schwarz implies that \begin{align*} |\langle\phi(u_k)-\phi(u),v\rangle| & \leq \int_{0}^{T}|D_xL(t,u_k(t),\dot{u}_k(t))-D_xL(t,u(t),\dot{u}(t))||v|dt\\ & + \int_{0}^{T}|D_yL(t,u_k(t),\dot{u}_k(t)-D_yL(t,u(t),\dot{u}(t))||\dot{v}|dt\\ & \leq \|v\|_\infty\int_{0}^{T}\Big|a(|u_k|)(b(t)+|\dot{u}_k(t)|^p)-a(|u|)(b(t)+|\dot{u}(t)|^p)\Big|dt\\ & + \|\dot{v}\|_p\left(\int_{0}^{T}\Big|a(|u_k(t)|)(c(t)+|\dot{u}_k(t)|^{p-1})-a(|u(t)|)(c(t)+|\dot{u}(t)|^{p-1})\Big|^{q}\right)^{1/q} \end{align*}
Now I can bound both norms in $v$ by the Sobolev norm however how can one show that the integrals converge to $0$?
Maybe one can use that every subsequence has itself a subsequence which converges and that such a limit is unique:
By a result of Riesz we can always extract a subsequence of $u_k$ such that $\dot{u}_k(t)\rightarrow \dot{u}(t)$ for almost every $t$. Since $$\Big|a(|u_k|)(b(t)+|\dot{u}_k(t)|^p)-a(|u|)(b(t)+|\dot{u}(t)|^p)\Big|\leq a(|u_k|)|\dot{u}_k(t)^p-\dot{u}(t)^p|+|a(|u_k|)-a(|u|)||b(t)+|\dot{u}(t)|^p|$$ which converges to zero in $L^1$ by Lebesgue's dominated convergence theorem since we assumed that $\dot{u}_k\rightarrow \dot{u}$ pointwise for almost every $t$.
The fact that $u_k\rightarrow u$ in $W_T^{1,p}$ implies that $\|u_k-u\|_{L^\infty}\rightarrow 0$. So in particular this value stays bounded. Furthermore by a result proven by Riesz we can extract a subsequence such that the derivatives converges pointwise. Assume that this has been done without changing the notation. Furthermore one can also choose the subsequence such that there is a dominating function $h\in L^p$ such that $\dot{u}_k(t)\leq h(t)$ (see Theorem 4.9 in Functional Analysis, Sobolev Spaces and Partial Differential Equations by Hair Brezis, reference courtesy of @daw).
Cauchy Schwarz implies that \begin{align*} |\langle\varphi'(u_k)-\varphi'(u),v\rangle| & \leq \int_{0}^{T}|D_xL(t,u_k(t),\dot{u}_k(t))-D_xL(t,u(t),\dot{u}(t))||v|dt\\ & + \int_{0}^{T}|D_yL(t,u_k(t),\dot{u}_k(t)-D_yL(t,u(t),\dot{u}(t))||\dot{v}|dt\\ & \leq \|v\|_{L^\infty}\int_{0}^{T}\Big|a(|u_k|)(b(t)+|\dot{u}_k(t)|^p)-a(|u|)(b(t)+|\dot{u}(t)|^p)\Big|dt\\ & + \|\dot{v}\|_{L^p}\left(\int_{0}^{T}\Big|a(|u_k(t)|)(c(t)+|\dot{u}_k(t)|^{p-1})-a(|u(t)|)(c(t)+|\dot{u}(t)|^{p-1})\Big|^{q}dt\right)^{1/q}. \end{align*} Clearly we can bound both of the norms in $v$ by a multiple of $\|v\|_{W_T^{1,p}}$ using Soblev and Poincar\'{e}'s inequalities so all that remains is to show that the integrals converge to $0$. Since $\|u_k\|_{L^\infty}$ is bounded we can find $M>0$ such that \begin{align*} \Big|a(|u_k|)(b(t)+|\dot{u}_k(t)|^p)-a(|u|)(b(t)+|\dot{u}(t)|^p)\Big|&\leq \underbrace{a(|u_k|)}_{\leq M}|\dot{u}_k(t)^p-\dot{u}(t)^p|+\underbrace{|a(|u_k|)-a(|u|)|}_{\leq M}|b(t)+|\dot{u}(t)|^p|\\ &\leq M\Big[|h(t)|^p+|\dot{u}(t)|^p\Big]+M|b(t)+|\dot{u}(t)|^p|\in L^1. \end{align*} Since also the pointwise limit is $0$ we conclude by Lebesgue's dominated convergence theorem that \begin{equation*} \int_{0}^{T}\Big|a(|u_k|)(b(t)+|\dot{u}_k(t)|^p)-a(|u|)(b(t)+|\dot{u}(t)|^p)\Big|dt\rightarrow 0. \end{equation*} We now consider the second equation. By rewriting similarly to before it is clear that the pointwise limit is $0$. Since \begin{equation*} a(|u_k(t)|)(c(t)+|\dot{u}_k(t)|^{p-1})\leq M(c(t)+h(t)^{p-1})\in L^q \end{equation*} we conclude again by Lebesgue's dominated convergence theorem that we may take the pointwise limit obtaining that \begin{equation*} \left(\int_{0}^{T}\Big|a(|u_k(t)|)(c(t)+|\dot{u}_k(t)|^{p-1})-a(|u(t)|)(c(t)+|\dot{u}(t)|^{p-1})\Big|^{q}dt\right)^{1/q}\rightarrow 0. \end{equation*} This shows that \begin{equation*} \frac{|\langle\varphi'(u_k)-\varphi'(u),v\rangle|}{\|v\|_{W_T^{1,p}}} \leq \epsilon(k)\rightarrow 0 \end{equation*} as $k\rightarrow \infty$ independently of $v$ so that $\|\varphi(u_k)-\varphi(u)\|\leq \epsilon(k)\rightarrow 0$.