I want to prove that the following function is continuous :

$$ V(\phi)=\int_0^1\left[ \frac{a(x)}{2}\phi_x^2(x)-F\left(\phi(x)\right)\right] dx $$ where:

  1. $\phi\in H_0^1([0,1)]$
  2. $\phi_x=\frac{\partial \phi}{\partial x}$
  3. $a\in C^1([0,1],\mathbb R)$ and $a(x)>0$ pour tout $x\in[0,1]$.
  4. $f\in C^2(\mathbb R,\mathbb R)$ and $\lim\sup_{|u|\to+\infty}\frac{f(u)}{u}\leq 0$
  5. $F(u)=\int_0^u f(s)ds$

Since the function $V$ is not linear it's hard for me to prove the continuity, I tried to use sequences i.e. $\phi_n\to \phi$ but I'm stuck in proving that $\phi_{n_x}^2\to \phi_x^2$ since the derivative is not continuous. Please do you have an idea how to prove it ? thank you for your time :)


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