The problem:
Define
$$F(x,z,p) := |p|^q - z^2 + \log(1+z^2)$$
where $\log$ denotes the natural logarithm and $q > 1$. Define
$$\mathscr{F}(u) := \int_0^1F(x,u(x),u'(x)) \, {\rm d} x$$
for every $u \in W^{1,1}([0,1])$ such that $u(0)=0$ and $u(1)=1$.
I would like to know if the minimum of $\mathscr{F}$ exists.
My attempt:
First of all $F \in C^{\infty}$ and is convex in $p$.
I tried defining $F_k(x,z,p)=F(x,z,p)$ if $|z|\leq k$ and $F(x,k,p)$ if $|z|\geq k$ and studying first the functional $F_k$ but I cannot reach any conclusion.
Any hint or help will be appreciated.