= Solution
A scalar <Tonelli existence theorem> states: if $f(x,y,p)$ is continuous, convex in $p$, and has a uniform superlinear lower bound $f(x,y,p)\geq\Theta(|p|)-C$ with $\Theta(r)/r\to\infty$, and at least one fixed-endpoint admissible <absolutely continuous function> has finite energy, then the energy attains its <infimum> in the fixed-endpoint $W^{1,1}$ class. One may take $\Theta$ nonnegative and convex. The lower bound makes derivatives uniformly integrable; endpoint control then gives uniform compactness of a minimizing sequence, and <convexity> supplies the required <lower semicontinuity>.
For the specified functional, the integrand is continuous, nonnegative, and convex in the derivative, with
$$
f(x,y,p)\geq\frac{|p|^{52}}{52},\qquad
\frac{|p|^{52}/52}{|p|}\longrightarrow\infty.
$$
The affine function $u_0(x)=2x-1$ has the required endpoints and finite energy. Thus \b[a global minimizer exists]. In fact any finite-energy minimizer belongs to $W^{1,52}$, since its derivative's 52nd power is integrable.
It is \b[unique]. The admissible class is convex, and the integrand is strictly convex in $(y,p)$: $|p|^{52}$ and $|y|^{2011}$ are strictly convex, with the latter multiplied by $e^x>0$. If two distinct admissible finite-energy functions minimized the energy, their midpoint would have strictly smaller energy. Distinct continuous representatives differ on a set of positive measure, so the strict <integral> inequality really applies. This contradiction proves <uniqueness>.
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