Let $u ∈ W^{1,1} (]0, 1[)$ and $$J_\varepsilon(u) =\frac{1}{2}\int_0^1(\varepsilon+x^\alpha)u'^2dx+\frac{1}{4}\int_0^1u^4 dx-\int_0^¹uf dx.$$
Prove $J(u)$ is convex.
My argument is sum of 3 functions.
The first is quadratic with strictly positive coefficient $\frac{1}{2}(\epsilon+x^\alpha)$, so it is strictly convex.
The second is also quadratic and equal to $\|u^4\|_{L_4}^4$ so also strictly convex.
The third is linear so convex.
As a sum of convex and strictly convex functions, $J$ is strictly convex.
Is that a correct proof?
Thank you for your help.