Consider the functional
$$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$.
Given $a<b$, find the value of $$\inf\{F(u): u\in W^{1,p}(0,1), u(0)=a, u(1)=b\}$$
Can any one help me with this problem ?