How can I find the supremum value of this equation? In the following equation:$$f(y)=\sup_{x>0}\bigl(\exp(|y|-|y-x|)\bigr)$$
How can I find the value of supremum? can anyone help me to find it?Thank you.

You want to maximize $|y| - |y-x|$ and since the contribution of $|y-x|$ is always $\leq 0$ while that of $|y|$ is always $> 0$ you see that the supremum is attained when $x = y$, and hence corresponds to $\exp(|y|)$. 
EDIT: I haven't noticed the condition $x > 0$. As pointed out by Henry for $y < 0$ the supremum is attained when $x \rightarrow 0$ and is $\exp(0) = 1$. 
