The Question to which you were replying does not refer to "physics" or invite an interpretation in terms of "an energy functional". Making this interpretation might be useful as content for Math.SE, but your presentation merely mentions such an interpretation and immediately transitions to "Putting it all together..."
If you have in mind some definite way to interpret the functional to be minimized as "energy", it might be a welcome supplement to the existing upvoted Accepted Answer to the Question. But the present wording does not really expose your thinking to Readers (such as myself) with a casual familiarity with mathematical physics, and minimum energy formulations in particular.
Note that you state your conclusion as finding a limit $r$ which is "not equal to $0$", and thus in apparent disagreement with the Accepted Answer. The Comments from that Answer's author on your post point to your having misunderstood what the Question asks for (an infimum), and you should make an effort to resolve the discrepancy before claiming (as above) "My answer is correct however from a mathematical-physics standpoint."