You cannot win a challenge if all your participating characters strength has been reduced to zero, no claim no unopposed bonus, no triggers from winning a challenge. The opponent does not even need to defend.
Technically, you can win a challenge if all of your participating characters' STR have been reduced to 0. You cannot win a challenge of your total STR has been reduced to 0. RR, p. 28 (2ed, but this rule is functionally the same in 1ed):
The player whose side has the higher total STR wins the challenge (or in the case of a tie, the challenge is won by the attacking player). If this player does not have a total STR of 1 or higher, and/or if there are no participating characters on this player’s side when this step (i.e., the STR comparison) occurs, neither player wins (or loses) the challenge.
If, for example, you have The Red Keep (2ed, but plenty of such cards exist in 1ed) in play, you have a +2 modifier in any a Power challenge in which you have a participating character. You could therefore win the challenge even if your character's STR is reduced to 0 (so long as the character is still participating; having its STR reduced to 0 does not remove it from the challenge).