〜ないものか・〜ないものだろうか
isn't there some way to …?, if only
Verb ない-form (often potential) + ものか / ものだろうか
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ないものか expresses a strong wish that something could somehow happen, phrased as a question: もっと安く行ける方法はないものか, "isn't there some cheaper way to get there?"
It's often used with potential verbs and 何とか ("somehow"): 何とかならないものか, "isn't there something that can be done?" The speaker is searching for a solution, not really asking.
ないものだろうか is softer and more polite.
Don't confuse it with ものか (N2, "as if I would!"), which after a positive verb is a strong refusal. With a negative, the meaning flips into a longing wish.
Examples
もっと安く行ける方法はないものか。
Isn't there some cheaper way to get there?
何とかならないものか。
Isn't there something that can be done?
この状況を変えられないものだろうか。
Is there really no way to change this situation?
早く夏休みにならないものか。
If only the summer holidays would hurry up and come.
彼の病気が治らないものかと、毎日祈っている。
Every day I pray that somehow his illness will be cured.