ないものか / ないものだろうか

ないものか and ないものだろうか voice an earnest wish for something hard to come about: "isn't there some way…?", "can't something be done?". They follow the ない-form of a verb, very often a potential one or ならない, and the speaker is thinking aloud more than asking anyone.

The wish is for something difficult but possible. Words like なんとか, どうにか and もうすこし少し go with it.

ないものか or ものか

Without ない, dictionary form + ものか is the opposite: a flat refusal, "as if I would". The ない is what makes it a wish.

ないものか or ないものだろうか

The meaning is the same. ないものだろうか is softer and more reflective; ないものか sounds more impatient.

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Both belong to writing, speeches and inner monologue. In conversation the same wish is 〜～たらいいのに or なんとかならない何とかならないかな.