poświęcić wiele czasu na szukanie jakiegoś rozwiązania
http://sjp.pwn.pl/sjp/;2486315

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rack one's brain

(idiomatic) To struggle to think of or remember something.
I have been racking my brain all day trying to think of a birthday present for her.
https://en.wiktionary.org/wiki/rack_one's_brain





I racked my brain over this problem
http://www.dailywritingtips.com/wracking-or-racking-your-bra...

If you rack your brain, you think very hard when trying to remember something or think hard to solve a problem,
http://www.usingenglish.com/reference/idioms/rack your brain...




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search high and low (for something)
to try very hard to find something
look high and low (for something) Janet searched high and low, but she couldn't find the kitten and finally had to ask the man.
http://idioms.thefreedictionary.com/search high and low

hunt/search high and low
› to ​search ​everywhere for something:
I've been ​hunting high and ​low for that ​certificate and I still can't ​find it!
http://dictionary.cambridge.org/us/dictionary/english/hunt-s...

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Some examples:

I have searched high and low for a solution.
https://qlc.intuit.com/questions/1070339-why-does-quicken-20...
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“I've been searching high and low for the right pair of shoes for the party
http://www.wisegeek.com/what-does-it-mean-to-search-high-and...
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We search high and low to find used junior equipment so that we might ensure a
http://anymountain.net/services/junior-trade-program/