I trust you give my application serious consideration.
The verb "trust" is not used in combination with a bare infinitive.
You have created a subjunctive form: I trust you give (I trust that you give) that is rarely if ever used in English.
A subjunctive form would mean that there was some doubt in your trust.
Better would be: I trust you to give my application serious consideration.
or as Steve suggested using the future tense, which is more polite:
I trust that you will give my application serious consideration.
I hope that you will give my application serious consideration......even more polite...
You might want to stick with the tried and true universal formula:
Thank you in advance for considering my application.
No hoping or trusting needed! :)