Finding weak form of equation with two different functions

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There are two different functions f and g in an equation. I would like to make this equation into a weak formulation.

$$ \nabla^2f+\nabla^2 \ln g + a=0 $$

For example, if this equation is defined in x dimension, how can I make this to weak form? I'm not sure if there are two different test functions needed for each function. Please feel free to set a convenient boundary condition for solving