How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$.

Question

How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$. For some question we can make the $\nabla$ term dissapear by transformation, then using varaition method, for example, Yamabe problem, but how can we write the Euler-Langrange equation of elliptic PDE containing $\nabla$.