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[Evans PDE Problem 6.6.1] Laplace's equation with potential function

Consider Laplace's equation with potential function   :

and the divergence structure equation:

where the function   is postive.

(1)、 Show that if   solves  and   also solves  , then   solves  for   .

(2)、 Conversely, show that if   solves  , then   solves  for some potential   .

Proof.

(1)、 Assume   . Then

(2)、 Assume

Then

Choosing   , then   . $$