The purpose of this note is to derive some integral functionals associated to the following
in the weak form in the sense that each critical point of is a weak solution for equation
For simplicitly, we denote
where is a Riemannian manifold with metric and a function sitting in an appropriate Sobolev space. To be exact, we shall find a functional so that its first variation, denoted by , equals .
Type 1. We shall find of the following form
for some constant to be determined later. To this purpose, we try to calculate the first variation of in the direction . Indeed,
Therefore, we may choose .
Type 2. We shall find of the following form
for some constant to be determined later. The first variation of in the direction can be computed as below
Thus, in this case, we have to choose