I am frequently confused the definition of sup- and super-solutions so yesterday I tried to figure out a way to remember those things. Fortunately, I think I got a very simple way to remember. The point is, just denote by and the sup- and super-solutions respectively to the very simple PDE

,

which one is true

or

and similarly to .

Let us consider a general case. Assume we are working with a general second-order elliptic operator in non-divergence form, say

where are positive coefficients. Besides, coefficients of are assumed to satisfy several conditions like symmetry, etc. but it is not considered here.

Concerning the following PDE

,

we say

function (resp. ) is said to be a super-solution (resp. sub-solution) to the PDE if (resp. ).

Observe that is positive lets us think that has positive spectrum. So once is a super-solution, the inequality should be where the left hand side should be an operator with positive spectrum. Similarly, concerning the sub-solutions, we need .

Let us go back to the Laplacian operator . So has negative spectrum, this yields

and

.

Equivalently,

and

since has positive spectrum.