where is the Laplace operator, λ is a constant that expresses the "screening", f is an arbitrary function of position (known as the "source function") and u is the function to be determined.
In the homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets.
Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension , is a superposition of 1/r functions weighted by the source function f:
On the other hand, when λ is extremely large, u approaches the value f/λ², which goes to zero as λ goes to infinity. As we shall see, the solution for intermediate values of λ behaves as a superposition of screened (or damped) 1/r functions, with λ behaving as the strength of the screening.
The screened Poisson equation can be solved for general f using the method of Green's functions. The Green's function G is defined by
where δ3 is a delta function with unit mass concentrated at the origin of R3.
As stated above, this is a superposition of screened 1/r functions, weighted by the source function f and with λ acting as the strength of the screening. The screened 1/r function is often encountered in physics as a screened Coulomb potential, also called a "Yukawa potential".