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Potentials play an important role in solving boundary value problems forelliptic equations. In the middle of the last century, a potential theory wasconstructed for a two-dimensional elliptic equation with one singularcoefficient. In the study of potentials, the properties of the fundamentalsolutions of the given equation are essentially and fruitfully used. At thepresent time, fundamental solutions of a three-dimensional elliptic equationwith one degeneration line are already known. In this paper, we investigate thedouble- and simple-layer potentials for this kind of elliptic equations.Results from potential theory allow us to represent the solution of theboundary value problems in integral equation form. By using some properties ofGaussian hypergeometric function, we prove limiting theorems and deriveintegral equations concerning a densities of the double- and simple-layerpotentials. The obtained results are applied to find an explicit solution ofthe Dirichlet and Holmgren problems for the three-dimensional singular ellipticequation in the half of the ball.

Solving the Dirichlet and Holmgren problems for a three-dimensional elliptic equation by the potential method