Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

Abstract

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. Our results depend on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter we prove existence of solutions to this boundary value problem. For large values of the same parameter we prove nonexistence of solutions. We also provide rigorous bounds for the values of this parameter which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions.