Large radial solutions of a polyharmonic equation with superlinear growth

Abstract

This paper concerns the equation ∆mu = |u| p, where m ∈ N, p ∈ (1, ∞), and ∆ denotes the Laplace operator in RN, for some N ∈ N. Specifically, we are interested in the structure of the set L of all large radial solutions on the open unit ball B in RN . In the well-understood second-order case, the set L consists of exactly two solutions if the equation is subcritical, of exactly one solution if it is critical or supercritical. In the fourth-order case, we show that L is homeomorphic to the unit circle S 1 if the equation is subcritical, to S 1 minus a single point if it is critical or supercritical. For arbitrary m ∈ N, the set L is a full (m − 1)-sphere whenever the equation is subcritical. We conjecture, but have not been able to prove in general, that L is a punctured (m − 1)-sphere whenever the equation is critical or supercritical. These results and the conjecture are closely related to the existence and uniqueness (up to scaling) of entire radial solutions. Understanding the geometric and topological structure of the set L allows precise statements about the existence and multiplicity of large radial solutions with prescribed center values u(0), ∆u(0), . . . , ∆m−1u(0).