Radial solutions of a polyharmonic equation with power nonlinearity Original Research Article

We classify the regular radial solutions of the equation Δmu=u|u|p−1Δmu=u|u|p−1 with mm a positive integer and p∈(1,∞)p∈(1,∞). Any such solution uu is uniquely determined by the center values of uu and its iterated Laplacians up to order m−1m−1, that is, by the “center point” (u(0),Δu(0),…,Δm−1u(0))(u(0),Δu(0),…,Δm−1u(0)). Most solutions have finite exit radius (that is, they are unbounded and defined on open balls of finite radius), are eventually monotonic with respect to the radial variable, and diverge to ∞∞ or −∞−∞. The center points of these solutions form two open half-spaces in RmRm, separated by an (m−1)(m−1)-dimensional manifold MM. Solutions with center points on MM are either global, or they have finite exit radius and oscillate about 0 with unbounded amplitude. Our results yield precise information regarding the existence, multiplicity, and nature of large radial solutions of Δmu=u|u|p−1Δmu=u|u|p−1 on a ball of given radius.