Existence and nonexistence results of polyharmonic boundary value problems with supercritical growth

Consider the following polyharmonic problems under the Dirichlet or the Navier boundary conditions (-Delta)mu=f(u)+ l |u|(p-1)u,in Omega, where Omega subset of R-N is a bounded smooth domain and p > 1 is a subcritical power. We examine the effect of the positive parameter lambda to study the existence and the nonexistence of regular solutions. When lambda is large enough, we establish an existence result only undersuitable growth condition on f at zero. Our approach is based on truncation argument as well as L-infinity-bounds. Also, by virtue of Pucci-Serrin's variational identity [Pucci P, Serrin J. A general variational identity. Indiana Univ Math J. 1986;35:681-703.],, we provide a nonexistence result when f has a supercritical growth at infinity and lambda is small enough.