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

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