We give a new bound on the exponent for the nonexistence of stable solutions to the biharmonic problem ∆u = u, u > 0 in R where p > 1, n ≥ 20.

In this work we study the existence of solutions for the nonlinear eigenvalue problem with p-biharmonic ∆pu = λm(x)|u|p−2u in a smooth bounded domain under Neumann boundary conditions.

In this note we use the Nehari manifold and fibering maps to show existence of positive solutions for a nonlinear biharmonic equation in a bounded smooth domain in Rn, when n = 5, 6, 7. Mathematics Subject Classification: 35J35, 35J40

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

In this article, a Picone-type identity for the weighted p-biharmonic operator is established and comparison results for a class of half-linear partial differential equations of fourth order based on this identity are derived.

In this paper we consider the Balmuş-Montaldo-Oniciuc's conjecture in case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface hemisphere Sn+1 must be small hypersphere Sn(1/2), provided n2−H2 does not change sign.