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.

A. Both bi-harmonic map and f -harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f -bi-harmonic maps as the critical points of the f -bi-energy functional 1 2 ∫ M f |τ(φ)| dvg. This class of maps generalizes both concepts of harmonic maps and biharmonic maps. We first derive the f -biharmonic map ...

A new very high-order finite volume method to solve the harmonic and the biharmonic operators for one-dimensional geometries is proposed. The main ingredient is the polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic ope...