In this article we deal with the existence questions to the nonlinear biharmonic systems. Using theory of monotone operators, we show the existence of a unique weak solution to the weighted biharmonic systems. We also show the existence of a positive solution to weighted biharmonic systems in the unit ball in Rn , using Leray Schauder fixed point theorem. In this study we allow sign-changing we...

Abstract In this paper we consider the lower order eigenvalues of biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a type of general inequalities for them. In particular, we study the lower order eigenvalues of biharmonic operator on compact submanifolds of Euclidean spaces, of spheres, and of projective spaces. We obtain some estimates for lower orde...

We consider the existence, the nonexistence, and the uniqueness of solutions of some special systems of nonlinear elliptic equations with boundary conditions. In a particular case, the system reduces to the homogeneous Dirichlet problem for the biharmonic equation ∆ 2 u = |u| p in a ball with p > 0.

In this article, we show the existence of at least three solutions to a Navier boundary problem involving the p(x)-biharmonic operator. The technical approach is mainly base on a three critical points theorem by Ricceri.

We investigate biharmonic submanifolds of the product of two space forms. We prove a necessary and sufficient condition for biharmonic submanifolds in these product spaces. Then, we obtain mean curvature estimates for proper-biharmonic submanifold of a product of two unit spheres. We also prove a non-existence result in the case of the product of a sphere and a hyperbolic space.

This paper studies some properties of F-biharmonic maps between Riemannian manifolds. By considering the first variation formula of the F-bienergy functional, F-biharmonicity of conformal maps are investigated. Moreover, the second variation formula for F-biharmonic maps is obtained. As an application, instability and nonexistence theorems for F-biharmonic maps are given.

This report investigates the paper Lipman et al. Biharmonic distance, who introduced a new distance measure, called biharmonic. We examine the approach of the distance, which is based on the Green’s kernel of the Bi-Laplacian in the continuous and the discrete setting. For the discrete setting we have two different methods of calculation, an approximate as well as an exact computation. Furtherm...

We classify the space-like biharmonic surfaces in 3dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.

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 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.