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 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 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 consider the nonlinear biharmonic Schr\"odinger equation $$i\partial_tu+(\Delta^2+\mu\Delta)u+f(u)=0,\qquad (\text{BNLS})$$ in critical Sobolev space $H^s(\R^N)$, where $N\ge1$, $\mu=0$ or $-1$, $0<s<\min\{\fc N2,8\}$ and $f(u)$ is a function that behaves like $\lambda\left|u\right|^{\alpha}u$ with $\lambda\in\mathbb{C},\alpha=\frac{8}{N-2s}$. prove existence uniqueness of global solutions t...

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies following: $(1)$ biharmonic equation $\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes circle), $(3)$ $f(0)=0$. The purpose this paper to prove Lipschitz continuos, and, furthe...