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.

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of regular solutions tow...

A new mimetic iterative scheme for solving general biharmonic equations under Robin’s conditions is presented. This approach combines recently developed mimetic techniques for partial differential equations (PDEs) with an efficient iterative scheme based on a global conjugate gradient and a local preconditioned biconjugate gradient methods. The elegant matrix formulation of mimetic methods allo...

We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions a few reference biharmonic problems elasticity elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that challenging solve using classical numerical methods have not been addressed PINNs. Our...

A factoring and block elimination method for the fast numerical solution of block five diagonal linear algebraic equations is described. Applications of the method are given for the numerical solution of several boundary-value problems involving the biharmonic operator. In particular, 22 eigenvalues and eigenfunctions of the clamped square plate are computed and sketched.

*Correspondence: [email protected] School of Mathematics and Physics, University of South China, Hengyang, P.R. China Abstract In this paper, we study a class of biharmonic equations with a singular potential inRN . Under appropriate assumptions on the nonlinearity, we establish some existence results via the Morse theory and variational methods. We significantly extend and complement some re...

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristi...