On a bi-nonlocal fourth order elliptic problem

A Semilinear Fourth Order Elliptic Problem with Exponential Nonlinearity

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section. Mathematics Subject Classification: 35J65; 35J40.

Multiple solutions for a fourth-order nonlinear elliptic problem

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation β∆u−τ∆u = λ (1−u)2 on a ball B ⊂ R , under Navier boundary conditions u = ∆u = 0 on ∂B, where λ > 0 is a parameter, while τ > 0, β > 0 are fixed constants. It is known that there exists a λ∗ such that for λ > λ∗ there is no solution while for λ < λ∗ there is a branch of minimal solutions. Our main result asse...

A third-order nonlocal problem with nonlocal conditions

Fast Algorithm for the Fourth-Order Elliptic Problem Based on Orthogonal Matrix Decomposition

A fast algorithm for solving the first biharmonic boundary problem on a rectangular domain is presented. It is based on splitting the fourth-order problem into a coupled system of two-order problems, whose finite-di↵erence approximations are solved iteratively with linear algebra routines. There, a crucial role plays the orthogonal eigenvalue decomposition of the iteration matrix, which leads t...