We consider a sequence of multi-bubble solutions $u_k$ of the following fourth order equation$${\Delta }^2{u}_k={\rho }_k\frac{h\left(x\right)e^{u_k}}{{\int }_{\Omega }h{e}^{u_k}}\text{in}\Omega ,u_k=\Delta u_k=0\text{on}\partial \Omega ,(*)$$where $h$ is a $C^{2,\beta }$ positive function, $\Omega$ is a bounded and smooth domain in ${ℝ}^4$, and ${\rho }_k$ is a constant such that ${\rho }_k\le C$. We show that (after extracting a subsequence), ${lim}_{k\to +\infty }{\rho }_k=32{\sigma }_{3}m$ for some positive integer $m\ge 1$, where ${\sigma }_3$ is the area of the unit sphere in ${ℝ}^4$. Furthermore, we obtain the following sharp estimates for ${\rho }_k$:$$\begin{array}{cc}\hfill {\rho }_k-32{\sigma }_{3}m& =c_0\sum _{j=1}^m{ϵ}_{k,j}^2\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\hfill \\ & +o\left(\sum _{j=1}^m{ϵ}_{k,j}^2\right)\hfill \end{array}$$where $c_0\>0$, $log\frac{64}{{ϵ}_{k,j}^4}=\underset{x\in B_{\delta }\left(p_j\right)}{max}{u}_k\left(x\right)-log\left(\underset{\Omega }{\int }h{e}^{u_k}\right)$ and $u_k\to 32{\sigma }_3\sum _{j=1}^m{G}_4(·,p_j)$ in $C_{\mathrm{loc}}^4(\Omega \setminus \{p_1,...,p_m\})$. This yields a bound of solutions as ${\rho }_k$ converges to $32{\sigma }_{3}m$ from below provided that$$\sum _{j=1}^m\left(\sum _{l\ne j}\Delta G_4(p_j,p_l)+\Delta R_4(p_j,p_j)+\frac{1}{32{\sigma }_3}\Delta logh\left(p_j\right)\right)\>0.$$The analytic work of...

Applying the method of normalized systems of functions we construct solutions of the generalized Dirichlet problem for the iterated slice Dirac operator in Clifford analysis. This problem is a natural generalization of the Dirichlet problem.

This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $\Omega \subset {ℝ}^n$, $u=Du=...=D^{m-1}u$ on ∂Ω in the Sobolev space $W_0^{m,2}\left(\Omega \right)$, where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.

The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector....

The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples....

A family of linear homogeneous 4th order elliptic differential operators $L$ with real constant coefficients, and bounded nonsmooth convex domains $\Omega$ are constructed in ${ℝ}^6$ so that the $L$ have no constant coefficient coercive integro-differential quadratic forms over the Sobolev spaces $W^{2,2}\left(\Omega \right)$.