Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

Abstract

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of R n (n ≥ 2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions K m,n containing properly the classical Kato class K n . We exploit properties of functions belonging to K m,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.

Existence results for some polyharmonic problems in the half-space

“…Using (1.2), the authors in [5] proved some estimates for G m,n , which are parallel to those satisfied by the Green function H m,n in B (see [4,7]). …”

Positive bounded solutions for nonlinear polyharmonic problems in the unit ball

“…Finally, we mention that the letter c will be a positive generic constant which may vary from line to line. To make the paper self contained, this section is devoted to recall some results established in [2,4,8,10] that will be useful for our study.…”

“…where k m,n is a positive constant, ∂ ∂ν is the outward normal derivative and for x, y in B, [x, y] 2 = |x − y| 2 + 1 − |x| 2 1 − |y| 2 . In [2], the estimates on the Green function G m,n of (−∆) m on B and particulary the 3G−theorem (see [2], Theorem 2.8), allowed the authors to introduce a large functional class called Kato class denoted by K m,n (see Definition 1 below). This class plays a key role in the study of some nonlinear polyharmonic equations (see [2,3,6,10]).…”

“…In [2], the estimates on the Green function G m,n of (−∆) m on B and particulary the 3G−theorem (see [2], Theorem 2.8), allowed the authors to introduce a large functional class called Kato class denoted by K m,n (see Definition 1 below). This class plays a key role in the study of some nonlinear polyharmonic equations (see [2,3,6,10]). For related results we refer to the recent monograph [12] and the papers [7,9,11,13,14,15].…”

On the very weak solvability of the beam equation

“…After the obtention of the rsults of this Note, lectured by the author on September 16th, 2010, in the meeting "The Eleventh International Conference Zaragoza-Pau on Applied Mathematics and Statistics", Jaca (Spain) the author was informed by J.M. Rakotoson of the paper [1] in which the authors study the case of Dirichlet boundary conditions on a ball of R n with n ≥ 2 and under symmetry conditions on the data. Some of their results are close to conclusions b) and c) of Theorem 1 but the methos of proof are different and, as indicated before, the conclusions of Theorem 1 remains valid for the many other boundary conditions (for instance for the case of Dirichlet conditions at x = 0 and Neumann at x = L) which cannot be related to formulations obtained trough radially symmetry solutions on balls of higher dimensions.…”