Full paper in PDF:
$%H. Brézis and J.L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,
Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469.%$
Laboratoire d’Analyse Numérique Univ. Pierre et Marie Curie 4 Pl. Jussieu 75252 Paris Cedex 05 — France | Departamento de Matemáticas Fac. de Ciencias Univ. Autónoma de Madrid 28049 Madrid — Spain |
We consider the semilinear elliptic equation
posed in a bounded domain
of
with smooth boundary
with
Dirichlet data
, and a continuous, positive, increasing and convex
function
on
such that
as
. Under these
conditions there is a maximal or extremal value of the parameter
such
that the problem has a solution. We investigate the existence and properties of
the corresponding extremal solutions when they are unbounded (i.e., singular or
blow-up solutions). We characterize the singular
extremal solutions and the
extremal value by a criterion consisting of two conditions: (i) they must be energy
solutions, not in
; (ii) they must satisfy a Hardy inequality which translates
the fact that the first eigenvalue of the linearized operator is nonnegative.
In order to apply this characterization to the typical examples arising in the
literature we need an improved version of the classical Hardy inequality with
best constant. We establish such a result as a simultaneous generalization of
Hardy’s and Poincaré’s inequalities for all dimensions
.
A striking property of some examples of unbounded extremal solutions is the fact that the linearization of the problem around them happens to be formally invertible and nevertheless the application of the Inverse and Implicit Function theorems fails to produce the usual existence or continuation results. We consider this question and explain the phenomenon as a lack of appropriate functional setting.
1991 Mathematics Subject Classification: 35J60.