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$%H. Brézis and J.L. Vázquez, Blow-up solutions of some nonlinear elliptic problems,
Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 443–469.%$

Blow-Up Solutions of Some Nonlinear
Elliptic Problems

Haïm BRÉZIS and Juan Luis VÁZQUEZ

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 |

Revised: December 2, 1996

ABSTRACT

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.