BLOW-UP SOLUTIONS OF SOME NONLINEAR ELLIPTIC PROBLEMS


Haim BREZIS \& Juan Luis VAZQUEZ
   

R\'esum\'e : On consid\`ere l'\'equation elliptique semilin\'eaire
$- \Delta u = \lambda f(u) \; \hbox{sur} \; \Omega;
\; u = 0 \; \hbox{sur} \; \partial\Omega$ dans un domaine
born\'e r\'egulier $\Omega$ de $\R^n$ où $f$ est une fonction
convexe, croissante, strictement positive sur
$[0,+\infty[$ avec $f(s)/s \rightarrow +
\infty$ quand $s \rightarrow + \infty$. On sait qu'il
existe une valeur $\lambda^\star$ maximale (on dira aussi
extr\'emale) du param\`etre $\lambda$ telle que le probl\`eme
admette une solution. On s'int\'eresse \`a la situation o\`u la
solution extr\'emale correspondante est singuli\`ere. On
caract\'erise les solutions $u^\star$ extr\'emales
singuli\`eres dans $H^1$ par deux conditions : (i) $u^\star
\in H^1$, $u^\star
\not\in L^\infty$, (ii) on a une in\'egalité de type Hardy
$\int |\nabla \varphi|^2 - \lambda^\star \int
f'(u^\star)\varphi^2 \geq 0, \; \forall \varphi \in H^1_0$
qui exprime que l'op\'erateur lin\'earis\'e a une premi\`ere
valeur propre $\geq 0$. L'\'etude d'exemples classiques
nous conduit \`a une version am\'elior\'ee de l'in\'egalit\'e de
Hardy. Une autre propri\'et\'e surprenante est que
l'op\'erateur lin\'earis\'e est formellement inversible et
n\'eanmoins on ne peut appliquer ni le th\'eor\`eme des
fonctions inverses, ni le th\'eor\`eme des fonctions
implicites.


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Abstract: We consider the semilinear elliptic
equation $ - \Delta u = \lambda f(u)$, posed in a bounded
domain $\Omega$ of $\R^n$ with smooth boundary
$\partial\Omega$ with Dirichlet data $u_{|\partial\Omega}
= 0$, and a continuous, positive, increasing and convex
function $f$ on $[0,\infty)$ such that $f(s)/s
\rightarrow \infty$ as $s \rightarrow
\infty$. Under these conditions there is a maximal or
extremal value of the parameter
$\lambda > 0$ 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 $H^1$ extremal solutions and the extremal value
by a criterion consisting of two conditions : (i) they
must be energy solutions, not in $L^\infty$, (ii) they
must satisfy a Hardy inequality which translates the fac
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 $n \geq 2$. 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.


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