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Résumé: Dans cette Note nous considérons une classe de problèmes non linéaires et non coercifs dont le prototype est $$- \Delta_p u + b(x)|\nabla u|^\lambda = \mu \; \hbox{dans} \; \Omega, \quad u = 0 \; \hbox{sur} \; \partial\Omega,$$ où $\Omega$ est un ouvert borné de $\R^N$ $(N \geq 2)$, $\Delta_p$ est le $p$-Laplacien $(1 1 /2$, alors il existe $\lambda_0 >0$ tel qu'il y a une unique solution pour $\lambda\lambda_0$. La preuve repose sur des estimations a priori, la construction de barrières et des arguments de degré topologique.
Abstract: We consider the problem $\Delta u = \lambda f(u)$ in $\Omega$, $ u(x)$ tends to $+\infty$ as $x$ approaches $\partial\Omega$. Here $\Omega$ is a bounded, star shaped, smooth domain in $\R^N$, $N \geq 1$ and $\lambda$ a positive parameter. In this paper, we are interested in analyzing the role of the {\sl sign changes} of the function $f$ in the number of solutions of this problem. Our main result states that if $\Omega$ is star-shaped and, $f$ behaves like $f(u) = u(u-a)(u-1)$ with $a>1 /2$, then there is a $\lambda_0 >0$ such that the problem has a unique solution for $\lambda\lambda_0$. The proof is based on a priori estimates, the construction of barriers and topological degree arguments.
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Date: 2001-09-12