Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores

Garcia-Melian, Jorge

Electronic Journal of Differential Equations, Tome 2004 (2004).

Résumé

Summary: In this work we study the problem $$ -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda f(u) $$ in the unit ball of $\mathbb{R}^N$, with $u=0$ on the boundary, where $p$, and $\lambda$ is a real parameter. We assume that the nonlinearity $f$ has a zero $\bar{u}_0$ of order $k\ge p-1$. Our main contribution is showing that there exists a unique positive solution of this problem for large enough $\lambda$ and maximum close to $\bar{u}_0$. This will be achieved by means of a linearization technique, and we also prove the new result that the inverse of the $p$-Laplacian is differentiable around positive solutions.

Détail

Classification : 35J60, 35J70
Keywords: p-Laplacian, linearization, uniqueness