Electron. J. Diff. Eqns., Vol. 2004(2004), No. 110, pp. 1-16

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

Jorge Garcia-Melian

Abstract:
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 $p$ greater than 2, and $\lambda$ is a real parameter. We assume that the nonlinearity $f$ has a zero $\bar{u}_0$ greater than 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.

Submitted July 5, 2004. Published September 21, 2004.
Math Subject Classifications: 35J60, 35J70.
Key Words: p-Laplacian; linearization; uniqueness

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Jorge Garcia-Melian
Dpto. de Analisis Matematico
Universidad de La Laguna
38271 La Laguna - Tenerife, Spain
email: jjgarmel@ull.es

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