Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 110, pp. 1-16
Title: Uniqueness for degenerate elliptic sublinear problems
in the absence of dead cores
Author: Jorge Garcia-Melian (Univ. de La Laguna, Tenerife, Spain)
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 $p>2$, and $\lambda$ is a real parameter.
We assume that the nonlinearity $f$ has a zero
$\bar{u}_0>0$ 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