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