Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores
Electronic journal of differential equations, Tome 2004 (2004)
In this work we study the problem
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.
| $ -\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)=\lambda f(u) $ |
@article{EJDE_2004__2004__a50,
author = {Garcia-Melian, Jorge},
title = {Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores},
journal = {Electronic journal of differential equations},
year = {2004},
volume = {2004},
zbl = {1129.35378},
language = {en},
url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a50/}
}
Garcia-Melian, Jorge. Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a50/