Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 249-265

In this paper, for $0<{m}_{1}⩽m\left(x\right)⩽{m}_{2}$ and positive parameters λ and p, we study the existence of positive solution for the quasilinear model problem $\left\{\begin{array}{cc}-\Delta u+m\left(x\right)\frac{{|\nabla u|}^{2}}{1+u}=\lambda {\left(1+u\right)}^{p}\hfill & \text{in}\phantom{\rule{4pt}{0ex}}\Omega ,\hfill \\ u=0\hfill & \text{on}\phantom{\rule{4pt}{0ex}}\partial \Omega .\hfill \end{array}$ We prove that the maximal set of λ for which the problem has at least one positive solution is an interval $\left(0,{\lambda }^{⁎}\right]$, with ${\lambda }^{⁎}>0$, and there exists a minimal regular positive solution for every $\lambda \in \left(0,{\lambda }^{⁎}\right)$. We also prove, under suitable conditions depending on the dimension N and the parameters p, ${m}_{1}$, ${m}_{2}$, that for $\lambda ={\lambda }^{⁎}$ there exists a minimal regular positive solution. Moreover we characterize minimal solutions as those solutions satisfying a stability condition in the case ${m}_{1}={m}_{2}$.

DOI : https://doi.org/10.1016/j.anihpc.2013.03.002
Keywords: Gelfand problem, Quasilinear elliptic equations, Quadratic gradient, Stability condition, Extremal solutions
@article{AIHPC_2014__31_2_249_0,
author = {Arcoya, David and Carmona, Jos\'e and Mart\'\i nez-Aparicio, Pedro J.},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {2},
year = {2014},
pages = {249-265},
doi = {10.1016/j.anihpc.2013.03.002},
zbl = {1300.35044},
mrnumber = {3181668},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_2_249_0}
}

Arcoya, David; Carmona, José; Martínez-Aparicio, Pedro J. Gelfand type quasilinear elliptic problems with quadratic gradient terms. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, pp. 249-265. doi : 10.1016/j.anihpc.2013.03.002. http://www.numdam.org/item/AIHPC_2014__31_2_249_0/

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