Gelfand type quasilinear elliptic problems with quadratic gradient terms
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 2, p. 249-265
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In this paper, for 0<m 1 m(x)m 2 and positive parameters λ and p, we study the existence of positive solution for the quasilinear model problem {-Δu+m(x)|u| 2 1+u=λ(1+u) p inΩ,u=0onΩ. We prove that the maximal set of λ for which the problem has at least one positive solution is an interval (0,λ ], with λ >0, and there exists a minimal regular positive solution for every λ(0,λ ). We also prove, under suitable conditions depending on the dimension N and the parameters p, m 1 , m 2 , that for λ=λ 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.},
     title = {Gelfand type quasilinear elliptic problems with quadratic gradient terms},
     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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