Gelfand type quasilinear elliptic problems with quadratic gradient terms

Arcoya, David; Carmona, José; Martínez-Aparicio, Pedro J.

Arcoya, David ; Carmona, José ; Martínez-Aparicio, Pedro J.

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 249-265 / Harvested from Numdam

Résumé

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 ${\begin{matrix} - Δ u + m (x) \frac{{| \nabla u |}^{2}}{1 + u} = λ {(1 + u)}^{p} & in Ω, \\ u = 0 & on \partial Ω . \end{matrix}$ 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 $λ \in (0, λ^{⁎})$ . 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 $m_{1} = m_{2}$ .

Publié le : 2014-01-01

MR 3181668 | Zbl 1300.35044

DOI : https://doi.org/10.1016/j.anihpc.2013.03.002

@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},
     volume = {31},
     year = {2014},
     pages = {249-265},
     doi = {10.1016/j.anihpc.2013.03.002},
     mrnumber = {3181668},
     zbl = {1300.35044},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 31 (2014) pp. 249-265. doi : 10.1016/j.anihpc.2013.03.002. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_2_249_0/

Bibliographie

[1] D. Arcoya, J. Carmona, P.J. Martínez-Aparicio, Radial solutions for a Gelfand type quasilinear elliptic problem with quadratic gradient terms, Contemp. Math., to appear. | MR 3155963

[2] D. Arcoya, J. Carmona, P.J. Martínez-Aparicio, Bifurcation for quasilinear elliptic singular BVP, Comm. Partial Differential Equations 36 (2011), 1-23 | MR 2763328 | Zbl 1239.35046

[3] D. Arcoya, S. Segura De León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var. 2 (2010), 327-336 | Numdam | MR 2654196 | Zbl 1189.35109

[4] C. Bandle, Sur un problème de Dirichlet non linéaire, C. R. Acad. Sci. Paris 276 (1973), 1155-1157 | MR 316888 | Zbl 0249.35031

[5] G. Barles, A.P. Blanc, C. Georgelin, M. Kobylanski, Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 no. 3 (1999), 381-404 | Numdam | Zbl 0940.35078

[6] G. Barles, F. Murat, Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions, Arch. Ration. Mech. Anal. 133 (1995), 77-101 | MR 1367357 | Zbl 0859.35031

[7] L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM Control Optim. Calc. Var. 14 (2008), 411-426 | Numdam | MR 2434059 | Zbl 1147.35034

[8] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), 581-597 | MR 1183665 | Zbl 0783.35020

[9] L. Boccardo, F. Murat, J.P. Puel, Existence de solutions non bornees pour certaines équations quasi-linéaires, Port. Math. 41 (1982), 507-534 | MR 766873 | Zbl 0524.35041

[10] L. Boccardo, F. Murat, J.P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Ann. Mat. Pura Appl. (4) 152 (1988), 183-196 | MR 980979 | Zbl 0687.35042

[11] L. Boccardo, L. Orsina, M.M. Porzio, Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources, Adv. Calc. Var. 4 no. 4 (2011), 397-419 | MR 2844511 | Zbl 1232.35068

[12] H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa, Blow up for $u_{t} - Δ u = g (u)$ revisited, Adv. Differential Equations 1 (1996), 73-90 | MR 1357955 | Zbl 0855.35063

[13] H. Brezis, J.L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 no. 2 (1997), 443-469 | MR 1605678 | Zbl 0894.35038

[14] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension 4, Comm. Pure Appl. Math. 63 no. 10 (2010), 1362-1380 | MR 2681476 | Zbl 1198.35094

[15] M. Crandall, P.H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal. 58 no. 3 (1975), 207-218 | MR 382848 | Zbl 0309.35057

[16] L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. vol. 143, Chapman and Hall/CRC (2011) | MR 2779463 | Zbl 1228.35004

[17] J. García Azorero, I. Peral, J.P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal. 22 no. 4 (1994), 481-498 | MR 1266373 | Zbl 0804.35037

[18] I.M. Gelfand, Some problems in the theory of quasi-linear equations, Amer. Math. Soc. Transl. 29 no. 2 (1963), 295-381 | MR 153960 | Zbl 0127.04901

[19] D.D. Joseph, T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1972/1973), 241-269 | MR 340701 | Zbl 0266.34021

[20] H.B. Keller, D.S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361-1376 | MR 213694 | Zbl 0152.10401

[21] F. Mignot, J.P. Puel, Sur une classe de problèmes nonlinéaires avec nonlinéarité positive, croissante, convexe, Comm. Partial Differential Equations 5 (1980), 791-836 | MR 583604 | Zbl 0456.35034

[22] L. Orsina, J.P. Puel, Positive solutions for a class of nonlinear elliptic problems involving quasilinear and semilinear terms, Comm. Partial Differential Equations 26 (2001), 1665-1689 | MR 1865941 | Zbl 1242.35126

[23] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus, Les Presses de l'Université de Montréal, Montréal (1966) | MR 251373 | Zbl 0151.15501