Gelfand type quasilinear elliptic problems with quadratic gradient terms
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 249-265.

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 : 10.1016/j.anihpc.2013.03.002
Mots clés : 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},
     pages = {249--265},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.03.002},
     mrnumber = {3181668},
     zbl = {1300.35044},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.002/}
}
TY  - JOUR
AU  - Arcoya, David
AU  - Carmona, José
AU  - Martínez-Aparicio, Pedro J.
TI  - Gelfand type quasilinear elliptic problems with quadratic gradient terms
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
SP  - 249
EP  - 265
VL  - 31
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.002/
DO  - 10.1016/j.anihpc.2013.03.002
LA  - en
ID  - AIHPC_2014__31_2_249_0
ER  - 
%0 Journal Article
%A Arcoya, David
%A Carmona, José
%A Martínez-Aparicio, Pedro J.
%T Gelfand type quasilinear elliptic problems with quadratic gradient terms
%J Annales de l'I.H.P. Analyse non linéaire
%D 2014
%P 249-265
%V 31
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.002/
%R 10.1016/j.anihpc.2013.03.002
%G en
%F 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) no. 2, pp. 249-265. doi : 10.1016/j.anihpc.2013.03.002. http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.002/

[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

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

[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 | EuDML | Numdam | MR | Zbl

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

[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 | EuDML | Numdam | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

Cité par Sources :