Gelfand type elliptic problems under Steklov boundary conditions

Berchio, Elvise; Gazzola, Filippo; Pierotti, Dario

doi:10.1016/j.anihpc.2009.09.011

Berchio, Elvise ; Gazzola, Filippo ; Pierotti, Dario

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 315-335

Résumé

For a Gelfand type semilinear elliptic equation we extend some known results for the Dirichlet problem to the Steklov problem. This extension requires some new tools, such as non-optimal Hardy inequalities, and discovers some new phenomena, in particular a different behavior of the branch of solutions and three kinds of blow-up for large solutions in critical growth equations. We also show that small values of the boundary parameter play against strong growth of the nonlinear source.

MR Zbl

DOI : 10.1016/j.anihpc.2009.09.011

@article{AIHPC_2010__27_1_315_0,
     author = {Berchio, Elvise and Gazzola, Filippo and Pierotti, Dario},
     title = {Gelfand type elliptic problems under {Steklov} boundary conditions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {315--335},
     year = {2010},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     doi = {10.1016/j.anihpc.2009.09.011},
     mrnumber = {2580512},
     zbl = {1184.35132},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.09.011/}
}

TY  - JOUR
AU  - Berchio, Elvise
AU  - Gazzola, Filippo
AU  - Pierotti, Dario
TI  - Gelfand type elliptic problems under Steklov boundary conditions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 315
EP  - 335
VL  - 27
IS  - 1
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2009.09.011/
DO  - 10.1016/j.anihpc.2009.09.011
LA  - en
ID  - AIHPC_2010__27_1_315_0
ER  -

%0 Journal Article
%A Berchio, Elvise
%A Gazzola, Filippo
%A Pierotti, Dario
%T Gelfand type elliptic problems under Steklov boundary conditions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 315-335
%V 27
%N 1
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2009.09.011/
%R 10.1016/j.anihpc.2009.09.011
%G en
%F AIHPC_2010__27_1_315_0

Berchio, Elvise; Gazzola, Filippo; Pierotti, Dario. Gelfand type elliptic problems under Steklov boundary conditions. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 315-335. doi: 10.1016/j.anihpc.2009.09.011

Bibliographie
Cité par

[1] Adimurthi, Hardy–Sobolev inequality in $H^{1} (Ω)$ and its applications, Commun. Contemp. Math. 4 (2002), 409-434 | MR | Zbl

[2] G. Barbatis, S. Filippas, A. Tertikas, Critical heat kernel estimates for Schrodinger operators via Hardy–Sobolev inequalities, J. Funct. Anal. 208 (2004), 1-30 | MR | Zbl

[3] 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 | Zbl

[4] H. Brezis, T. Kato, Remarks on the Schrodinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), 137-151 | MR | Zbl

[5] H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443-469 | MR | EuDML | Zbl

[6] X. Cabré, A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), 709-733 | MR | Zbl

[7] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297 | MR | Zbl

[8] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publ. Inc. (1985) | MR | JFM

[9] M.G. Crandall, P. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Ration. Mech. Anal. 58 (1975), 207-218 | MR | Zbl

[10] S. Eidelman, Y. Eidelman, On regularity of the extremal solution of the Dirichlet problem for some semilinear elliptic equations of the second order, Commun. Contemp. Math. 9 (2007), 31-39 | MR | Zbl

[11] H. Fujita, On the nonlinear equations $Δ u + e^{u} = 0$ and $\partial v / \partial t = Δ v + e^{u}$ , Bull. Amer. Math. Soc. 75 (1969), 132-135 | MR | Zbl

[12] J. García Azorero, I. Peral Alonso, J.P. Puel, Quasilinear problems with exponential growth in the reaction term, Nonlinear Anal. 22 (1994), 481-498 | MR | Zbl

[13] F. Gazzola, Critical growth quasilinear elliptic problems with shifting subcritical perturbation, Differential Integral Equations 14 (2001), 513-528 | MR | Zbl

[14] F. Gazzola, A. Malchiodi, Some remarks on the equation $- Δ u = λ {(1 + u)}^{p}$ for varying λ, p and varying domains, Comm. Partial Differential Equations 27 (2002), 809-845 | MR | Zbl

[15] I.M. Gelfand, Some problems in the theory of quasi-linear equations, Amer. Math. Soc. Transl. 29 (1963), 295-381, Uspekhi Mat. Nauk 14 (1959), 87-158 | MR

[16] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, University Press, Cambridge (1934) | MR | Zbl

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

[18] D. Joseph, E.M. Sparrow, Nonlinear diffusion induced by nonlinear sources, Quart. Appl. Math. 28 (1970), 327-342 | MR | Zbl

[19] Y. Kabeya, E. Yanagida, S. Yotsutani, Global structure of solutions for equations of Brezis–Nirenberg type on the unit ball, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 647-665 | 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] W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa 17 (1963), 43-77 | MR | EuDML | Zbl | Numdam

[22] Y. Martel, Uniqueness of weak extremal solutions for nonlinear elliptic problems, Houston J. Math. 23 (1997), 161-168 | MR | Zbl

[23] 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

[24] F. Mignot, J.P. Puel, Solution radiale singulière de $- Δ u = λ e^{u}$ , C. R. Acad. Sci. Paris Sér. I 307 (1988), 379-382 | MR | Zbl

[25] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I 330 (2000), 997-1002 | MR | Zbl

[26] X.J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), 283-310 | MR | Zbl

[27] Z.Q. Wang, M. Zhu, Hardy inequalities with boundary terms, Electron. J. Differential Equations 43 (2003), 1-8 | MR | EuDML

Cité par Sources :