Gelfand type elliptic problems under Steklov boundary conditions

Berchio, Elvise ; Gazzola, Filippo ; Pierotti, Dario

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 315-335 / Harvested from Numdam

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.

Détail

Publié le : 2010-01-01

MR 2580512 | Zbl 1184.35132

DOI : https://doi.org/10.1016/j.anihpc.2009.09.011

Bibliographie

[1] Adimurthi, Hardy–Sobolev inequality in $ {H}^{1}(\Omega )$ and its applications, Commun. Contemp. Math. 4 (2002), 409-434 | MR 1918752 | Zbl 1005.35072

[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 2034290 | Zbl 1058.35049

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

[4] H. Brezis, T. Kato, Remarks on the Schrodinger operator with singular complex potentials, J. Math. Pures Appl. 58 (1979), 137-151 | MR 539217 | Zbl 0408.35025

[5] H. Brezis, J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443-469 | MR 1605678 | Zbl 0894.35038

[6] X. Cabré, A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, J. Funct. Anal. 238 (2006), 709-733 | MR 2253739 | Zbl 1130.35050

[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 982351 | Zbl 0702.35085

[8] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover Publ. Inc. (1985) | JFM 65.1543.02 | MR 92663

[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 382848 | Zbl 0309.35057

[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 2293559 | Zbl 1134.35053

[11] H. Fujita, On the nonlinear equations $ \mathrm{\Delta }u+{e}^{u}=0$ and $ \partial v/\partial t=\mathrm{\Delta }v+{e}^{u}$, Bull. Amer. Math. Soc. 75 (1969), 132-135 | MR 239258 | Zbl 0216.12101

[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 1266373 | Zbl 0804.35037

[13] F. Gazzola, Critical growth quasilinear elliptic problems with shifting subcritical perturbation, Differential Integral Equations 14 (2001), 513-528 | MR 1824741 | Zbl 1030.35076

[14] F. Gazzola, A. Malchiodi, Some remarks on the equation $ -\mathrm{\Delta }u=\lambda {(1+u)}^{p}$ for varying λ, p and varying domains, Comm. Partial Differential Equations 27 (2002), 809-845 | MR 1900564 | Zbl 1010.35042

[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 110868

[16] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, University Press, Cambridge (1934) | MR 46395 | Zbl 0010.10703

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

[18] D. Joseph, E.M. Sparrow, Nonlinear diffusion induced by nonlinear sources, Quart. Appl. Math. 28 (1970), 327-342 | MR 272272 | Zbl 0208.13006

[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 1838505 | Zbl 1114.35320

[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] W. Littman, G. Stampacchia, H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa 17 (1963), 43-77 | Numdam | MR 161019 | Zbl 0116.30302

[22] Y. Martel, Uniqueness of weak extremal solutions for nonlinear elliptic problems, Houston J. Math. 23 (1997), 161-168 | MR 1688823 | Zbl 0884.35037

[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 583604 | Zbl 0456.35034

[24] F. Mignot, J.P. Puel, Solution radiale singulière de $ -\mathrm{\Delta }u=\lambda {e}^{u}$, C. R. Acad. Sci. Paris Sér. I 307 (1988), 379-382 | MR 965802 | Zbl 0683.35032

[25] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I 330 (2000), 997-1002 | MR 1779693 | Zbl 0955.35029

[26] X.J. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations 93 (1991), 283-310 | MR 1125221 | Zbl 0766.35017

[27] Z.Q. Wang, M. Zhu, Hardy inequalities with boundary terms, Electron. J. Differential Equations 43 (2003), 1-8 | MR 1971109