Navegar

Colección

Datos Estadísticas

Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

This work deals with the system (-Δ) mu = a(x) v p, (-Δ) mv = b(x) u q with Dirichlet boundary condition in a domain Ω ⊂ ℝ n, where Ω is a ball if n ≥ 3 or a smooth perturbation of a ball when n = 2. We prove that, under appropriate conditions on the parameters (a, b, p, q, m, n), any nonnegative solution (u, v) of the system is bounded by a constant independent of (u, v). Moreover, we prove that the conditions are sharp in the sense that, up to some border case, the relation on the parameters are also necessary. The case m = 1 was considered by Souplet (Nonlinear Partial Differ Equ Appl 20:464-479, 2004). Our paper generalize to m ≥ 1 the results of that paper. © 2011 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.

Registro:

Documento: Artículo
Título:On the existence of bounded solutions for a nonlinear elliptic system
Autor:Durán, R.G.; Sanmartino, M.; Toschi, M.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 1900 La Plata, Buenos Aires, Argentina
Palabras clave:A priori estimates; Critical exponents; Elliptic systems; Weighted Sobolev spaces
Año:2012
Volumen:191
Número:4
Página de inicio:771
Página de fin:782
DOI: http://dx.doi.org/10.1007/s10231-011-0205-2
Título revista:Annali di Matematica Pura ed Applicata
Título revista abreviado:Ann. Mat. Pura Appl.
ISSN:03733114
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v191_n4_p771_Duran

Referencias:

  • Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A., up for u t-Δu = g(u) revisited (1996) Adv. Differ. Equ., 1 (1), pp. 73-90
  • Brézis, H., Turner, R.E.L., On a class of superlinear elliptic problems (1977) Comm. Partial Differ. Equ., 2 (6), pp. 601-614
  • Brezis, H., Cabré, X., Some simple nonlinear PDE's without solutions (1998) Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8, 1 (2), pp. 223-262
  • Cabré, X., Martel, Y., Weak eigenfunctions for the linearization of extremal elliptic problems (1998) J. Funct. Anal., 156 (1), pp. 30-56
  • Clément, P., Sweers, G., Uniform anti-maximum principle for polyharmonic boundary value problems (2001) Proc. Am. Math. Soc., 129 (2), pp. 467-474. , [Electronic]
  • Dall'Acqua, A., Sweers, G., Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems (2004) J. Differ. Equ., 205 (2), pp. 466-487
  • Dall'Acqua, A., Sweers, G., On domains for which the clamped plate system is positivity preserving (2004) Partial Differential Equations and Inverse Problems, 362, pp. 133-144. , of Contemp. Math, Amer. Math. Soc., Providence, RI
  • Domingos, A.R., Guo, Y., A note on a Liouville-type result for a system of fourth-order equations in ℝ N Electron (2002) J. Differ. Equ, pp. 1-20. , No. 99, [electronic]
  • Durán, R.G., Sanmartino, M., Toschi, M., Weighted a priori estimates for solutions of (-Δ) mu = f with homogeneous dirichlet conditions poisson equation (2010) Analysis in Theory and Applications, 26 (4)
  • García-Huidobro, M., Manásevich, R., Mitidieri, E., Yarur, C.S., Existence and nonexistence of positive singular solutions for a class of semilinear elliptic systems (1997) Arch. Ration. Mech. Anal., 140 (3), pp. 253-284
  • Grunau, H.-C., Sweers, G., Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions (1997) Math. Ann., 307 (4), pp. 589-626
  • Hulshof, J., van der Vorst, R., Differential systems with strongly indefinite variational structure (1993) J. Funct. Anal., 114 (1), pp. 32-58
  • Liu, J., Guo, Y., Zhang, Y., Liouville-type theorems for polyharmonic systems in ℝ N (2006) J. Differ. Equ., 225 (2), pp. 685-709
  • Mitidieri, E., A Rellich type identity and applications (1993) Comm. Partial Differ. Equ., 18 (1-2), pp. 125-151
  • Mitidieri, E., Nonexistence of positive solutions of semilinear elliptic systems in R N (1996) Differ. Integral Equ., 9 (3), pp. 465-479
  • Quittner, P., Souplet, P., A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces (2004) Arch. Ration. Mech. Anal., 174 (1), pp. 49-81
  • Souplet, P., A survey on L p δ spaces and their applications to nonlinear elliptic and parabolic problems. Nonlinear Partial Differ (2004) Equ. Appl. 20 GAKUTO Internat. Ser. Math. Sci. Appl, pp. 464-479. , Gakkōtosho, Tokyo
  • Souplet, P., Optimal regularity conditions for elliptic problems via L p δ-spaces (2005) Duke Math. J., 127 (1), pp. 175-192
  • Zhang, Y., A Liouville type theorem for polyharmonic elliptic systems (2007) J. Math. Anal. Appl., 326 (1), pp. 677-690

Citas:

---------- APA ----------
Durán, R.G., Sanmartino, M. & Toschi, M. (2012) . On the existence of bounded solutions for a nonlinear elliptic system. Annali di Matematica Pura ed Applicata, 191(4), 771-782.
http://dx.doi.org/10.1007/s10231-011-0205-2
---------- CHICAGO ----------
Durán, R.G., Sanmartino, M., Toschi, M. "On the existence of bounded solutions for a nonlinear elliptic system" . Annali di Matematica Pura ed Applicata 191, no. 4 (2012) : 771-782.
http://dx.doi.org/10.1007/s10231-011-0205-2
---------- MLA ----------
Durán, R.G., Sanmartino, M., Toschi, M. "On the existence of bounded solutions for a nonlinear elliptic system" . Annali di Matematica Pura ed Applicata, vol. 191, no. 4, 2012, pp. 771-782.
http://dx.doi.org/10.1007/s10231-011-0205-2
---------- VANCOUVER ----------
Durán, R.G., Sanmartino, M., Toschi, M. On the existence of bounded solutions for a nonlinear elliptic system. Ann. Mat. Pura Appl. 2012;191(4):771-782.
http://dx.doi.org/10.1007/s10231-011-0205-2