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Abstract:

In this paper, we study the existence of viscosity solutions to the Gelfand problem for the 1-homogeneous p-Laplacian in a bounded domain ω ⊃ ℝ N , that is, we deal with (equation presented) in ω with u = 0 on δ ω. For this problem we show that, for p ϵ [2, ∞], there exists a positive critical value λ ∗ = λ ∗ (ω, N, p) such that the following holds: • If λ λ ∗ , the problem admits a minimal positive solution wλ ∗ • If λ > λ ∗ , the problem admits no solution. Moreover, the branch of minimal solutions {wλ} is increasing with λ ∗ In addition, using degree theory, for fixed p we show that there exists an unbounded continuum of solutions that emanates from the trivial solution u = 0 with λ = 0, and for a small fixed λ we also obtain a continuum of solutions with p ϵ [2, ∞]. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.

Registro:

Documento: Artículo
Título:The Gelfand problem for the 1-homogeneous p-Laplacian
Autor:Tapia, J.C.; Salas, A.M.; Rossi, J.D.
Filiación:Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano, Almería, 04120, Spain
Departamento de Análisis Matemático, Campus Fuentenueva S/N, Universidad de Granada, Granada, 18071, Spain
Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina
Palabras clave:elliptic equations; Gelfand problem; viscosity solutions
Año:2019
Volumen:8
Número:1
Página de inicio:545
Página de fin:558
DOI: http://dx.doi.org/10.1515/anona-2016-0233
Título revista:Advances in Nonlinear Analysis
Título revista abreviado:Adv. Nonlinear Anal.
ISSN:21919496
Registro:http://digital.bl.fcen.uba.ar/collection/paper/document/paper_21919496_v8_n1_p545_Tapia

Referencias:

  • Ambrosetti, A., Arcoya, D., An introduction to nonlinear functional analysis and elliptic problems (2011) Progr. Nonlinear Differential Equations Appl., 82. , Birkhäuser, Boston
  • Arcoya, D., Carmona, J., Martínez-Aparicio, P.J., Gelfand type quasilinear elliptic problems with quadratic gradient terms (2014) Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2), pp. 249-265
  • Barles, G., Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term (2001) Comm. Partial Differential Equations, 26 (11-12), pp. 2323-2337
  • Cabré, X., Capella, A., Regularity of radial minimizers and extremal solutions of semilinear elliptic equations (2006) J. Funct. Anal., 238 (2), pp. 709-733
  • Cabré, X., Sanchón, M., Geometric-type Sobolev inequalities and applications to the regularity of minimizers (2013) J. Funct. Anal., 264 (1), pp. 303-325
  • Caffarelli, L.A., Cabré, X., Fully nonlinear elliptic equations (1995) Amer. Math. Soc. Colloq. Publ., 43. , American Mathematical Society, Providenc
  • Charro, F., De Philippis, G., Di Castro, A., Máximo, D., On the aleksandrov-bakelman-pucci estimate for the infinity laplacian (2013) Calc. Var. Partial Differential Equations, 48 (3-4), pp. 667-693
  • Chen, Y.G., Giga, Y., Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations (1991) J. Differential Geom., 33 (3), pp. 749-786
  • Crandall, M.G., Ishii, H., Lions, P.-L., User's guide to viscosity solutions of second order partial differential equations (1992) Bull. Amer. Math. Soc. (N. S.), 27 (1), pp. 1-67
  • Gel'Fand, I.M., Some problems in the theory of quasilinear equations (1963) Amer. Math. Soc. Transl. (2), 29, pp. 295-381
  • Gilbarg, D., Trudinger, N.S., (1983) Elliptic Partial Differential Equations of Second Order, , 2nd ed., Grundlehren Math. Wiss. 224, Springer, Berlin
  • Imbert, C., Lin, T., Silvestre, L., (2016) Hölder Gradient Estimates for A Class of Singular or Degenerate Parabolic Equations, , http://arxiv.org/abs/1609.01123, preprint
  • Jacobsen, J., Schmitt, K., The Liouville-Bratu-Gelfand problem for radial operators (2002) J. Differential Equations, 184 (1), pp. 283-298
  • Joseph, D.D., Lundgren, T.S., Quasilinear Dirichlet problems driven by positive sources (1972) Arch. Rational Mech. Anal., 49, pp. 241-269
  • Juutinen, P., Lindqvist, P., Manfredi, J.J., On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation (2001) SIAM J. Math. Anal., 33 (3), pp. 699-717
  • Leray, J., Schauder, J., Topologie et équations fonctionnelles (1934) Ann. Sci. École Norm. Sup., 51 (3), pp. 45-78
  • Lu, G., Wang, P., A PDE perspective of the normalized infinity Laplacian (2008) Comm. Partial Differential Equations, 33 (10-12), pp. 1788-1817
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., On the definition and properties of p-harmonious functions (2012) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2), pp. 215-241
  • Martínez-Aparicio, P.J., Pérez-Llanos, M., Rossi, J.D., The limit as pfor the eigenvalue problem of the 1-homogeneous p-Laplacian (2014) Rev. Mat. Complut., 27 (1), pp. 241-258
  • Martínez-Aparicio, P.J., Pérez-Llanos, M., Rossi, J.D., The sublinear problem for the 1-homogeneous p-Laplacian (2014) Proc. Amer. Math. Soc., 142 (8), pp. 2641-2648
  • Molino, A., Gelfand type problem for singular quadratic quasilinear equations (2016) NoDEA Nonlinear Differential Equations Appl., 23 (5), p. 56
  • Ohnuma, M., Sato, K., Singular degenerate parabolic equations with applications to the p-Laplace diffusion equation (1997) Comm. Partial Differential Equations, 22 (3-4), pp. 381-411
  • Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B., Tug-of-war and the infinity Laplacian (2009) J. Amer. Math. Soc., 22 (1), pp. 167-210
  • Peres, Y., Sheffield, S., Tug-of-war with noise: A game-theoretic view of the p-Laplacian (2008) Duke Math. J., 145 (1), pp. 91-120
  • Ros-Oton, X., Regularity for the fractional Gelfand problem up to dimension 7 (2014) J. Math. Anal. Appl., 419 (1), pp. 10-19
  • Schmitt, K., Analysis methods for the study of nonlinear equations (1995) Leture Notes, , University of Utah

Citas:

---------- APA ----------
Tapia, J.C., Salas, A.M. & Rossi, J.D. (2019) . The Gelfand problem for the 1-homogeneous p-Laplacian. Advances in Nonlinear Analysis, 8(1), 545-558.
http://dx.doi.org/10.1515/anona-2016-0233
---------- CHICAGO ----------
Tapia, J.C., Salas, A.M., Rossi, J.D. "The Gelfand problem for the 1-homogeneous p-Laplacian" . Advances in Nonlinear Analysis 8, no. 1 (2019) : 545-558.
http://dx.doi.org/10.1515/anona-2016-0233
---------- MLA ----------
Tapia, J.C., Salas, A.M., Rossi, J.D. "The Gelfand problem for the 1-homogeneous p-Laplacian" . Advances in Nonlinear Analysis, vol. 8, no. 1, 2019, pp. 545-558.
http://dx.doi.org/10.1515/anona-2016-0233
---------- VANCOUVER ----------
Tapia, J.C., Salas, A.M., Rossi, J.D. The Gelfand problem for the 1-homogeneous p-Laplacian. Adv. Nonlinear Anal. 2019;8(1):545-558.
http://dx.doi.org/10.1515/anona-2016-0233