DOI: http://dx.doi.org/10.18273/revint.v34n2-2016007
On the uniqueness of sign-changing solutions
to a semipositone problem in annuli
HUGO ADUÉNa, SIGIFREDO HERRÓNb *
a Universidad de Córdoba, Departamento de Matemáticas y Estadística, Montería, Colombia.
b Universidad Nacional de Colombia, Escuela de Matemáticas, Medellín, Colombia.
Abstract. In this paper we establish the uniqueness of radial solutions for a semipositone Dirichlet problem in an annulus, having a prescribed large number of nodal regions. Shooting method and Prüfer transformation are the main tools used in this work.
Keywords: Semipositone, nonhomogeneous problem, uniqueness of sign hanging solution, weighted Dirichlet problem, nonlinear elliptic problem.
MSC2010: 34B15, 35J25, 35J60, 35J61, 35J66.
Resumen. En este artículo establecemos la unicidad de soluciones radiales para un problema de Dirichlet, de tipo Semipositone, en un anillo, con un número prescrito (grande) de regiones nodales. Las principales herramientas usadas en este trabajo son el método del disparo y la transformación de Prüfer.
Palabras clave: Semipositone, problema no homogéneo, unicidad de soluciones que cambian de signo, problemas de Dirichlet con peso, problemas elípticos no lineales.
Texto Completo disponible en PDF
Referencias
[1] Aduén H., Castro A. and Cossio J., "Uniqueness of large radial solutions and existence of nonradial solutions for a superlinear Dirichlet problem in annulii", J. Math. Anal. Appl. 337 (2008), No. 1, 348-359. [ Links ]
[2] Aftalion A. and Pacella F., "Uniqueness and nondegenera y for some nonlinear elliptic problems in a ball", J. Differential Equations 195 (2003), No. 2, 380-397. [ Links ]
]]>[3] Bae S. and Ni W-M., "Existence and infi nite multiplicity for an inhomogeneous semilinear elliptic equation on Rn", Math. Ann. 320 (2001), No. 1, 191-210. [ Links ]
[4] Berestycki H., Caffarelli L.A. and Nirenberg L., "Inequalities for second-order elliptic equations with applications to unbounded domains, I, A celebration of John F. Nash, Jr", Duke Math. J. 81 (1996), No. 2, 467-494. [ Links ]
[5] Bernard G., "An inhomogeneous semilinear equation in entire space", J. Differential Equations 125 (1996), No. 1, 184-214. [ Links ]
[6] Brezis H. and Oswald L.,"Remarks on sublinear elliptic equations", Nonlinear Anal. 10 (1986), No. 1, 55-64. [ Links ]
[7] Castro A., Chhetri M. and Shivaji R., "Nonlinear eigenvalue problems with semipositone structure", Electron. J. Differ. Equ. Conf., 5 (2000), 33-49. [ Links ]
]]>[8] Castro A. and Kurepa A., "In nitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball", Proc. Amer. Math. Soc. 101 (1987), No. 1, 57-64. [ Links ]
[9] Coffman C.V., "Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for Δu - u + u3 = 0", J. Differential Equations 128 (1996), No. 2, 379-386. [ Links ]
[10] Coffman C.V. and Marcus M., "Existence and uniqueness results for semi-linear Dirichlet problems in annuli", Arch. Rational Mech. Anal. 108 (1989), No. 4, 293-307. [ Links ]
[11] Cortázar C., Dolbeault J., García-Huidobro M. and Manásevich R., "Existence of sign changing solutions for an equation with a weighted p-Laplace operator", Nonlinear Anal. 110 (2014), 1-22. [ Links ]
[12] Cortázar C., Elgueta M. and Felmer P., "Uniqueness of positive solutions of Δu + ƒ(u) = 0 in RN,N ≥ 3", Arch. Rational Mech. Anal. 142 (1998), No. 2, 127-141. [ Links ]
]]>[13] Cossio J., Herrón S. and Vélez C., "Infi nitely many radial solutions for a p-Laplacian problem p-superlinear at the origen", J. Math. Anal. Appl. 376 (2011), No. 2, 741-749. [ Links ]
[14] Dambrosio W., "Nodal solutions to semilinear elliptic equations in a ball", Differential Integral Equations 15 (2002), No. 8, 945-972. [ Links ]
[15] Dambrosio W., "On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains", J. Differential Equations 196 (2004), No. 1, 91-118. [ Links ]
[16] Esteban M.J., "Multiple solutions of semilinear elliptic problems in a ball", J. Differential Equations 57 (1985), No. 1, 112-137. [ Links ]
[17] Felmer P., Martínez S. and Tanaka K., "Uniqueness of radially symmetric positive solutions for - Δu + u = up in an annulus", J. Differential Equations 245 (2008), No.5, 1198-1209. [ Links ]
]]>[18] Ferreira L.C.F and Montenegro M., "Existene and asymptotic behavior for elliptic equations with singular anisotropic potentials", J. Differential Equations 250 (2011), No. 4, 2045-2063. [ Links ]
[19] García-Huidobro M., Manásevich R. and Zanolin F., "Infi nitely many solutions for a Dirichlet problem with a nonhomogeneous p-Laplacian like operator in a ball", Adv. Differential Equations 2 (1997), No. 2, 203-230. [ Links ]
[20] Hartman P., Ordinary differential equations, Second ed. SIAM, Philadelphia, PA, 2002. [ Links ]
[21] Hastings S.P. and McLeod J.B., Classical methods in ordinary differential equations: with applications to boundary value problems, AMS, Providence, RI, 2012. [ Links ]
[22] Kajikiya R., "Necessary and suffient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations", Adv. Differential Equations 6 (2001), No. 11, 1317-1346. [ Links ]
]]>[23] Kajikiya R., "Sobolev norm of radially symmetric oscillatory solutions for super-linear elliptic equations", Hiroshima Math. J. 20 (1990), No. 2, 259-276. [ Links ]
[24] Kwong M.K., "Uniqueness of positive solutions of Δu - u + up = 0 in Rn", Arch. Rational Mech. Anal. 105 (1989), No. 3, 243-266. [ Links ]
[25] Kwong M.K. and Zhang L.Q., "Uniqueness of the positive solution of Δu + ƒ(u) = 0 in an annulus", Differential Integral Equations 4 (1991), No. 3, 583-599. [ Links ]
[26] Lions P.-L., "On the existence of positive solutions of semilinear elliptic equations", SIAM Rev. 24 (1982), No. 4, 441-467. [ Links ]
[27] McLeod K., "Uniqueness of positive radial solutions of Δu + ƒ(u) = 0 in Rn, II", Trans. Amer. Math. Soc. 339 (1993), No. 2, 495-505. [ Links ]
]]>[28] McLeod K. and Serrin J., "Uniqueness of positive radial solutions of Δu + ƒ(u) = 0 in Rn", Arch. Rational Mech. Anal. 99 (1987), No. 2, 115-145. [ Links ]
[29] Naito Y., "Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation", Hiroshima Math. J. 24 (1994), No. 1, 177-220. [ Links ]
[30] Ni W-M., "Uniqueness of solutions of nonlinear Dirichlet problems", J. Differential Equations 50 (1983), No. 2, 289-304. [ Links ]
[31] Ni W-M. and Nussbaum R.D., "Uniqueness and nonuniqueness for positive radial solutions of Δu + ƒ(u, r) = 0", Comm. Pure Appl. Math. 38 (1985), No. 1, 67-108. [ Links ]
[32] Sankar L., Sasi S. and Shivaji R., "Semipositone problems with falling zeros on exterior domains", J. Math. Anal. Appl. 401 (2013), No. 1, 146-153. [ Links ]
]]>[33] Tanaka S., "Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball", Nonlinear Anal. 71 (2009), No. 11, 5256-5267. [ Links ]
[34] Tanaka S., "Uniqueness of nodal radial solutions of superlinear elliptic equations in a ball", Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), No. 6, 1331-1343. [ Links ]
[35] Tanaka S., "Uniqueness of sign-changing radial solutions for Δu - u + |u|p-1u = 0 in some ball and annulus", J. Math. Anal. Appl. 439 (2016), No. 1, 154-170. [ Links ]
[36] Tang M., "Uniqueness of positive radial solutions for Δu - u + up = 0 on an annulus", J. Differential Equations 189 (2003), No. 1, 148-160. [ Links ]
[37] Walter W., Ordinary differential equations, Springer-Verlag, New York, 1998. [ Links ]
]]>[38] Yadava S.L., "Uniqueness of positive radial solutions of a semilinear Dirichlet problem in an annulus", Proc. Roy. Soc. Edinburgh A 130 (2000), No. 6, 1417-1428. [ Links ]
[39] Yanagida E., "Uniqueness of positive radial solutions of Δu + g(r)u + h(r)up = 0 in Rn", Arch. Rational Mech. Anal. 115 (1991), No. 3, 257-274. [ Links ]
[40] Yanagida E., "Uniqueness of positive radial solutions of Δu + ƒ(u, |x|) = 0", Nonlinear Anal. 19 (1992), No. 12, 1143-1154. [ Links ]
*E-mail: sherron@unal.edu.co.
Recceived: 22 August 2016, Acepted: 31 October 2016.
To cite this article: H. Aduén, S. Herrón, On the uniqueness of sign-changing solutions to a semipositone problem in annuli, Rev. Integr. Temas Mat. 34 (2016), No. 2, 207-224.