Published 2016-12-12
Keywords
- Semipositone,
- nonhomogeneous problem,
- uniqueness of sign changing solution,
- weighted Dirichlet problem,
- nonlinear elliptic problem
How to Cite
Copyright (c) 2016 Hugo Aduén, Sigifredo Herrón
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
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.
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.
Downloads
References
- 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.
- Aftalion A. and Pacella F., "Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball", J. Differential Equations 195 (2003), No. 2, 380-397.
- Bae S. and Ni W-M., "Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on R^n", Math. Ann. 320 (2001), No. 1, 191-210.
- 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.
- Bernard G., "An inhomogeneous semilinear equation in entire space", J. Differential Equations 125 (1996), No. 1, 184-214.
- Brezis H. and Oswald L.,"Remarks on sublinear elliptic equations", Nonlinear Anal. 10 (1986), No. 1, 55-64.
- Castro A., Chhetri M. and Shivaji R., "Nonlinear eigenvalue problems with semipositone structure", Electron. J. Differ. Equ. Conf., 5 (2000), 33-49.
- Castro A. and Kurepa A., "Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball", Proc. Amer. Math. Soc. 101 (1987), No. 1, 57-64.
- Coffman C.V., "Uniqueness of the positive radial solution on an annulus of the Dirichlet problem for Δu - u+ u^3=0", J. Differential Equations 128 (1996), No. 2, 379-386.
- 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.
- 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.
- Cortázar C., Elgueta M. and Felmer P., "Uniqueness of positive solutions of Δu+ f(u) =0 in R^N,N ≥ 3", Arch. Rational Mech. Anal. 142 (1998), No. 2, 127-141.
- Cossio J., Herrón S. and Vélez C., "Infinitely many radial solutions for a p-Laplacian problem p-superlinear at the origen", J. Math. Anal. Appl. 376 (2011), No. 2, 741-749.
- Dambrosio W., "Nodal solutions to semilinear elliptic equations in a ball", Differential Integral Equations 15 (2002), No. 8, 945-972.
- Dambrosio W., "On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains", J. Differential Equations 196 (2004), No. 1, 91-118.
- Esteban M.J., "Multiple solutions of semilinear elliptic problems in a ball", J. Differential Equations 57 (1985), No. 1, 112-137.
- Felmer P., Martínez S. and Tanaka K., "Uniqueness of radially symmetric positive solutions for -Δu+ u=u^p in an annulus", J. Differential Equations 245 (2008), No.5, 1198-1209.
- Ferreira L.C.F and Montenegro M., "Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials", J. Differential Equations 250 (2011), No. 4, 2045-2063.
- García-Huidobro M., Manásevich R. and Zanolin F., "Infinitely 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.
- Hartman P., Ordinary differential equations, Second ed. SIAM, Philadelphia, PA, 2002.
- Hastings S.P. and McLeod J.B., Classical methods in ordinary differential equations: with applications to boundary value problems, AMS, Providence, RI, 2012.
- Kajikiya R., "Necessary and sufficient condition for existence and uniqueness of nodal solutions to sublinear elliptic equations", Adv. Differential Equations 6 (2001), No. 11, 1317-1346.
- Kajikiya R., "Sobolev norm of radially symmetric oscillatory solutions for super-linear elliptic equations", Hiroshima Math. J. 20 (1990), No. 2, 259-276.
- Kwong M.K., "Uniqueness of positive solutions of Δu - u+ u^p=0 in R^n", Arch. Rational Mech. Anal. 105 (1989), No. 3, 243-266.
- Kwong M.K. and Zhang L.Q., "Uniqueness of the positive solution of Δu+ f(u)= 0 in an annulus", Differential Integral Equations 4 (1991), No. 3, 583-599.
- Lions P.-L., "On the existence of positive solutions of semilinear elliptic equations", SIAM Rev. 24 (1982), No. 4, 441-467.
- McLeod K., "Uniqueness of positive radial solutions of Δu+ f(u)= 0 in R^n, II", Trans. Amer. Math. Soc. 339 (1993), No. 2, 495-505.
- McLeod K. and Serrin J., "Uniqueness of positive radial solutions of Δu+ f(u)= 0 in R^n", Arch. Rational Mech. Anal. 99 (1987), No. 2, 115-145.
- Naito Y., "Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation", Hiroshima Math. J. 24 (1994), No. 1, 177-220.
- Ni W-M., "Uniqueness of solutions of nonlinear Dirichlet problems", J. Differential Equations 50 (1983), No. 2, 289-304.
- Ni W-M. and Nussbaum R.D., "Uniqueness and nonuniqueness for positive radial solutions of Δu+ f(u,r)= 0", Comm. Pure Appl. Math. 38 (1985), No. 1, 67-108.
- 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.
- Tanaka S., "Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball", Nonlinear Anal. 71 (2009), No. 11, 5256-5267.
- 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.
- Tanaka S., "Uniqueness of sign-changing radial solutions for Δu-u+ |u|^(p-1) u = 0 in some ball and annulus", J. Math. Anal. Appl. 439 (2016), No. 1, 154--170.
- Tang M., "Uniqueness of positive radial solutions for Δu- u+ u^p=0 on an annulus", J. Differential Equations 189 (2003), No. 1, 148--160.
- Walter W., Ordinary differential equations, Springer-Verlag, New York, 1998.
- 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.
- Yanagida E., "Uniqueness of positive radial solutions of Δu+g(r)u+h(r)u^p=0 in R^n", Arch. Rational Mech. Anal. 115 (1991), No. 3, 257-274.
- Yanagida E., "Uniqueness of positive radial solutions of Δu+ f(u,|x|)= 0", Nonlinear Anal. 19 (1992), No. 12, 1143-1154.