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November  2014, 13(6): 2713-2731. doi: 10.3934/cpaa.2014.13.2713

## Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions

 1 Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762 2 Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, United States 3 Department of Mathematics Education, Pusan National University, Busan, 609-735 4 Department of Mathematics Education, Pusan National University, Busan, South Korea

Received  November 2013 Revised  March 2014 Published  July 2014

We study positive radial solutions to the boundary value problem \begin{eqnarray} -\Delta u = \lambda K(|x|)f(u), \quad x \in \Omega, \\ \frac{\partial u}{\partial \eta}+\tilde{c}(u)u = 0, \quad |x|=r_0, \\ u(x) \rightarrow 0, \quad |x|\rightarrow \infty, \end{eqnarray} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x \in \mathbb{R}^N| N>2, |x|> r_0 \mbox{ with }r_0>0\}$, $K:[r_0, \infty)\rightarrow(0,\infty)$ is a continuous function such that $\lim_{r \rightarrow \infty}K(r)=0$, $\frac{\partial}{\partial \eta}$ is the outward normal derivative, and $\tilde{c}:[0,\infty) \rightarrow (0,\infty)$ is a continuous function. We consider various $C^1$ classes of the reaction term $f:[0,\infty) \rightarrow \mathbb{R}$ that are sublinear at $\infty$ $(i.e. \lim_{s \rightarrow \infty}\frac{f(s)}{s}=0)$. In particular, we discuss existence and multiplicity results for classes of $f$ with $(a)$ $f(0)>0$, $(b)$ $f(0)<0$, and $(c)$ $f(0)=0$. We establish our existence and multiplicity results via the method of sub-super solutions. We also discuss some uniqueness results.
Citation: Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713
##### References:
 [1] I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball,, \emph{Proc. Amer. Math. Soc.}, 117 (1993), 775. doi: 10.2307/2159143. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces,, \emph{SIAM Rev.}, 18 (1976), 620. [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory,, \emph{Differential Integral Equations}, 7 (1994), 655. [4] K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves,, \emph{Nonlinear Anal. TMA}, 5 (1981), 475. doi: 10.1016/0362-546X(81)90096-1. [5] A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity,, \emph{Comm. Partial Differential Equations}, 20 (1995), 1927. doi: 10.1080/03605309508821157. [6] A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains,, \emph{J. Math. Anal. Appl.}, 394 (2012), 432. doi: 10.1016/j.jmaa.2012.04.005. [7] A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems,, \emph{Proc. Amer. Math. Soc.}, 106 (1989), 735. doi: 10.2307/2047429. [8] A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 98 (1984), 267. doi: 10.1017/S0308210500013445. [9] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 4 (1987), 97. [10] D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory,, \emph{J. Differential Equations}, 7 (1970), 217. [11] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large,, \emph{Proc. London Math. Soc.}, 53 (1986), 439. doi: 10.1112/plms/s3-53.3.429. [12] E. N. Dancer and J. Shi, Uniqueness and nonexistence of positive solutions to semipositone problems,, \emph{Bull. London Math. Soc.}, 38 (2006), 1033. doi: 10.1112/S0024609306018984. [13] P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion,, \emph{Nonlinear Anal. Real World Appl.}, 15 (2014), 51. doi: 10.1016/j.nonrwa.2013.05.005. [14] F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions,, \emph{Indiana Univ. Math. J.}, 31 (1982), 213. doi: 10.1512/iumj.1982.31.31019. [15] H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation,, \emph{J. Math. Mech.}, 16 (1967), 1361. [16] E. Ko, E. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain,, \emph{Discrete Contin. Dyn. Syst}, 33 (2013), 5153. doi: 10.3934/dcds.2013.33.5153. [17] C. Maya and R. Shivaji, Multiple positive solutions for a class of semilinear elliptic boundary value problems,, \emph{Nonlinear Anal. TMA}, 38 (1999), 497. doi: 10.1016/S0362-546X(98)00211-9. [18] S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. I. Steady states,, \emph{Trans. Amer. Math. Soc.}, 354 (2002), 3601. doi: 10.1090/S0002-9947-02-03005-2. [19] P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, \emph{Indiana Univ. Math. J.}, 23 (): 174. [20] L. Sankar, S. Sasi and R. Shivaji, Semipositone problems with falling zeros on exterior domains,, \emph{J. Math. Anal. Appl.}, 401 (2013), 146. doi: 10.1016/j.jmaa.2012.11.031. [21] R. Shivaji, A remark on the existence of three solutions via sub-super solutions,, \emph{Nonlinear analysis and applications}, 109 (1987), 561. [22] R. Shivaji, Uniqueness results for a class of positone problems,, \emph{Nonlinear Anal. TMA}, 7 (1983), 223. doi: 10.1016/0362-546X(83)90084-6.

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##### References:
 [1] I. Ali, A. Castro and R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball,, \emph{Proc. Amer. Math. Soc.}, 117 (1993), 775. doi: 10.2307/2159143. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces,, \emph{SIAM Rev.}, 18 (1976), 620. [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory,, \emph{Differential Integral Equations}, 7 (1994), 655. [4] K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves,, \emph{Nonlinear Anal. TMA}, 5 (1981), 475. doi: 10.1016/0362-546X(81)90096-1. [5] A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity,, \emph{Comm. Partial Differential Equations}, 20 (1995), 1927. doi: 10.1080/03605309508821157. [6] A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains,, \emph{J. Math. Anal. Appl.}, 394 (2012), 432. doi: 10.1016/j.jmaa.2012.04.005. [7] A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems,, \emph{Proc. Amer. Math. Soc.}, 106 (1989), 735. doi: 10.2307/2047429. [8] A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 98 (1984), 267. doi: 10.1017/S0308210500013445. [9] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, 4 (1987), 97. [10] D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory,, \emph{J. Differential Equations}, 7 (1970), 217. [11] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large,, \emph{Proc. London Math. Soc.}, 53 (1986), 439. doi: 10.1112/plms/s3-53.3.429. [12] E. N. Dancer and J. Shi, Uniqueness and nonexistence of positive solutions to semipositone problems,, \emph{Bull. London Math. Soc.}, 38 (2006), 1033. doi: 10.1112/S0024609306018984. [13] P. V. Gordon, E. Ko and R. Shivaji, Multiplicity and uniqueness of positive solutions for elliptic equations with nonlinear boundary conditions arising in a theory of thermal explosion,, \emph{Nonlinear Anal. Real World Appl.}, 15 (2014), 51. doi: 10.1016/j.nonrwa.2013.05.005. [14] F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions,, \emph{Indiana Univ. Math. J.}, 31 (1982), 213. doi: 10.1512/iumj.1982.31.31019. [15] H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation,, \emph{J. Math. Mech.}, 16 (1967), 1361. [16] E. Ko, E. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain,, \emph{Discrete Contin. Dyn. Syst}, 33 (2013), 5153. doi: 10.3934/dcds.2013.33.5153. [17] C. Maya and R. Shivaji, Multiple positive solutions for a class of semilinear elliptic boundary value problems,, \emph{Nonlinear Anal. TMA}, 38 (1999), 497. doi: 10.1016/S0362-546X(98)00211-9. [18] S. Oruganti, J. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting. I. Steady states,, \emph{Trans. Amer. Math. Soc.}, 354 (2002), 3601. doi: 10.1090/S0002-9947-02-03005-2. [19] P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, \emph{Indiana Univ. Math. J.}, 23 (): 174. [20] L. Sankar, S. Sasi and R. Shivaji, Semipositone problems with falling zeros on exterior domains,, \emph{J. Math. Anal. Appl.}, 401 (2013), 146. doi: 10.1016/j.jmaa.2012.11.031. [21] R. Shivaji, A remark on the existence of three solutions via sub-super solutions,, \emph{Nonlinear analysis and applications}, 109 (1987), 561. [22] R. Shivaji, Uniqueness results for a class of positone problems,, \emph{Nonlinear Anal. TMA}, 7 (1983), 223. doi: 10.1016/0362-546X(83)90084-6.
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