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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 and 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, Proc. Amer. Math. Soc., 117 (1993), 775-782. doi: 10.2307/2159143. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620-709. [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations, 7 (1994), 655-663. [4] K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal. TMA, 5 (1981), 475-486. 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, Comm. Partial Differential Equations, 20 (1995), 1927-1936. doi: 10.1080/03605309508821157. [6] A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437. doi: 10.1016/j.jmaa.2012.04.005. [7] A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc., 106 (1989), 735-740. doi: 10.2307/2047429. [8] A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 267-269. doi: 10.1017/S0308210500013445. [9] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1987), 97-121. [10] D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations, 7 (1970), 217-226. [11] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 439-452. doi: 10.1112/plms/s3-53.3.429. [12] E. N. Dancer and J. Shi, Uniqueness and nonexistence of positive solutions to semipositone problems, Bull. London Math. Soc., 38 (2006), 1033-1044. 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, Nonlinear Anal. Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005. [14] F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221. doi: 10.1512/iumj.1982.31.31019. [15] H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16 (1967), 1361-1376. [16] E. Ko, E. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Contin. Dyn. Syst, Ser. A, 33 (2013), 5153-5166. 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, Nonlinear Anal. TMA, 38 (1999), 497-504. 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, Trans. Amer. Math. Soc., 354 (2002), 3601-3619. doi: 10.1090/S0002-9947-02-03005-2. [19] P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J., 23 (1973/74), 174-186. [20] L. Sankar, S. Sasi and R. Shivaji, Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl., 401 (2013), 146-153. doi: 10.1016/j.jmaa.2012.11.031. [21] R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear analysis and applications, Lecture Notes in Pure and Appl. Math., 109 (1987), 561-566. [22] R. Shivaji, Uniqueness results for a class of positone problems, Nonlinear Anal. TMA, 7 (1983), 223-230. 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, Proc. Amer. Math. Soc., 117 (1993), 775-782. doi: 10.2307/2159143. [2] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces, SIAM Rev., 18 (1976), 620-709. [3] A. Ambrosetti, D. Arcoya and B. Buffoni, Positive solutions for some semi-positone problems via bifurcation theory, Differential Integral Equations, 7 (1994), 655-663. [4] K. J. Brown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal. TMA, 5 (1981), 475-486. 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, Comm. Partial Differential Equations, 20 (1995), 1927-1936. doi: 10.1080/03605309508821157. [6] A. Castro, L. Sankar and R. Shivaji, Uniqueness of nonnegative solutions for semipositone problems on exterior domains, J. Math. Anal. Appl., 394 (2012), 432-437. doi: 10.1016/j.jmaa.2012.04.005. [7] A. Castro and R. Shivaji, Nonnegative solutions for a class of radially symmetric nonpositone problems, Proc. Amer. Math. Soc., 106 (1989), 735-740. doi: 10.2307/2047429. [8] A. Castro and R. Shivaji, Uniqueness of positive solutions for a class of elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A, 98 (1984), 267-269. doi: 10.1017/S0308210500013445. [9] P. Clément and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1987), 97-121. [10] D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory, J. Differential Equations, 7 (1970), 217-226. [11] E. N. Dancer, On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 439-452. doi: 10.1112/plms/s3-53.3.429. [12] E. N. Dancer and J. Shi, Uniqueness and nonexistence of positive solutions to semipositone problems, Bull. London Math. Soc., 38 (2006), 1033-1044. 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, Nonlinear Anal. Real World Appl., 15 (2014), 51-57. doi: 10.1016/j.nonrwa.2013.05.005. [14] F. Inkmann, Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 31 (1982), 213-221. doi: 10.1512/iumj.1982.31.31019. [15] H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech., 16 (1967), 1361-1376. [16] E. Ko, E. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Contin. Dyn. Syst, Ser. A, 33 (2013), 5153-5166. 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, Nonlinear Anal. TMA, 38 (1999), 497-504. 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, Trans. Amer. Math. Soc., 354 (2002), 3601-3619. doi: 10.1090/S0002-9947-02-03005-2. [19] P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations, Indiana Univ. Math. J., 23 (1973/74), 174-186. [20] L. Sankar, S. Sasi and R. Shivaji, Semipositone problems with falling zeros on exterior domains, J. Math. Anal. Appl., 401 (2013), 146-153. doi: 10.1016/j.jmaa.2012.11.031. [21] R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear analysis and applications, Lecture Notes in Pure and Appl. Math., 109 (1987), 561-566. [22] R. Shivaji, Uniqueness results for a class of positone problems, Nonlinear Anal. TMA, 7 (1983), 223-230. doi: 10.1016/0362-546X(83)90084-6.
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