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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. Google Scholar

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces,, \emph{SIAM Rev.}, 18 (1976), 620. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory,, \emph{J. Differential Equations}, 7 (1970), 217. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation,, \emph{J. Math. Mech.}, 16 (1967), 1361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, \emph{Indiana Univ. Math. J.}, 23 (): 174. Google Scholar

[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. Google Scholar

[21]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions,, \emph{Nonlinear analysis and applications}, 109 (1987), 561. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces,, \emph{SIAM Rev.}, 18 (1976), 620. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

D. S. Cohen and T. W. Laetsch, Nonlinear boundary value problems suggested by chemical reactor theory,, \emph{J. Differential Equations}, 7 (1970), 217. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation,, \emph{J. Math. Mech.}, 16 (1967), 1361. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

P. H. Rabinowitz, Pairs of positive solutions of nonlinear elliptic partial differential equations,, \emph{Indiana Univ. Math. J.}, 23 (): 174. Google Scholar

[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. Google Scholar

[21]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions,, \emph{Nonlinear analysis and applications}, 109 (1987), 561. Google Scholar

[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. Google Scholar

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