Positive solutions for a class of superlinear semipositone systems on exterior domains
© Abebe et al.; licensee Springer 2014
Received: 8 May 2014
Accepted: 5 August 2014
Published: 25 September 2014
We study the existence of a positive radial solution to the nonlinear eigenvalue problem in , in , if (>0), , as , where is a parameter, is the Laplace operator, , and ; are such that as . Here are functions such that they are negative at the origin (semipositone) and superlinear at infinity. We establish the existence of a positive solution for λ small via degree theory and rescaling arguments. We also discuss a non-existence result for for the single equations case.
MSC: 34B16, 34B18.
where is a parameter, is the Laplace operator, and is an exterior domain. Here the nonlinearities are functions which satisfy:
(H1): and (semipositone).
(H2): For there exist and such that , and .
Further, for , the weight functions are such that as . In particular, we are interested in the challenging case, where do not decay too fast. Namely, we assume
We then establish the following.
Let (H1)-(H3) hold. Then (1.1) has a positive radial solution (, in) when λ is small, and, as.
where Ω is a bounded domain in , , and establish an existence result when λ is small. The main motivation of this paper is to extend this study in the case of exterior domains (see Theorem 1.1).
for large values of λ, when , satisfy the following hypotheses:
(H4):, for all , , and there exists such that .
(H5): The weight function is such that is decreasing for .
We establish the following.
Let (H3)-(H5) hold. Then (1.2) has no nonnegative radial solution for.
We establish Theorem 1.2 by recalling various useful properties of solutions established in , where the authors prove a uniqueness result for for such an equation in the case when is sublinear at ∞. However, the properties we recall from  are independent of the growth behavior of at ∞. Non-existence results for such superlinear semipositone problems on bounded domain also have a considerable history starting from the work in the 1980s in  leading to the recent work in . Here we discuss such a result for the first time on exterior domains.
via the Kelvin transformation , where , (see ).
The assumption (H3) implies that , for , , and there exist , such that for , and for . When in addition (H5) is satisfied, is decreasing in .
2 Existence result
where is a parameter. (Clearly, any solution of (2.1) for must satisfy , for . This is also true for any nontrivial solution when .) We prove the following.
Proof of (i)
In particular, this implies . Since is independent of l, clearly this is a contradiction for , and hence there must exists an such that for , (2.1) has no solution.
Proof of (ii)
This is a contradiction since and as . Thus (ii) holds. □
Proof of Theorem 1.1
(which is the same as (2.1) with ) has a positive solution , in that persists for small .
We first establish the following.
There existssuch thatfor allwithand.
Next we establish the following.
There existssmall enough such thatfor allwithand.
and hence (2.3) has a solution with , in , and . Now we show that the solution obtained above (when ) persists for small and remains positive componentwise.
for all .
If for with , then , in .
Proof of (i)
for all .
Proof of (ii)
Arguing as before, with (up to a subsequence). Note that since . By the strong maximum principle , , , , and . Now suppose there exists with and . Then must have a subsequence (renamed as itself) such that . But in implies that . Suppose . Since and , there exists such that , and hence taking the limit as we will have , which is a contradiction since . A similar contradiction follows if , using the fact that . Further, contradictions can be achieved if there exists with and using the facts that and . This completes the proof of the lemma. □
We now easily conclude the proof of Theorem 1.1. From Lemma 2.4, since is a positive solution of (2.2) for γ small, with is a positive solution of (1.3) for where . Further, since and in for , and as . This completes the proof of Theorem 1.1. □
3 Non-existence result
Proof of Theorem 1.2
But , and for . Thus clearly (3.1) can hold when , only if with . But by (H4), this is not possible since . Hence the nonnegative solution cannot exist for . □
The third author is funded by the project EXLIZ - CZ.1.07/2.3.00/30.0013, which is co-financed by the European Social Fund and the state budget of the Czech Republic.
- Ambrosetti A, Arcoya D, Buffoni B: Positive solutions for some semi-positone problems via bifurcation theory. Differ. Integral Equ. 1994, 7(3-4):655-663.MathSciNetGoogle Scholar
- Maya C, Girg P: Existence and nonexistence of positive solutions for a class of superlinear semipositone systems. Nonlinear Anal. 2009, 71(10):4984-4996. 10.1016/j.na.2009.03.070MathSciNetView ArticleGoogle Scholar
- Berestycki H, Caffarelli LA, Nirenberg L: Inequalities for second-order elliptic equations with applications to unbounded domains. I. Duke Math. J. 1996, 81(2):467-494. A celebration of John F. Nash, Jr 10.1215/S0012-7094-96-08117-XMathSciNetView ArticleGoogle Scholar
- Lions P-L: On the existence of positive solutions of semilinear elliptic equations. SIAM Rev. 1982, 24(4):441-467. 10.1137/1024101MathSciNetView ArticleGoogle Scholar
- Brown KJ, Shivaji R: Simple proofs of some results in perturbed bifurcation theory. Proc. R. Soc. Edinb., Sect. A 1982/1983, 93(1-2):71-82. 10.1017/S030821050003167XMathSciNetView ArticleGoogle Scholar
- Castro A, Shivaji R: Nonnegative solutions for a class of nonpositone problems. Proc. R. Soc. Edinb., Sect. A 1988, 108(3-4):291-302. 10.1017/S0308210500014670MathSciNetView ArticleGoogle Scholar
- Castro A, Shivaji R: Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric. Commun. Partial Differ. Equ. 1989, 14(8-9):1091-1100. 10.1080/03605308908820645MathSciNetView ArticleGoogle Scholar
- Hai DD, Sankar L, Shivaji R: Infinite semipositone problems with asymptotically linear growth forcing terms. Differ. Integral Equ. 2012, 25(11-12):1175-1188.MathSciNetGoogle Scholar
- Lee EK, Sankar L, Shivaji R: Positive solutions for infinite semipositone problems on exterior domains. Differ. Integral Equ. 2011, 24(9-10):861-875.MathSciNetGoogle Scholar
- Lee EK, Shivaji R, Ye J: Classes of infinite semipositone systems. Proc. R. Soc. Edinb., Sect. A 2009, 139(4):853-865. 10.1017/S0308210508000255MathSciNetView ArticleGoogle Scholar
- Sankar L, Sasi S, Shivaji R: Semipositone problems with falling zeros on exterior domains. J. Math. Anal. Appl. 2013, 401(1):146-153. 10.1016/j.jmaa.2012.11.031MathSciNetView ArticleGoogle Scholar
- Maya C, Girg P: Existence of positive solutions for a class of superlinear semipositone systems. J. Math. Anal. Appl. 2013, 408(2):781-788. 10.1016/j.jmaa.2013.06.041MathSciNetView ArticleGoogle Scholar
- Castro A, Sankar L, Shivaji R: Uniqueness of nonnegative solutions for semipositone problems on exterior domains. J. Math. Anal. Appl. 2012, 394(1):432-437. 10.1016/j.jmaa.2012.04.005MathSciNetView ArticleGoogle Scholar
- Brown KJ, Castro A, Shivaji R: Nonexistence of radially symmetric nonnegative solutions for a class of semi-positone problems. Differ. Integral Equ. 1989, 2(4):541-545.MathSciNetGoogle Scholar
- Shivaji R, Ye J:Nonexistence results for classes of elliptic systems. Nonlinear Anal. 2011, 74(4):1485-1494. 10.1016/j.na.2010.10.021MathSciNetView ArticleGoogle Scholar
- Ko E, Lee EK, Shivaji R: Multiplicity results for classes of singular problems on an exterior domain. Discrete Contin. Dyn. Syst. 2013, 33(11-12):5153-5166. 10.3934/dcds.2013.33.5153MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.