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Positive solutions for a class of superlinear semipositone systems on exterior domains
Boundary Value Problems volume 2014, Article number: 198 (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.
We consider the nonlinear elliptic boundary value problem
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
(H3): There exist , , such that for
We then establish the following.
Let (H1)-(H3) hold. Then (1.1) has a positive radial solution (, in) when λ is small, and, as.
We prove this result via the Leray-Schauder degree theory, by arguments similar to those used in  and . The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see , ), yet a rich history is developing starting from the 1980s (see –) until recently (see –). In ,  the authors studied such superlinear semipositone problems on bounded domains. In particular, in  the authors studied the system
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).
We also discuss a non-existence result for the single equation model:
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.
Finally, we note that the study of radial solutions (with ) of (1.1) corresponds to studying
which can be reduced to the study of solutions ; to the singular system:
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 .
We will prove Theorem 1.1 in Section 2 by studying the singular system (1.3), and Theorem 1.2 in Section 3 by studying the corresponding single equation
2 Existence result
We first establish some useful results for solutions to the system
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.
There exists such that 2.1 has no solution if .
For each , there exists (independent of l) such that if is a solution of (2.1), then .
Proof of (i)
Let , . Here is the principal eigenvalue and a corresponding eigenfunction of in with . Let , be such that for all and for . Now let be a solution of (2.1). Multiplying (2.1) by and integrating, we obtain
By Remark 1.3, , and for . Then from the above inequalities we obtain
Hence we deduce that
where , and . This implies
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)
Assume the contrary. Then without loss of generality we can assume there exists such that as . Clearly , and for all . Let , be the points at which and attain their maximums. Now since for all , we have
Hence , and in particular, for ,
Let be such that , and . Now for ,
where (>0), and G is the Green’s function of with . In particular, . Similarly . Hence, there exists a constant such that
This is a contradiction since and as . Thus (ii) holds. □
Proof of Theorem 1.1
We first extend f and g as even functions on ℝ by setting and . Then we use the rescaling, , , and with , , and . With this rescaling, (1.3) reduces to
Note that by our hypothesis (H2), and as . Hence we can continuously extend and to and , respectively. Note that proving (1.3) has a positive solution for λ small is equivalent to proving (2.2) has a solution with , in for small . We will achieve this by establishing that the limiting equation (when )
(which is the same as (2.1) with ) has a positive solution , in that persists for small .
Let be the Banach space equipped with , where denotes the usual supremum norm in . Then for fixed , we define the map by
We first establish the following.
There existssuch thatfor allwithand.
for . (Note .) By Lemma 2.1, if then and if for , then . This implies that there exists such that for for any . Also, since (2.1) has no solution for , . Hence, using the homotopy invariance of degree with the parameter we get
Next we establish the following.
There existssmall enough such thatfor allwithand.
for . Clearly , and is the identity operator. Note that if is a solution of
Then for some constant independent of . Similarly for some constant . This implies that
for some constant . But , and hence this is a contradiction if is small. Thus there exists small such that (2.4) has no solution with for all . Now using the homotopy invariance of degree with the parameter , in particular using the values and , we obtain
By Lemma 2.2 and Lemma 2.3, with , we conclude that
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.
Let R, r be as in Lemmas 2.2, 2.3, respectively. Then there existssuch that:
for all .
If for with , then , in .
Proof of (i)
We first show that there exists such that for all with , for all . Suppose to the contrary that there exists with , and . Since is compact, and are bounded in , (up to a subsequence) with or r and . This is a contradiction to Lemma 2.2 or 2.3 and hence there exists a small satisfying the assertions. Now, by the homotopy invariance of degree with respect to ,
for all .
Proof of (ii)
Assume to the contrary that there exists and a corresponding solution such that and
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
We first recall from  that, when (H5) is satisfied, one can prove via an energy analysis that a nonnegative solution u of (1.4) must be positive in and have a unique interior maximum with maximum value greater than θ, where θ is the unique positive zero of . Further, for and such that , (see Figure 1), where is the unique zero of , there exists a constant C such that and . Hence we can assume for . Now we provide the proof of Theorem 1.2.
Proof of Theorem 1.2
Let . Then in and satisfies
Note that in , , and it satisfies in . Hence using the fact that , we obtain
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 . □
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Abebe, A., Chhetri, M., Sankar, L. et al. Positive solutions for a class of superlinear semipositone systems on exterior domains. Bound Value Probl 2014, 198 (2014). https://doi.org/10.1186/s13661-014-0198-z
- positive solutions
- exterior domains