Abstract
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.
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1 Introduction
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.
Theorem 1.1
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 [1] and [2]. The study of such eigenvalue problems with semipositone structure has been documented to be mathematically challenging (see [3], [4]), yet a rich history is developing starting from the 1980s (see [5]–[7]) until recently (see [8]–[12]). In [1], [2] the authors studied such superlinear semipositone problems on bounded domains. In particular, in [12] 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.
Theorem 1.2
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 [13], 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 [13] 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 [14] leading to the recent work in [15]. 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 [16]).
Remark 1.3
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.
Lemma 2.1
-
(i)
There exists such that 2.1 has no solution if .
-
(ii)
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
and
By Remark 1.3, , and for . Then from the above inequalities we obtain
and
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
where
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
where . Note that are continuous and is compact. Hence is a compact perturbation of the identity. Clearly for , if , then is a solution of (2.2), and if , then is a solution of (2.3).
We first establish the following.
Lemma 2.2
There existssuch thatfor allwithand.
Proof
Define by
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.
Lemma 2.3
There existssmall enough such thatfor allwithand.
Proof
Define by
for . Clearly , and is the identity operator. Note that if is a solution of
and for , (2.4) coincides with (2.3). Assume to the contrary that (2.4) has a solution with . Without loss of generality assume . Now,
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.
Lemma 2.4
Let R, r be as in Lemmas 2.2, 2.3, respectively. Then there existssuch that:
-
(i)
for all .
-
(ii)
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 [13] 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
In particular,
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.
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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
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DOI: https://doi.org/10.1186/s13661-014-0198-z