Let be a bounded domain with boundary. We consider a uniformly elliptic nondivergence structure second order differential operator, namely,
In (1.1) and throughout this paper, the summation convention over repeated indices from 1 to n is in effect. We will assume that is an symmetric matrix of continuous real-valued functions on Ω such that
for some constant .
In this paper we wish to study the question of existence and uniqueness of solutions to the following problem:
Here is a continuous function which may be unbounded on Ω, and is a continuous function with appropriate conditions to be specified later.
for all , with and for all .
In (DB) and (DC) we have used to denote the distance of from the boundary , a notation we will continue to use throughout the paper. In (DC), is a positive constant and is an increasing function that satisfies the Dini condition
Following the pioneering works of Keller  and Osserman , problem (1.3) has been studied extensively by numerous authors when L is the Laplacian and . The reader is referred to the monograph  and the references therein for more discussion related to such problems. In , Bandle and Marcus investigated existence and asymptotic boundary behavior of solutions to (1.3) when and , , for some . Concerning problem (1.3) with on Ω, Véron in , and Díaz and Letelier in , established the existence and uniqueness of positive solutions to (1.3) for the nonlinearity , , and for a non-positive unbounded inhomogeneous term with appropriate growth condition on the boundary.
As far as we are aware, it was García-Melián who first studied problem (1.3) for a sign-changing and unbounded inhomogeneous term h in the recent paper . He studied the existence of a solution to problem (1.3) when L is the Laplacian, for and h belonged to a large class of unbounded and sign-changing functions on Ω. He also obtained uniqueness of positive solutions of (1.3) for with appropriate growth condition on the boundary, but bounded on Ω from above.
In this paper we wish to continue the aforementioned investigations with the objective of extending the results in several fronts. In all cases the class of inhomogeneous terms h we consider will include sign-changing and unbounded functions in Ω having appropriate growth conditions near the boundary. The necessity of some restriction on the growth of h near the boundary has already been noted in [9, Theorem 3]. As our first main result we will show the existence of solutions to (1.3), where the lower-order coefficients are allowed to be unbounded in Ω, and f comes from a wide class of nonlinearities. In addition, to the usual Keller–Osserman, we will require some mild conditions on f. As it turns out, if the inhomogeneous term h grows no faster than a suitable multiple of near the boundary of Ω, then problem (1.3) admits a solution. Here ϕ is a decreasing function on , for some , that is associated with the nonlinearity f. For instance, when , , for . Our second main result concerns the asymptotic boundary estimates for solutions of (1.3) when the coefficients of L are bounded in Ω and, as a consequence, the uniqueness of solutions is obtained. These results are shown to hold for a large class of inhomogeneous terms h, which may change sign and be unbounded on Ω. In this regard, the asymptotic estimate and uniqueness results of this paper are new even when L is the Laplacian and for , as we do not require h to be bounded from above.
We point out that problem (1.3) was also considered in [10, 25]. We direct the reader to  for a discussion of problem (1.3) in these papers. The reader is referred to the recent papers [2, 4, 5, 8, 21, 26], and references therein, on asymptotic behavior and uniqueness of singular solutions related to the content of this paper. In particular, we draw attention to the paper  in which the authors make a systematic use of Karamata variation theory to study uniqueness of boundary blow-up solutions. To the best of our knowledge,  is the first paper to investigate existence of boundary blow-up solutions of equations with nonmonotonic nonlinearity.
The paper is organized as follows. In Section 2, we state the main conditions used on the nonlinearity f to study problem (1.3). In particular, we recall a lemma that will be useful in establishing the existence of solutions in the case when L has singular lower-order term coefficients. Section 3 is devoted to the study of existence of a solution to (1.3). In Section 4, we establish boundary asymptotic estimates of solutions to problem (1.3). Existence of positive solutions and uniqueness of solutions is investigated in Section 5. Finally, we have included an appendix where we prove some technical results that are used in the paper.
Let be a bounded domain with boundary. Throughout the paper, it will be convenient to use the following notations for a given :
Since Ω is a bounded domain, we note that there exists such that and on . See [12, Lemma 14.16] for a proof.
By modifying the distance function d appropriately we can suppose that d is a positive function on Ω. For instance, one can use instead of d, where is a cut-off function with on Ω, on for some , and on . Therefore, hereafter, we will always suppose that d is this modified distance function and that d is in with on .
By a solution u of we mean a twice weakly differentiable function on Ω such that
We start with the following extension of the classical maximum principle. We assume that is non-decreasing on for each .
Lemma 2.1 (Comparison Principle).
Let , and assume that in Ω and in Ω. If on , then in Ω.
Let sufficiently small such that on . Suppose that the open set
is non-empty. Since in and the coefficients of L are bounded on , the maximum principle applies (see [12, Theorem 9.1]) and we conclude that on . This is an obvious contradiction to the assumption that is non-empty. Therefore, we must have in . Since is arbitrary, we conclude that in Ω. Since is arbitrary, we find that on Ω, as desired. ∎
We consider the following conditions on the nonlinearity f in (1.3):
is a non-decreasing continuous function such that with for .
f satisfies the Keller–Osserman condition, namely,
We make a few remarks about the above conditions.
Any regularly varying function at infinity with index satisfies conditions (A2) and (A3). We recall that f is said to be regularly varying at infinity of index if f is a measurable function defined on for some and
We refer the reader to  for a proof of the above lemma. We let ϕ to be the non-increasing function such that
It follows that
For later use, let us compute Lv, where for some :
3 On existence of solutions to problem (1.3)
Throughout this section, we assume that the lower order coefficients () and of L satisfy conditions (DB) and (DC), respectively.
We prove the first limit and omit the second as the proof is similar. We have
On noting that , the claim follows from Lemma 2.3 (ii).∎
It is clear that (DB) and (DC) hold when the coefficients and c of L are bounded on Ω. Therefore, Corollary 3.1 holds when the coefficients of L are bounded. In fact, in this case, condition (A3) is not needed. One only needs to recall (2.1).
Let and , where
Here, Λ is the ellipticity constant of L as noted in condition (1.2).
As an example, we observe that for any regularly varying function (at infinity) f of index , it can be easily seen that
We need some conditions on f and h in order to prove the existence of a solution to (1.3). We require the following assumption on f:
The main result of this section gives the existence of a solution to problem (1.3). We employ the sub-solution and super-solution technique to establish the result. In preparation for this, let us consider a function which is non-decreasing in in the second variable for each , and for each .
From  we recall the following result on the solvability of a class of Dirichlet problems with continuous boundary data.
Given , the following Dirichlet problem admits a solution for each :
Suppose that all coefficients of L are bounded and belong to for some . If, in addition to the hypotheses on H, we suppose , then problem (3.2) admits a solution . This is a consequence of the elliptic regularity theory, see [12, Theorem 9.19] with .
We will use Lemma 3.4 to study the following infinite boundary value problem.
As a simple consequence of Lemma 3.4 we obtain the following.
Let for be such that
Assume that or on . If in Ω, then there exists a solution of (3.3) with a.e. in Ω.
Let us first consider the case when . For each positive integer , by Lemma 3.4, we let be a solution of
By the Comparison Principle, Lemma 2.1, we have
Now assume that on . Let for , and let be a solution of
By the Comparison Principle, we see that on for each . Consequently,1 we also note that in . Thus, we have
We are now ready to state the following theorem on the existence of a solution to (1.3).
There is no loss of generality in supposing that . Let be the sequence of open subsets of Ω introduced in the proof of Lemma 3.6 above. Let us first show that the following has a solution:
To this end, let be a solution of in and on . If in , then is a super-solution of (3.4). Otherwise, let be a solution of in with on . Note that . Then satisfies
In the last inequality, we have used the fact that for any . By (h1), since , we let A such that . Let us choose sufficiently small so that
Therefore, there exists such that for , we have
Likewise, by the definition of , we have (by shrinking δ further, if necessary)
Next we choose α so that w is a sub-solution of on . For this, let
We should point out that η is independent of α. The hypothesis on f shows that
Thus, we can choose sufficiently large so that
Having fixed such α, we see that in the following holds:
Thus, we have shown that w is a sub-solution of (1.3) in Ω. Moreover, for each positive integer j, the Comparison Principle shows that in . Therefore, by Lemma 3.6, problem (3.4) has a solution such that in . Since is bounded on , the Comparison Principle shows that on , that is, is a decreasing sequence in . Let us set
Since is locally uniformly bounded, by standard Schauder estimates, we see that satisfies in Ω. From this and the inequality in Ω, we see that u is a solution of (1.3), as desired.
4 On the boundary asymptotic estimates of solutions to problem (1.3)
To discuss asymptotic boundary estimates of solutions to (1.3), we need an additional condition on h which we now make explicit:
If h is bounded from above on Ω, then .
We have the following lemma on asymptotic boundary estimates of solutions to (1.3).
For any , let us consider the sets
Proof of (4.1). For an appropriate choice of a positive constant , we will show that
Let , to be specified later. By (2.1), we can take sufficiently small so that for all ,
Now, let ε be chosen so that . We pick a number such that
We also suppose that μ is sufficiently small, so that for ,
Let us now suppose that . In this case, for , we have
from the definition of . Let . Then on and
By the Comparison Principle, Lemma 2.1, we conclude that in . Therefore,
On letting , we see that the following holds on :
Now, let to obtain (4.1) when .
We now consider the case . Then h is non-negative near . Let be the sequence of open subsets of Ω defined as in the proof of Lemma 3.6 and . We note that . Let be a solution of (3.4) with . If u is any solution of (1.3), then . By the Comparison Principle, in for all j. Proceeding as in the proof of Theorem 3.7, one can show that is a solution of (1.3) on Ω with on Ω. Clearly, in Ω, and since v is a large solution of on Ω and , the case considered above applied to v shows that
where , and therefore (4.1) holds.
Proof of (4.2).
We consider the function
For an appropriate choice of , we show that is a sub-solution in for all , assuming that μ is sufficiently small. We assume first that . Let us fix such that . Then, by definition, there exists a positive real number such that . That is,
Therefore, assuming that μ is sufficiently small, the following holds in for all :
By shrinking μ if necessary, we can invoke (2.1) to estimate
Thus, for any , the following chain of inequalities hold on :
Recall that . Since h is bounded from above on Ω, we can assume μ is small enough so that
Thus, for and sufficiently small , we have shown that
Now let us suppose that . Then there exists such that for and small, we have
which can be rewritten as
Using this in estimate (4.5), we find
On noting that , and using (4.7), we find
Choosing in the above inequality, we find
Inserting the latter estimate into (4.6), for sufficiently small , yields
In conclusion, we have shown that in either of the cases, the following holds:
Let . Note that on and
Therefore, in and
On letting , we see that
On recalling that as , we get
and the proof is complete. ∎
In Lemma 4.2, we required h in (1.3) to be bounded from above in order to get the asymptotic estimate (4.2). Next, we wish to remove this restriction to allow h to be unbounded on Ω. However, the growth of h near the boundary of Ω needs to be constrained in such a way that a Dirichlet problem with zero boundary data is solvable. We use the notation .
We assume that is bounded in Ω and that the following Dirichlet problem admits a solution :
Suppose that satisfies (h3). Then we note that there exists a positive constant C such that in Ω. Therefore, we have
By Lemma 2.3 (ii), we note that
Consequently, we see that
The following result complements Lemma 4.2.
The Comparison Principle shows that is an increasing sequence. Let w be the unique solution of (4.8). Since on Ω, we find that
Now, if u is any solution of (1.3) in Ω, then again by the Comparison Principle, we find
Letting , we obtain
Then v is a large solution of on Ω.
Since h satisfies (h1) and (h3), we see that . Since, in addition, is bounded from above we invoke Lemma 4.2 (ii) to infer that there exists a constant such that v satisfies the asymptotic estimate (4.2). Since w is bounded on Ω, we have
and this concludes the proof of (4.2). ∎
The following corollary is noteworthy.
Suppose that satisfies for all , with , for some , where is an increasing function that satisfies (1.4). According to [1, Theorem 4] or [16, Theorem 5.2], we note that satisfies condition (h3). Therefore, if f satisfies (A1) and is regularly varying at infinity of order , then (3.1), together with Corollary 4.4, shows that any solution u of (1.3) satisfies (4.12) for some constants .
5 On uniqueness and existence of positive solutions to problem (1.3)
In this section, we continue to assume that all the coefficients of L are bounded in Ω.
Our next result shows that problem (1.3) admits at most one non-negative solution even when h is allowed to be unbounded in Ω.
The proof is omitted since it is similar to that of [18, Theorem 4.4], see also [7, 8, 9, 17]. In particular, we mention  in which the authors prove uniqueness to the boundary blow-up problem associated with in balls, where f is non-decreasing in and convex at infinity.
For our next result, we suppose that . If , then . This follows from [12, Theorem 7.8]. Furthermore, we suppose that the coefficients of L are Hölder continuous in Ω. If is a solution of (1.3) with , then, since , from the elliptic regularity theory, we conclude that , see [12, Theorem 9.19].
In Theorem 3.7 we showed that problem (1.3) has a solution. This solution may change sign in Ω. In the next theorem, we consider the existence of a positive solution to (1.3). We use an adaptation of the argument in .
Let . There exists a constant , depending on β and Ω, such that for every satisfying
problem (1.3) admits a positive solution in .
Let be the maximal solution of (1.3) with on Ω (see the remark right after Theorem 5.1). By the tangency principle (see [20, Theorem 2.1.3]), we note that . Let be the unique solution of in Ω with on . The existence of such a solution is justified by [1, Theorem 4]. Moreover, in Ω and there exists a positive constant such that (see [1, Remark 6.2])
Therefore, we have
We choose in (5.1) so that
Thus, in Ω, and
Let be the sequence of solutions of (3.4) constructed in the proof of Theorem 3.7. Then, by the Comparison Principle, we note that in for all j. Therefore, in Ω, showing that u is a positive solution of (1.3). ∎
We use this appendix to prove two lemmas and make some remarks, which may be of independent interest.
Given , we see that
Now assume, contrary to (A.1), that . We choose so that
This shows that (A2) fails. ∎
(i) We remark that if is non-decreasing at infinity, then
To see this, suppose is non-decreasing on for some . Then
Therefore, we have
from which the claim follows.
(ii) The converse of Lemma A.1 is false, as the example shows.
As a consequence of (A.5), for , we find
Therefore, for , we have shown that
For an arbitrary, but fixed , we take in (A.7) to find
Consequently, we get
Therefore, since , we note that the right-hand side tends to 0 as . This shows that
and the proof is complete. ∎
We complement the above lemma with the following remark.
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About the article
Published Online: 2017-06-08
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 517–532, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0065.
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