The primary objective of the paper is to study the existence, asymptotic boundary estimates and uniqueness of large solutions to fully nonlinear equations in bounded domains . Here H is a fully nonlinear uniformly elliptic differential operator, f is a non-decreasing function that satisfies appropriate growth conditions at infinity, and h is a continuous function on Ω that could be unbounded either from above or from below. The results contained herein provide substantial generalizations and improvements of results known in the literature.
Let be a bounded open set with boundary . We consider the infinite boundary value problem
where is a fully nonlinear uniformly elliptic operator. For , as usual, Du stands for the gradient of u while denotes the Hessian matrix of u.
Let be the set of real symmetric matrices. Throughout this paper, we fix constants and we set .
To specify our assumptions on H, we first recall the so-called Pucci extremal operators and (see ) defined by
Here stands for the trace of .
Given non-negative functions let us set
for , where .
The class of functions considered in this work will include
where is a continuous function such that
for some non-negative .
In this paper, we will consider mappings such that for all , , and the following hold:
H is continuous, and
, where is a constant and such that .
Here and throughout, stands for the set of positive real numbers.
We now turn to the nonlinearity f in (1.1). Throughout this paper we will assume that is a continuous function that satisfies both of the following conditions:
f is non-decreasing, positive in and .
f satisfies the Keller–Osserman condition; namely
Further conditions on f, as well as on , that will be needed in this work will be explained later.
The study of large solutions has a long history. Perhaps a systematic study of large solutions started with the works of Keller  and Osserman . Since then a huge amount of work has emerged focusing on existence and uniqueness of large solutions. An exhaustive list on large solutions is impossible and we only list [1, 5, 11, 10, 17, 18, 22, 24, 30, 31, 32, 33, 41, 42, 47, 21] and refer the interested reader to the references therein. We wish to single out the papers of García-Melián , López-Gómez and Luis Maire , and Marcus and Véron  on uniqueness of large solutions of on smooth bounded domains under some general conditions on f. However, in order to put the problems we wish to consider in this paper in perspective, let us recall some works that are directly related to problem (1.1) with on Ω. In , Verón studied the existence and uniqueness of solutions to problem (1.1) when is non-positive, is uniformly elliptic, and for some . Likewise, in , Diaz and Letelier investigated larges solutions of , , in bounded domains when f is a non-decreasing function that satisfies a condition of Keller–Osserman type suited for the p-Laplace operator and is non-positive. In , Alarcón and Quaas study existence, asymptotic boundary behavior and uniqueness of solutions to problem (1.1). In the paper , the authors consider the case when depends on only, f satisfies the usual Keller–Osserman condition and is non-positive. In a related work , one of the authors, Amendola and Galise show that has at most one positive large solution on a bounded domain with “local graph property” introduced by Marcus and Verón . Here , , on Ω and H is a uniformly elliptic operator that is “homogeneous” of degree which satisfies appropriate structural conditions. We refer to  for more details on the results and conditions imposed.
In a recent paper, García-Melián studied existence and uniqueness of large solutions to in bounded domains, where is allowed to change sign. See also . To the best of our knowledge, this more challenging case of a sign-changing inhomogeneous term h is investigated for the first time in the paper . In , the author obtains existence of large solutions to the aforementioned equation for a large class of unbounded and uniqueness result is proven under the restriction that h is bounded on Ω from above. Motivated by the work of , one of the authors and Porru  extended the work of  to large solutions of , where L is a linear uniformly elliptic equations in non-divergence form with possibly unbounded lower-order terms. In , existence and uniqueness results are obtained when the inhomogeneous term is allowed to be unbounded from above but with some restrictions, and with bounded coefficients for the first-order and zero-order terms.
The main objective of the present paper is to extend many of the above results to solutions of problem (1.1) by relaxing the conditions used in most of the aforementioned papers. In fact, the results contained herein are new for solutions of (1.1) when h is unbounded from above, even when . Another feature of the current work is that we obtain existence of solutions not only when h is unbounded on Ω, but also when the coefficients of H are unbounded on Ω.
The paper is organized as follows. In Section 2, we state the main results of the paper. These results discuss existence, asymptotic boundary estimates and uniqueness of solutions to problem (1.1). Section 3 presents some basic facts that are consequences of the assumptions made in the Introduction. We also recall several useful results from the literature that will be used in our work. The ABP maximum principle will play a recurring role in our work. In Section 4 we will develop several existence results. Depending on the rate of growth of h near the boundary, we will either relax the conditions needed on f or require more restriction.
Asymptotic boundary estimates of solutions to problem (1.1) will be developed in Section 5. In the investigation of such estimates, a condition on f introduced by Martin Dindoš in  will have a prominent role.
Uniqueness of solution to problem (1.1) will be investigated in Section 6. In its most general form, the uniqueness result will use a condition on h that manifests through the growth of a solution ψ of . In particular, our uniqueness result allows in problem (1.1).
Finally, we have included an Appendix where some useful results on existence to boundary value problems involving H with unbounded coefficients are studied. These results are used in the main body of the paper and are of independent interest.
In this section we state the main results of the paper. To avoid use of technicalities, we have chosen to present these results in less general terms than given in the main body of the paper.
We begin by considering the non-increasing function such that
Our first existence result, as well as many others, will depend on the sizes of
where is the distance of to the boundary of Ω. We will denote (2.2) simply as when there is no ambiguity concerning f.
Another important feature of our work is that we allow unbounded coefficients subject to the conditions:
To state our first existence result, we recall the following condition introduced by Dindoš in . There is such that
This condition, or a strengthened form thereof, will also appear in the study of boundary asymptotic estimates as well as in our uniqueness result. We remark that (f-theta), together with (f-1) implies (f-2). See Remark 3.6.
To obtain existence of solutions to problem (1.1) with the coefficients γ and χ allowed to be unbounded on Ω according to (C-γ) and (C-χ), we need control on the rate of growth of f at infinity and the following condition provides such control:
Perhaps a word on notational use is in order here. If we wish to use any condition (f-x) on a function , we will simply quote it as condition (g-x).
Let be a smooth bounded domain. Assume that (H-1), (H-2), (C-γ), (C-χ), (f-1), (f-2), (f-4) hold. Suppose that there is that satisfies (g-1), (g-3) and (g-θ) such that at infinity. Then there exists a constant such that problem (1.1) admits a maximal solution whenever satisfies as with . Here , where are the parameters in condition (g-3) and (g-θ).
A complementary existence result can be obtained by prescribing an indirect control on the size of . This control is imposed on the growth, near the boundary, of a non-negative solution ψ of a PDE related to the Pucci maximal operator as follows:
admits a non-negative solution .
We refer to Remark 4.8 for a discussion on this condition.
where is a non-decreasing function for some and satisfies the Dini condition
It will be convenient to refer to such a function as a Dini continuous function. Assuming that
Let be a bounded smooth domain. Assume (H-1), (H-2), (B-γ), (C- ), (f-1), (f-2). If is such that for some Dini continuous function η, then problem (1.1) admits a maximal solution. Moreover, there exists a constant such that the solution is positive whenever .
The proof relies on the existence of a positive solution ψ to problem (2.3). This approach based on condition (D-h) also proves to be useful in dealing with uniqueness for unbounded h, at least when the coefficients γ and χ are non-negative constants. The analysis on uniqueness will be carried out through boundary asymptotic estimates of solutions of (1.1).
To obtain boundary asymptotic estimates, we need Dindoš’ condition as well as control from below on h. In fact, we need to assume as , or equivalently . However, we need the coefficients γ and χ be bounded, which without loss of generality, we take to be non-negative constants.
Let be a bounded smooth domain. Assume (H-1) and (H-2) with γ and χ non-negative constants, (f-1) and (f-theta) for some . Let be such that and for some Dini continuous function η. There exist constants such that
for all solutions u of (1.1).
We need further assumptions on H and on f in order to get uniqueness. These are the sub-homogeneity property: For all
and the monotonicity condition
is non-decreasing at infinity.
The following uniqueness result holds.
Let be a bounded and smooth domain and assume (H-1)–(H-3) with non-negative constants. Assume that f satisfies conditions (f-1), (f-3), (f-theta) for all and (f-m). Suppose also that satisfies and as for some Dini continuous function η. Then problem (1.1) admits at most one solution.
We should point out that the above asymptotic estimate and uniqueness results, which are stated here with the condition as for some Dini continuous function η, have been established in this paper in a more general framework through control of the growth of the functions ψ given in condition (D-h). To the best of our knowledge, this approach appears here for the first time (see also ). The optimal growth on that this method leads to remains an open problem.
Throughout the entire paper we suppose that is a bounded open set with boundary. In this work it will be convenient to use the following notations. Given ,
where denotes the distance of to the boundary . Since Ω is a bounded domain, we note that there is such that and on . See [25, Lemma 14.16] for a proof. In fact, 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 .
It is helpful to keep in mind the following alternative description of the Pucci extremal operators:
where and are the positive and negative parts of X, respectively, and , , are the eigenvalues of X, counted according multiplicity, in non-decreasing order.
The positive homogeneity, duality, sub-additive and super-additive properties of the Pucci extremal operators (see ) lead to the following useful properties of :
for all and .
From (H-1) it follows that H is uniformly elliptic, that is,
Moreover, (H-1) implies that H is non-increasing in t:
Given , a function is said to be a classical solution of equation in Ω if and only if
However, in this paper we consider functions which are solutions in the viscosity sense, according to the following definition.
Let (upper semicontinuous in Ω), resp. (lower semicontinuous in Ω). Then u is said to be a viscosity subsolution (resp., supersolution) in Ω of (3.5) if and only if for each and such that has a local maximum (resp. minimum) at x we have
A function that is both a viscosity subsolution and viscosity supersolution in Ω of (3.5) is called a viscosity solution in Ω.
It is well known that a function is a classical subsolution (supersolution) of (3.5) if and only if u is a viscosity subsolution (supersolution) of (3.5). The forward implication follows directly from the definition. For the reverse, we refer to [8, Corollary 2.6].
We note the following consequence of condition (H-1):
for any function , where .
In the sequel we will make an extensive use of a fundamental tool for pointwise estimates of viscosity solutions of elliptic equations, known as the Alexandroff–Bakelman–Pucci maximum principle (see, for instance, [6, 9, 3, 43]). For the convenience of the reader we recall below the version needed here.
For this, we first remark that if and for some , then by (3.6) it follows that . Note also that the latter implies that satisfies . Therefore, setting
we also have
Consequently, the standard Alexandroff–Bakelman–Pucci maximum principle (see [6, Proposition 2.12]) leads to the following.
Let be a bounded domain with diameter R. Suppose that H satisfies condition (H-1), assuming . For , let be a viscosity subsolution of equation in . There is a non-negative constant C, depending only on and , such that
In particular, under the assumptions of Proposition 3.3, the following sign propagation property holds:
One then obtains a useful comparison principle by combining Proposition 3.3 and the following result which is based on [14, Proposition 2.1]. A justification for the reformulation presented below is sketched in [33, Lemma 2.5].
The non-increasing function defined in (2.1) satisfies
Here we mention some easy, but useful consequences of the Dindoš’ condition (f-theta).
for some . For a proof we refer to [35, Lemma 2.2].
Note that, by iterating (f-theta), we also have, for all ,
as well as
Assuming, in addition, (C-χ), we have
To show (3.8), observe that
Given any , there are positive constants and such that for all .
If, in addition, (f-theta) holds, then given , there are constants and such that for all .
We should point out that the constants and in Lemma 3.12 (i) depend on the parameter α in condition(f-3), while the constants and Lemma 3.12 (ii) also depend on θ and in condition (f-theta). See [35, Lemmas 2.12, 2.13 and 2.15].
The following condition which holds for any odd function f that satisfies (f-1) will be needed in one of our existence results:
We start this section with a result that shows the existence of supersolutions of (1.1) in balls of suitably small radii.
We may suppose that B is centered at the origin, that is, with .
Let us start with the case . We look for a solution of the form with . By (H-1), we have
where . Taking a smaller concentric ball B, and using Remark 3.5, we take R sufficiently small so that and then
Thus w solves problem (4.1) with .
Next, we consider an arbitrary . Suppose and in B, with . Set , with as considered above and to be suitably chosen. Then
We now shrink R further, if necessary, to have and then we take large enough in order that . Since , we see that v solves (4.1). ∎
For , let be a solution of
Therefore, assuming that is non-empty, we would have
By the maximum principle, Proposition 3.3, we have in which is a contradiction. It follows that is a non-decreasing sequence.
Hence is a sequence of locally uniformly bounded solutions of (4.2), and so via the Harnack inequality also locally equi-Hölder continuous in (see [7, 44]). By Ascoli–Arzelà and stability results on viscosity solutions (see ), and taking into account the monotonicity of the sequence , we have that
The following lemma shows that a maximal solution can be found for problem (1.1) if a subsolution exists.
has a solution w, then problem (1.1) has a maximal solution u such that in Ω. If , then the solution is non-negative in Ω.
Let be an exhaustion of Ω by smooth domains so that and . For each we take the solution of problem (1.1) with instead of Ω (h is bounded in ), provided by Theorem 4.2. On a fixed the solutions , with , are bounded and by the maximum principle on . Moreover, is a non-increasing sequence on . Let us set
Using the subsolution w in Ω of the hypothesis and supersolutions on balls of sufficiently small radius as provided by Lemma 4.1, we can show that the sequence is uniformly bounded on each domain . Consequently, the sequence is equi-Hölder continuous. Therefore is a continuous viscosity solution of equation in Ω (see the proof of Theorem 4.2). It is clear from the maximum principle that in for each j. Therefore in Ω and
To prove the second assertion in the lemma, suppose in Ω. Then by conditions (H-1) and (f-1) we note that is also a solution of (4.3). Therefore, comparison with each as in the above shows that maximal solution u constructed above satisfies in Ω, which was to be shown. ∎
We are ready to prove our first existence theorem for problem (1.1), where an auxiliary function g satisfying Dindoš’ condition will be employed. As mentioned in the Introduction, we will refer to conditions (f-1)–(f-4), (f-theta), respectively, as conditions (g-1)–(g-4), (g-θ) when we use g instead of f. Similarly, we denote by , , and , respectively, the function ϕ, the numbers in (f-theta) and Remark 3.7, and the quantities in (2.2) when we consider g instead of f.
In the statement of the theorem it will be convenient to use the following notation for any positive constant :
We recall that for g that satisfies (g-1) and Dindoš’ condition there is such that .
Then there exists a positive constant such that problem (1.1) admits a maximal solution provided satisfies as with . In fact, we may choose .
If g satisfies Dindoš’ condition with and , then (4.5) is equivalent to
In fact, suppose that (4.6) holds. In Remark 3.7 we have seen that the function g satisfies (3.7) with and for all . We choose large enough to have , obtaining (4.5) with . We also point out that if and holds at infinity, then we can take in the theorem.
Let us first observe that (f-2) and (4.5) show that g satisfies the Keller–Osserman condition (g-2). For notational simplicity, let us denote and by ϕ and , respectively. According to Lemma 4.3 it is enough to show that problem (4.3) admits a solution. To this end, we search for a solution in the form
with the constant to be suitably chosen.
Denoting by G the antiderivative of g vanishing at the origin, direct computation in , where , yields
In (4.7), given , we may pass to a smaller , if necessary, to ensure that
For small enough, the following chain of inequalities holds in :
Since , it follows that w is a subsolution in .
To finish the proof, we will choose large enough so that w is a subsolution also in . For this, let
By (f-4) we can choose A such that
so that in , once again by (H-1),
This concludes the proof. ∎
By assumption we note that , where ρ is the constant in (4.5). Since g satisfies condition (g-θ), we use Remark 3.7 to choose large enough such that . Therefore with we observe that (4.5) holds. Now we take Θ as in Theorem 4.4 to obtain the conclusion of Theorem 2.1. ∎
We also make note of the following special case.
Assume (H-1), (H-2), (f-1), (f-2), (f-4), (C-γ), (C-χ). If at infinity for some , then for any such that as , problem (1.1) admits a maximal solution. In particular, if at infinity for any , and for some constant as , then problem (1.1) has a maximal solution.
Let . Note that in this case (4.4) holds for any . In particular, the hypothesis on h allows us to choose small enough such that . Since in (4.5) we have , and , Theorem 4.4 shows that problem (1.1) admits a maximal solution. To prove the second assertion, it suffices to observe that given a constant we pick such that . ∎
Corollary 4.6 shows, for instance, that the following problem admits a solution for any such that is bounded in Ω for some :
Condition (D-h) is satisfied, for instance, if
admits a solution that is bounded in Ω from below. In fact, setting in this situation, then is a non-negative solution of (4.12). If, in addition, for some and for some , then the equation has a solution , in which case condition (D-h) holds. Here, is the Escauriaza exponent  (see also Crandall and Świȩch, ) such that for any with , solutions of satisfy the maximum principle. We refer the reader to [29, Theorem 7.1] for details.
In the present approach for existence of solutions to (1.1), we can relax condition (C-γ) by requiring the weaker condition (B-γ), while we need to strengthen condition (C-χ) to condition (C- ) as described in the Introduction.
We now have the following existence result.
Let v be a solution of
We direct the reader to the Appendix, Lemma A.4, for justification of the existence of such a solution.
The following gives a generalization of the existence results of Alarcón and Quaas .
As pointed out in the Introduction condition (2.4) allows us to show existence of solutions to (1.1), and the problem admits a positive solution provided is sufficiently small. This is the content of Theorem 2.2, which we now prove.
Let η be the Dini continuous function as provided in the hypothesis. Then according to , there is a positive function such that
We refer to the Appendix for how this assertion follows from the work of Ancona in . Therefore (2.4) together with (4.13) shows that condition (D-h) holds. Thus Theorem 4.9 shows that problem (1.1) admits a solution.
Let be a large solution of in Ω. We refer to the Appendix for the existence of such a large solution. By the Harnack inequality (see ) we note that actually in Ω. Let , where is chosen such that . It follows that in Ω and if , then (4.13) implies that
Consequently, we have
We have the following theorem on asymptotic boundary estimates of solutions to (1.1).
Suppose that satisfies . If there exists a positive constant such that
then for any continuous subsolution u of ( 1.1 ) we have
In the proof that follows it will be convenient to write for . For any let us consider the following subsets of Ω:
We start with the proof of (i), which will be carried out by showing that
Then, recalling (3.6) and the expression of , , computation shows that
On using the assumption that , we obtain the following estimates. Let us note that, according to (5.1), we can take μ sufficiently small that
By shrinking further, if necessary, the following hold in :
By (5.1), we can choose sufficiently small so that
Next, let be a subsolution of (1.1) and set . Then on and we note that the following inequalities hold on :
By the comparison principle we conclude that in . Therefore
On letting , we see that the following holds on :
from which we get (5.2) upon letting .
Now we turn to the proof of (ii). For this consider the function
and we wish to show that is a subsolution of equation in provided μ is sufficiently small. For to be chosen small enough, using Remark 3.5, we take a sufficiently small in order that
Then, on recalling (3.6) and the expression for , , direct computation in shows that
In the above, provided is small enough, by (5.3) we can make
Inserting this in (5.9), we get
and this shows that is a subsolution in as claimed.
Set . By the structure conditions,
and on . On the other hand, considering the function provided by condition (D-d), we also have in
and therefore we have
On letting , we see that the following holds on :
On recalling that , as , and using condition (5.3), we get
and this concludes the proof of the second part of the theorem with . ∎
for all solutions u of (1.1).
By hypothesis, f satisfies condition (f-theta) for some . We now take the smallest , depending on and Λ, such that
Then, recalling Remark 3.7, we see that inequality (5.1) holds with the choice . If we now also require , then both inequalities in (5.3) hold with the choice . We invoke Theorem 5.1 (i) and (ii) to complete the proof. ∎
In addition to requiring condition (D-h) in the statement of Theorem 4.9, we also needed the solution ψ of (2.3) to be bounded on Ω from above. Thanks to Theorem 5.1 (i), we can now relax this restriction as we now show.
Assume (H-1), (H-2) with satisfying (C-γ) and (C- ), respectively, (f-1) and (f-theta). There exists a positive constant such that if condition (D-h) holds with a solution ψ such that , then problem (1.1) admits a maximal solution.
By Lemma A.4 of the Appendix let v be a large solution of in Ω. Consider the function with ψ such that . Here is a constant to be suitably chosen soon. As in the proof of Theorem 4.9, we can show that .
It remains to prove that as , and then invoke Lemma 4.1 in order to complete the proof. To this end, we use (f-theta) to find large enough (see Remark 3.7) in order that , set and . If , then the assumption of Theorem 5.1 are satisfied and
As a consequence, we have
and when , as we wanted to show. ∎
The following is an immediate consequence of Theorem 5.3.
We conclude this section with the proof of Theorem 2.3.
We wish to show that