Nonexistence of positive supersolutions of elliptic equations via the maximum principle
We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the -Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.
Key words and phrases:Liouville theorem, semilinear equation, -Laplace equation, fully nonlinear equation, Lane-Emden system
2000 Mathematics Subject Classification:Primary 35B53, 35J60, 35J92, 35J47.
A well-studied problem in the theory of the elliptic partial differential equations is that of determining for which nonnegative, nonlinear functions there exists a positive solution or supersolution of the equation
in some subset of ; here denotes a second-order elliptic differential operator. A model case is the semilinear inequality
where is a positive continuous function defined on . There is a vast literature on the problem of obtaining sufficient conditions on to ensure the nonexistence of positive supersolutions of such equations, both in and in subsets of , which encompasses many different choices of operators and nonlinear functions .
In this paper we introduce a new method for proving the nonexistence of supersolutions in unbounded domains. It has the advantage of being both simple and robust, allowing us to prove new and essentially optimal results for wide classes of equations and systems of equations of type (1.1). In particular, we extend many of the previous Liouville results by substantially relaxing the hypotheses on required for nonexistence. Namely, we impose only “local” conditions on the behavior of , near or , and for large . Furthermore, our approach unites many previously known but seemingly disparate results by demonstrating that they follow from essentially the same argument.
Our method depends only on properties related to the maximum principle which are shared by many elliptic operators for which the solvability of (1.1) has been studied. Consequently, our technique applies to inequalities in both divergence and nondivergence forms, and interpreted in the appropriate (classical, weak Sobolev, or viscosity) sense.
To give a flavor of our results, let us consider the differential inequality (1.2) in an exterior domain , , where is any ball. Under only the hypotheses that is continuous, as well as
for each , we show that there does not exist a positive (classical, viscosity or weak Sobolev) solution of (1.2). Therefore in dimensions it is only the behavior of near that determines whether or not supersolutions exist, while in dimension it is the behavior of at infinity which determines solvability. These results are sharp and new.
Furthermore, we will see that if the inequality (1.2) is assumed to hold only on where is a proper cone of , then we must make assumptions on both at zero and at infinity in order to obtain a nonexistence result. Specifically, we exhibit exponents such that (1.2) has no positive solutions provided that is continuous and
It is usually thought that the most precise results for equations in divergence form like (1.2) are obtained by exploiting their integral formulation. A notable feature of this work is that we deduce new and optimal results for such equations by a method whose main ideas– in particular the use of the quantitative strong maximum principle (see (H3) and Theorem 3.3 below)– originate primarily from the theory of elliptic equations in nondivergence form.
We now give a rough list of the properties we assume the operator possesses, and on which our method relies:
satisfies a weak comparison principle;
the equations and have solutions in which are asymptotically homogeneous and positive (resp. negative) at infinity. Usually and are the fundamental solutions of ;
nonnegative solutions of have a lower bound (on compact subsets of the underlying domain) in terms of the measure of a set on which is greater than a positive constant;
nonnegative solutions of satisfy a weak Harnack inequality, or at least a “very weak” Harnack inequality; and
the operator possesses some homogeneity.
Specific details on these hypotheses and on some operators which satisfy them are given in Section 3. These properties are verified for instance by quasilinear operators of -Laplacian type with solutions interpreted in the weak Sobolev sense, and by fully nonlinear Isaacs operators with solutions interpreted in the viscosity sense.
We now make the following deliberately vague assertion:
Suppose has the properties (H1)-(H5) above, and the behavior of near and/or compares appropriately with that of the functions and for large . Then there does not exist a positive solution of the inequality (1.1) on any exterior domain in .
We prove a very general (and rigorous) version of this assertion in Section 4, see Theorem 4.1. The above statement is optimal in the sense that if a model nonlinearity does not satisfy its hypotheses, then (1.1) has positive supersolutions.
Obviously a nonexistence result in exterior domains implies nonexistence in as well as the absence of singular supersolutions in with arbitrary singularities in a bounded set. Another advantage of the technique we introduce here is that it applies very easily to systems of inequalities in unbounded domains.
Let us now give a brief account of the previous results on the subject. Due to the large number of works in the linear and quasilinear settings, we make no attempt to create an exhaustive bibliography here. Much more complete accounts can be found in the book of Veron  and the survey articles of Mitidieri and Pohozaev  and Kondratiev, Liskevich, and Sobol . Gidas  gave a simple proof of the fact that the equation has no solutions in , provided . Condition (1.3) appeared first in Ni and Serrin , where the nonexistence of decaying radial solutions to some quasilinear inequalities like in for was proved. In two important papers, Bidaut-Veron  and Bidaut-Veron and Pohozaev  extended these results by dropping the restrictions on the behavior of a supersolution and by showing that the same results hold in exterior domains of . For more nonexistence results for positive solutions of quasilinear inequalities with pure power right-hand sides, we refer to Serrin and Zou , Liskevich, Skrypnik, and Skrypnik . Liouville-type results for semilinear inequalities in nondivergence form can be found in the work by Kondratiev, Liskevich, and Sobol . Extensions to quasilinear inequalities in conical domains have been studied for instance by Bandle and Levine , Bandle and Essen , Berestycki, Capuzzo-Dolcetta, and Nirenberg , and Kondratiev, Liskevich, and Moroz .
Fully nonlinear inequalities of the form , where is an Isaacs operator, were first studied by Cutri and Leoni , and later by Felmer and Quaas , in the case of a rotationally invariant and a solution in the whole space (see also Capuzzo-Dolcetta and Cutri ). These results were recently extended in , by a different method, to arbitrary Isaacs operators and to exterior domains. In particular, the inequality has no positive solutions in any exterior domain in , provided that (or ), where characterizes the homogeneity of the upward-pointing fundamental solution of the operator (as found in ).
As far as systems of inequalities are concerned, Liouville results were obtained by Mitidieri , Serrin and Zou , for the case of a whole space, Bidaut-Veron  for quasilinear systems in exterior domains, Birindelli and Mitidieri , Laptev  for systems in cones, and Quaas and Sirakov  for fully nonlinear systems in the whole space. For elliptic systems, the literature is more sparse and concerns only systems with pure power right-hand sides such as the Lane-Emden system , .
Despite the great variety of approaches and methods, most of the previous results required a global hypothesis on the function , namely that be a power function or a combination of power functions. A notable exception is the very recent work of D’Ambrosio and Mitidieri , who obtained various nonexistence results for divergence-form quasilinear inequalities in the whole space with only a local hypothesis on the function near , as in (1.3). Their method is based on sophisticated integral inequalities and requires that the inequality holds in the whole space.
Finally, we note that there is a large literature concerning Liouville results for solutions (not supersolutions) of equations of the form in , which started with the well-known work by Gidas and Spruck . For instance, it is known that in has no classical positive solutions provided is an increasing function on ; see  and the references therein. These deep and important results are quite delicate, with the nonexistence range depending on the conformal invariance of the Laplacian, on the precise behavior of on the whole interval , on the differential equality being verified in the whole space, as well as on the solutions being classical.
This paper is organized as follows. In Section 2 we present the main ideas by proving the Liouville result we stated above in the simple particular case of (1.2) and . We collect some preliminary observations in Section 3, including a precise list of the properties (H1)-(H5) above as well as some estimates for the minima of positive supersolutions of over annuli. Our main results for scalar equations in exterior domains are presented in Section 4. We extend the results for equation (1.2) to conical domains in Section 5. We conclude in Section 6 with applications of our method to systems of inequalities.
2. A simple semilinear inequality
In this section, we illustrate our main ideas on the semilinear inequality
in exterior domains in dimension , and under the assumption that the nonlinearity is positive and continuous on . We will show that the additional hypothesis
implies that the inequality (2.1) has no positive solution in any exterior domain. Notice that we impose no requirements on the behavior of away from , apart from continuity and positivity. In particular, may have arbitrary decay at infinity.
It is easily checked that for , the function is a smooth supersolution of in , for each sufficiently small . Moreover, the function is a solution of the equation in , if the constant is chosen appropriately. Notice and decay to zero as , so having a hypothesis on the behaviour of as is unavoidable for a nonexistence result to hold. Thus the following theorem is seen to be optimal in a certain sense.
We have left the statement of Theorem 2.1 intentionally vague as to the notion of supersolution, since the result holds regardless of whether we consider supersolutions in the classical, weak, or viscosity sense.
Several easy facts regarding the Laplacian on annuli are required for the proof of Theorem 2.1, and we state them now.
The key ingredient in the proof of Theorem 2.1 is the following “quantitative” strong maximum principle.
Assume is nonnegative, and satisfies
There exists a constant depending only on such that for each
We denote with the essential infimum of on the set .
Lemma 2.2 is a simple consequence of the fact that Green’s function for the Laplacian with respect to any domain is strictly positive away from the boundary of the domain, which yields
for some depending only on the dimension . See for example [11, Lemma 3.2] and the references there for more precise statements on the Laplacian.
To show that it is only the behaviour of near zero which determines whether supersolutions of (2.1) exist, we use the following consequence of the mean value property.
For every , there exists a constant such that for any positive superharmonic function in and any , we have
We remark that Lemma 2.4 is clearly weaker than the weak Harnack inequality.
Applying the comparison principle to a positive superharmonic function and the fundamental solution of Laplace’s equation yields the following simple lemma, which is well-known. For the reader’s convenience, we recall an elementary proof. Here and throughout the paper, and denote positive constants which may change from line to line.
Suppose that is superharmonic in an exterior domain of , with . Then there are constants , depending only on and , such that
Fix such that . Select so small that in a neighbourhood of . Then for each , there exists such that in . Applying the maximum principle to
in , for each , we conclude that in . Letting we obtain in , which gives the first inequality in (2.3).
Let us now combine the three lemmas above into a proof of Theorem 2.1.
Proof of Theorem 2.1.
Let us suppose that is a supersolution of (2.1) in , for some . For each , denote and observe that is a supersolution of
For each , define the quantity
Thus applying Lemma 2.2 with and produces the estimate
We may rewrite this inequality as
Recall that by Lemma 2.5 we also have, for some ,
Let us now define the quantity
Observe that for every and , we may choose large enough that
By the maximum principle, in . Sending and then , we discover that
that is, . Therefore the map is nondecreasing. For every , define the function
Observe that by (2.9) we have in , and
where does not depend on . In particular,
That is, for all sufficiently large . Therefore we obtain that , which contradicts our inequality (2.7). ∎
Let be a positive, continuous function on which satisfies
Then the inequality (2.1) has no positive solution in any exterior domain of .
Observe that (2.11) is a condition on near , as opposed to near zero. This difference from condition (2.2) is due to the behavior of the fundamental solution of Laplace’s equation near infinity in dimension versus higher dimensions. See Section 4 for a much more detailed study of this phenomenon. In cone-like domains, one must impose conditions on both near and to obtain the nonexistence of supersolutions, as we will see in Section 5.
3.1. Several properties of supersolutions
In this section we state in detail and comment on the hypotheses (H1)-(H5) under which we prove our main Liouville results. We also confirm that these hypotheses are satisfied by the -Laplacian operator and fully nonlinear Isaacs operators.
Recall the -Laplacian is defined by
For the sake of simplicity, we do not consider more general quasilinear operators, although our techniques apply for instance to operators of the more general form , with satisfying hypotheses (1.1)-(1.4) in .
A uniformly elliptic Isaacs operator is a function satisfying the uniform ellipticity condition
and which is positively homogeneous of order one:
Here is the set of -by- symmetric matrices, and and are the Pucci extremal operators defined for instance in . Equivalent to (3.1) and (3.2) is the requirement that be an inf-sup or a sup-inf of linear uniformly elliptic operators
over a collection of matrices such that for all and . Consult  for more on fully nonlinear, uniformly elliptic equations.
Our notion of solution is chosen to suit the particular operator under consideration. The -Laplacian is of divergence form, and thus we use the weak integral formulation. More precisely, a weak supersolution of the quasilinear equation
in a domain is a function with the property that for all nonnegative we have
When for an Isaacs operator , the appropriate weak notion of solution is that of viscosity solution. Namely, satisfies the inequality
in the viscosity sense in if for each and for which the map has a local minimum (maximum) at , we have
Henceforth, when we write a differential inequality such as , we intend that it be interpreted in the appropriate sense.
We now present a list of properties which these operators share and upon which our method is based. We will confirm below that the following hold in the case that is the -Laplacian operator or an Isaacs operator:
satisfies a weak comparison principle: if in a bounded domain , and on , then in ;
has fundamental solutions: there exist functions which satisfy in , and are approximately homogeneous in the sense of (3.6) below;
satisfies a quantitative strong comparison principle: with , if in a bounded and some compact subset of positive measure, then in any , where depends only on , and a lower bound for ;
satisfies a very weak Harnack inequality: if in a bounded and , then for each there exists such that for any point , we have ;
has no zero order term and possesses some homogeneity: precisely, we have for each ; for some and every ; if satisfies in and we set , then in .
The hypotheses (H2) and (H5) can be weakened, as will be obvious from the proofs below. Namely, we can assume that if satisfies in , then on , for some operator which satisfies the same hypotheses as , with constants independent of ; and that for some the operator satisfies the same hypotheses as with constants independent of . We can also assume that the functions be only subsolutions in some exterior domain in (except for the last statement in Theorem 4.3).
Let us now recall that both the -Laplacian and Isaacs operators satisfy conditions (H1)-(H5). We begin by recalling the weak comparison principle. For -Laplacian type opertors, we refer for example to [36, Corollary 3.4.2], while for Isaacs operators, this is a particular case of the results in [14, 16].
Let denote the -Laplacian or an Isaacs operator. Suppose that is a bounded domain, and and satisfy the inequalities
and on . Then in .
Another important property for our purposes is the availability of solutions of with given behavior at infinity. For , we denote
Let denote the -Laplacian or an Isaacs operator. Then there exist numbers and functions such that
and for some positive constants ,
It is well-known (and can be easily checked) that the -Laplacian satisfies the statement above with and . For the reader interested in extending the results in this paper to more general quasilinear operators, we note that results on the existence and behavior of singular solutions of quasilinear equalities can be found in the classical work of Serrin .
For Isaacs operators the question of existence, uniqueness, and properties of fundamental solutions was studied in detail in the recent work . In particular, the result above is a consequence of Theorem 1.2 in that paper. We remark that for nonlinear Isaacs operators we have , except in very particular cases. This is due to the fact that Isaacs operators are not odd, in general. For the Pucci extremal operators, for example, we have
Central for our method is the following quantitative (uniform) strong maximum principle. This result, while well-known (and fundamental to the regularity theory of linear elliptic equations developed by Krylov and Safonov, see Section 4 in ) is surprisingly under-utilized in the theory of elliptic equations.
Let denote the -Laplacian or an Isaacs operator. Assume that and are compact subsets of a bounded domain , with . Suppose that is nonnegative in and satisfies
where denotes the characteristic function of .
Then there exists a constant such that
Suppose in addition that on , where is as in the previous theorem, and . Then there exists a constant such that
For an Isaacs operator, we have
Let us give a proof for the -Laplacian. Suppose that (i) or (ii) is false so that there exists a sequence of compact subsets with , and a sequence of positive functions such that in and
for some sequence of points . Let solve the Dirichlet problem
Then by Theorem 3.1, in , and so we can replace by . For all we have
According to the estimates for the -Laplace equation (see [43, 20, 31]), we deduce that is bounded in for some . Therefore we may extract a subsequence of which converges to a function in . We may pass to limits in (3.8) to obtain in , as well as on (or on ). By the strong maximum principle (see , or Theorem 3.4 below) we conclude that either or in . In the case (ii), by the strong comparison principle (see Theorem 1.4 in ), either or in . To apply the strong comparison principle here, we must note that the gradient of never vanishes in .
The final ingredient of our proofs of the Liouville results is the weak Harnack inequality. For weak solutions of degenerate quasilinear equations it is due to Serrin  and Trudinger . In the nondivergence framework it was proved by Krylov and Safonov for strong solutions (see ), see also , while for viscosity solutions of Isaacs equations it was obtained by Caffarelli ; see also Theorem 4.8 in .
Let and in a bounded domain , where is the -Laplacian or an Isaacs operator. Then there exists depending only on and , such that for each compact we have
for some positive constant , which depends only on .
In some cases the use of this theorem can be avoided, at the expense of strengthening the hypotheses on , see for instance Remark 2.7 after the proof of Theorem 2.1.
We actually use only the following weaker result: for each there exists a constant such that for any nonnegative weak supersolution of in the annulus , and any ,
This is a consequence of the “very weak” Harnack inequality, which states that for every , there exists a constant such that for any nonnegative weak supersolution of in ,
This fact, though a consequence of the weak Harnack inequality, is interesting in its own right. For instance, it admits a proof which is considerably simpler than the proof of the weak Harnack inequality while being sufficient to imply the Hölder estimates for solutions of .
The reader is advised that in the rest of the paper only properties (H1)-(H5) will be used. In other words, the Liouville theorems stated in Section 4 are proved for any such that the inequalities can be interpreted in such a way that properties (H1)–(H5), or a subset of them, are satisfied.
3.2. Properties of minima of supersolutions on annuli
Our method for proving nonexistence theorems is based on the study of minima of supersolutions in annuli. In this section we obtain some preliminary estimates by comparing supersolutions of with the fundamental solutions of from property (H2).
Note that, given , and can be assumed to never vanish in , since if needed we simply add or subtract a constant from these functions. With this in mind, let us define the quantities
Assume satisfies (H1) and (H2). Suppose that satisfies
Then for some ,
First consider the case . Then and as . Observe that for every and , we may choose large enough that
By the weak comparison principle,
Sending and then , we discover that
The desired monotonicity of follows.
Next, suppose that . Recall that for and as . Thus for every and , we can find so large that
Using the weak comparison principle and sending , we deduce that
Now let to obtain , and the monotonicity of on the interval .
Suppose that . Then , and we may normalize so that
For any , we clearly have
By the weak comparison principle,
for each . Recalling (3.6), if is fixed sufficiently large so that
we obtain the third statement in (3.10), for .
Finally, we consider the case . Observe that
for any . By the weak comparison principle,
By fixing sufficiently large so that the quantity in the last parentheses is larger than one (recall we are in a case when as ), the second part of (3.10) follows. The lemma is proved. ∎
The following bounds on are an immediate consequence of (3.10).
Assume satisfies (H1) and (H2). Suppose that and satisfy
Then for some depending on , , , and , but not on ,