Any real-valued harmonic function on () which is bounded either from above or below is a constant. This is perhaps the earliest formulation of what is commonly known as the Cauchy–Liouville theorem (or simply Liouville’s theorem). Today there is an extensive amount of work that extends this basic result in many different directions. A beautiful account of the history, techniques and some recent results on Liouville-type problems can be found in .
Our goal in this paper is to investigate classes of functions for which the quasilinear PDE
admits constants as the only possible non-negative solutions in the entire Euclidean space (). In this case we will say that the PDE has a Liouville-type property. Here is the so-called ϕ-Laplacian which reduces to the well-known p-Laplacian when for .
In our attempt to understand the Liouville-type property of (1.1), we will focus on two classes of functions f. In the first part of the paper we investigate problem (1.1) when f belongs to a class of non-decreasing and non-negative functions whose growth at infinity is dictated by a Keller–Osserman-type condition. To elaborate on this, suppose that f is a non-decreasing continuous real-valued function defined on such that and for . It is well known (see ) that the only non-negative solution of in is the trivial solution if and only if
This result is essentially due to Keller  and Osserman , obtained independently. We will use an adaptation of condition (1.2) to the ϕ-Laplacian to obtain a sufficient condition on f in order for (1.1) to have the Liouville-type property.
In the second part of the paper we study another class of functions f that will lead to the PDE (1.1) having a Liouville-type property. To motivate this aspect of our investigation, we mention the paper , where the Liouville-type property of the p-Laplacian has been discussed extensively. Among many important results developed therein, we would like to mention the following result which is relevant to the work at hand. Suppose is subcritical in the sense that f satisfies
for some , where is the critical Sobolev exponent of . If there exists such that at infinity, then any non-negative solution of in is the trivial solution, see . In the recent paper , an adaptation of a method in  was used to obtain a result which complements the aforementioned Liouville-type result. More specifically, the following result was proved. Suppose is a differentiable function that has a root and satisfies
Then any positive solution of in must be a constant. As pointed out above, the method used in  is based on the work of McCoy [21, 20], who investigated the Liouville-type property for . This result in  is just one among many other Liouville-type results studied for elliptic equations, including some fully nonlinear higher-order equations. We also call attention to the nice paper  which discusses, among other things, Liouville-type results for general anisotropic quasilinear equations. The paper  contains Liouville comparison principles for quasilinear singular parabolic second-order partial differential inequalities.
The paper is organized as follows. In Section 2 we begin by stating some general assumptions on ϕ and by recalling some results from the literature used in our work. In Section 3 we show that (1.1) has the Liouville-type property when ϕ meets a suitable condition and f is continuous and non-decreasing, and satisfies a generalized Keller–Osserman condition. Some examples will be presented to illustrate the applicability of the main result of the section. Finally, in Section 4, we discuss a Liouville-type property of (1.1) when f belongs to a class of differentiable functions , which are not necessarily monotonic, may change sign and satisfy a condition that generalizes (1.3). An appropriate condition on ϕ will also be required. We conclude the section with some illustrative examples.
Given , let us set
We make the following assumptions:
Ψ is a strictly increasing function in .
There exist constants such that
Equations involving the ϕ-Laplacian have been used to model different physical phenomena. For instance, they appear in quantum physics  and in the modeling of nonlinear elasticity problems or in plasticity . We refer to the papers [26, 2, 6, 10, 11, 24, 25, 28, 27] for more details and other applications.
We begin with inequalities which follow directly from (ϕ3), namely,
for some increasing functions . In fact,
Inequalities (2.1), in turn, imply
Another immediate consequence of (2.1) is
In fact, this is a direct consequence of (2.3).
Given an open set , the Orlicz space
is a Banach space under the so-called Luxemburg norm
The Orlicz–Sobolev space is defined as the set of all weakly differentiable such that for all multi-indices with . This is a Banach space under the norm
As with the usual Sobolev spaces, is defined as the closure of in .
We define the local spaces and by
The dual space is equal to , where is the N-function given by
Let be an open set and be a continuous function. We say a weakly differentiable function is a sub-solution of the PDE
if and only if for every open and bounded subset , we have , with , such that
A weakly differentiable function is said to be a super-solution of (2.5) in Ω if and only if for every open and bounded subset , we have , with , such that the reverse inequality holds in (2.6) for all non-negative . A weakly differentiable function is said to be a solution of (2.5) in Ω if u is both a sub-solution and a super-solution of (2.5) in Ω.
We follow common practice and write
to indicate that v is a sub-solution and w is a super-solution of (2.5), respectively, in Ω.
As noted in the introduction, in this paper we are interested in the investigation of Liouville-type property for the PDE
where the absorption term will be required to satisfy appropriate conditions.
We recall two results that will be useful in this paper. We begin with the following comparison principle, which is taken from [23, Theorem 2.4.1 and Proposition 2.4.2].
Theorem A (Comparison principle).
If on , then in Ω.
The following regularity result of Lieberman [17, Theorem 1.7], reminiscent of the regularity results for p-Laplacian-type quasilinear equations of DiBenedetto , Lieberman  and Tolksdorff , will also be useful.
3 Monotonic non-negative absorption term
Let be a non-decreasing continuous function such that and for . In this section we extend a classical result of Keller  and Osserman  who showed independently that can not have a non-trivial solution in if f satisfies a growth condition at infinity that has come to be known as the Keller–Osserman condition. We refer to the paper  for an excellent account on this and other related Liouville-type results.
Since , where σ and ρ are the parameters in condition (ϕ3), we note that for , we have
Therefore, for , we have
Since , we see that the right-hand side of (3.1) is less than one whenever . Now let us note that
so that for , we have
Applying this to (3.1), we find that
If , then it is clear that
On the other hand suppose . If, in addition, , then we have
Consequently, in this case we find
The above discussion leads us to consider the following condition on the parameters σ and ρ in condition (ϕ3):
For easy reference, let us summarize the above discussion with regard to condition (ϕ4) in the following remark.
Let us suppose that condition (ϕ4) holds. Then direct computations lead to the following conclusions:
If , then for each ,
If , then
Let us now consider a non-decreasing function such that
Condition (3.2) is easily recognized as a generalization of the well-known Keller–Osserman condition when .
We point out that condition (3.2) is equivalent to
Let and fix . Using (2.4), together with the above inequalities, we have
In this section we study Liouville-type properties of solutions to the following equation:
We assume that is a continuous function such that and for . Our first Liouville-type result in this section is provided by the following theorem.
Let and let be arbitrary. Let v be a solution of
Let be the maximal interval of existence of v. Note that v cannot be a constant. We shall show that . Note that
Therefore, in , and is a non-decreasing function in , and hence is non-decreasing on as well. On multiplying both sides of (3.6) by , for any , we estimate
which again is a consequence of (2.2), shows that
On noting that , we use (2.4) to get
Integrating this last inequality on yields
Consequently, we see that
is a non-increasing function. Therefore, since , we have . Hence, from inequality (3.9), we obtain
Now let us suppose that . Taking the limit in (3.10) as , we find that
Therefore, we conclude that indeed , and as a consequence as . Let us set for , and suppose that u is any non-negative sub-solution of (3.5) in . Then, for such that is sufficiently small, we have on . Since w is a solution of (3.5) in the ball we invoke the comparison principle, Theorem A, to conclude
In particular, we have . Since is arbitrary, we find that , and since is arbitrary, we conclude in . ∎
(a) Let u be any sub-solution of (3.5) in . To show that in , it suffices to demonstrate that is a sub-solution of (3.5) in . Once this is proven, then we can invoke Theorem 3.2 to conclude that , and hence in . First, since , let us notice that . To see that is a sub-solution of (3.5), let We suppose that Ω is non-empty for otherwise there is nothing to prove. For each positive integer j, set , , where , with
Given , let with in . We have
Let us note that and
since . Therefore, is a sub-solution of (3.5), as claimed.
(b) Suppose that f is an odd function, and that u is a solution of (3.5) in . By what was proved in (a), we observe that in . On the other hand, it is easily seen that is a solution of (3.5) in . Therefore, in again, completing the proof that in . ∎
3.1 Some examples
(2) Let us now consider for . Computation shows that , and
As a consequence, we see that
Therefore, if the right-hand side in the above inequality is finite for some , then condition (3.2) holds, and therefore Theorems 3.2 and 3.3 hold provided that condition (ϕ4) is satisfied. According to Remark 3.4, if
then (ϕ4) holds if and only if
On the other hand, if
then (ϕ4) holds if and only if .
(3) Let for and . Then and . Moreover, we see that . Note that given , there exists a constant , sufficiently large, such that
Therefore, if there exists such that
4 Sign-changing absorption term
This section is devoted to the study of the Liouville-type property of the following equation for a given :
We do not make any monotonicity assumption on f, but we require f to be a function that satisfies a sub-critical-type condition (see condition (Cf) below).
To study the Liouville-type property of (4.1), we start by making some suitable assumptions on ϕ. Specifically, we require that and satisfies the following condition:
and are the constants in condition (ϕ3).
The main result of this section is the following Liouville-type theorem for the solutions of (4.1). This theorem extends the result in , where the special case , , was considered. In the proof of the theorem we will observe the Einstein summation convention over repeated indices.
If is a positive solution of (4.1) in , then u is a constant in .
Suppose that ϕ satisfies the assumptions of Theorem 4.1. If for some , then f satisfies (Cf), and therefore the only positive solution of (4.1) in is . We should also note that if f is a non-negative and non-increasing function on , then f satisfies (Cf), and therefore in this case only constants are the possible positive solutions of (4.1).
Proof of Theorem 4.1.
Let us first note that, as a consequence of the continuity of u and Theorem B (see also [13, Lemma 3.3]), we conclude that . Therefore, it follows that equation (4.1) is uniformly elliptic on open sets that are compactly contained in . Hence, we observe that in any open set that is compactly contained in (see [18, Corollary 2.2], with ). See also .
Let be an arbitrary but fixed point. We wish to show . Given , let us set
We note that in and . Therefore, J attains its maximum value on at some interior point . Suppose . Then Θ (and hence J) will be zero at . But then J (and therefore Θ) will be zero on . This would imply in , and in particular . Therefore, we assume that at . We recall that u is , where .
Now, at , we have
Let us first show that the matrix
is positive definite. Here is the identity matrix, and stands for the transpose of the matrix A. To see that H is positive definite, we first note that for any ,
If , then . If, on the other hand, , then on noting that and therefore
we have (recalling condition ϕ4)
Thus, in any case, we have shown that .
Now recalling that at , we have at . Therefore, at , we have
where is the unit vector in with 1 in the jth position. In the last equation we have used the simple fact that , where is an matrix. Thus, at , we have
Recalling that (4.1) is invariant under rotation, we choose coordinates so that at the point , we have
Through direct computation, we note that
This, together with inequality (4.3), implies
To obtain (4.5), we have used the easily verifiable identities
Moreover, from (4.2), we obtain
Therefore, at , we have
Since in , we have
Let us now notice that for , we have
From the above computation we see that at ,
We now differentiate (4.11) with respect to the variable . On evaluating the resulting expression at , we find
Let us observe the following.
To proceed further, we need the following inequality.
Rearranging the terms in the last inequality, we find
We recall from (4.8) that for , we have , and therefore
Rearranging the terms in the above inequality yields
We recall from (4.8) that , that is
Inserting this in the last inequality gives
Further rearrangement leads to
To continue it will be convenient to introduce the following notations:
In other words, we have
From (4.2), we recall that at , we have
Therefore, at , we have the estimate
Since , we conclude that, at , we have
Thus, at , we have the estimate
Letting , we find that at . Since was arbitrary, we conclude that on , as desired. The proof is complete. ∎
4.1 An example
Let us illustrate the above theorem with a couple of examples. The simplest case occurs when for some . In this case, , and
Now let us consider for . Recall that in this case and . Let us note that2
We now proceed to find conditions under which (ϕ5) holds. We have
Let us observe that
Therefore, we see that
So let us first suppose that . Then inequality (4.22) reduces to
then condition (ϕ5) holds.
Now, given , let us set
We remark that for . We now summarize the above discussion in the following corollary.
Given , suppose that f satisfies condition (Cf) with . If , then any non-negative entire solution of
is a constant on .
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About the article
Published Online: 2017-08-24
This work was supported by ISP (International Science Program) of Uppsala University, Sweden.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 725–742, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0158.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0