Liouville's type results for singular anisotropic operators

1.1 A brief recall on the known regularity

Let aside the boundedness of solutions, Harnack inequality and oscillation estimates for operators with measurable and bounded coefficients as (1.3) are generally not at hand. For solutions to the prototype equation related to (1.3) (see (1.6)), requiring only the local boundedness of solutions, the result [5] is the most general in terms of sparseness of p i s (note that p max < p ¯ * is just a sufficient condition); on the other hand, still for the prototype equation, the intrinsic Harnack estimate of [10] hold in a parabolic range of finite speed of disturbances (see also [18]). The scenario is completely different when it comes to operators with measurable and bounded coefficients. A general Hölder continuity stability result was provided in [16], accompanied by specific Hölder continuity findings tailored to a narrower subset of exponents (and coefficients) for the elliptic equations (1.3). Specifically, in [34], the authors prove that solutions to A [ 2 , p ] u = 0 with p 1 = … = p N 1 = p > 2 and p N = 2 are locally Hölder continuous, while in [33], the focus shifts to the case where p 1 = 2 and p 2 = … = p N = p < 2 . Observe that in all these cases, s = 1 . Finally, in the special case of operator A [ 2 , p ] , in [11,32], the authors proved an intrinsic Harnack-type inequality (see Theorem 2.3) for local weak solutions to operators A [ 2 , p ] u = 0 , where s can vary in { 1 , … ( N − 1 ) } and 1 < p < 2 . The authors in [11] use a parabolic approach to the expansion of positivity, concluding interior Hölder continuity, integral, and pointwise Harnack inequalities. Then, in [3], following an approach of Moser [42], the authors show that the Harnack inequality can be proved without the continuity of solutions and that, in turn, this implies Hölder continuity of solutions. In the same study, authors showed that the solutions are automatically continuous because of their energy estimates, following the method presented in [30]. Continuity of local weak solutions has been shown for a general class of parabolic anisotropic operators in [17] with a similar strategy, while following the method of the study [32] in the articles [8,13] for doubly non-linear anisotropic equations. Further information can be found in [35].

1.2 Liouville-type results

Anisotropic operators as (1.1) do not behave like double-phase operators as

Indeed, the vanishing of a directional derivative ∣ ∂ i u ∣ provides a lack of information in the energy estimates; while in (1.5) all the gradient gets involved into a Caccioppoli-type inequality (that, then, has its own hard difficulties due to the ( p , q )-growth). From this respect, the operator prefers some directions and competes along other ones. This scheme is much more in line with the works of Lanconelli [20,23,24], where the diffusion is dictated along preferred vector fields, and the homogeneity of the operator with respect to the group of translations and dilations is strongly used. In this context, a Liouville-type property is generally found in the context of hypoelliptic operators (see, for instance, [26–28]) and in general linear operators that however move in a space-geometry dictated by non-canonical vector fields.

In the framework of local weak solutions to anisotropic operators as (1.3), there are very few rigidity results. The only one we are aware of, up to our knowledge, in the elliptic context, is the result in [10], which states that local weak solutions in R N of

where nonetheless the exponents are obliged to satisfy the interval 2 < p i < p ¯ ( N + 1 ) ⁄ N , for all i = 1 , … , N , are constant if they are bounded from below. In this article, we extend this result to local weak solutions in R N to the equation A [ 2 , p ] u = 0 , from one side allowing measurable coefficients in the formulation of A , while from the other side constraining the anisotropy to exponents 2 , p . Then, in the parabolic setting, the Liouville-type results available seem confined in [7,15], being naturally different in terms of assumptions because of the existence of nontrivial counter-examples. In the parabolic framework, a Liouville property is usually given by the global boundedness of solutions, or one bound (that, in line with [31], we call it one-side Liouville theorem), and an asymptotic condition for infinite future/past times. This is reflected, from the parabolic point of view, that in the context of hypoelliptic operators (see, for instance, [25]), it is possible to use the Harnack inequality along some preferred paths called ℒ -parabolic trajectories to infer that the solution tends (in time) asymptotically to its infimum. Similarly, this is done in the case of parabolic anisotropic equations (see [7]), where, in this case, the directions preferred are the ones of the self-similar geometry of the non-linear operator.

In this article, we use a similar approach for the elliptic non-linear operators (1.2), looking at the first s -variables as the time variables and developing the Harnack estimates found in [11] along the natural intrinsic geometry dictated by the equation.

1.3 Intrinsic geometry

We consider the prototype equation to (1.1) as a special case of the full anisotropic analogue (1.6), with p i = 2 for i = 1 , … , s and 1 < p i = p < 2 on the remaining components. This last equation suffers heavily from the combined effect of singular and degenerate behavior, even when for instance all p i s are greater than two. This is because the natural intrinsically scaled geometry of the equation that maintains invariant the volume ∣ K ∣ = ρ N can be shaped on anisotropic cubes as

where M is a number depending on the solution u itself (indeed the epithet intrinsic) that vanishes as soon as u vanishes. Therefore, when M approaches zero, for those directions whose index satisfies p i > p ¯ , the anisotropic cube K shrinks to a vanishing measure, while for the remaining ones, it stretches to infinity. For a detailed description of this geometry and its derivation through self-similarity, we refer to [8], where the evolutionary, fully anisotropic prototype equation is considered (see also [18] for the singular case).

Hence, the term intrinsic means that the geometry of the set depends on the function u itself. This technique is of vital importance in investigating the local behavior of anisotropic equations (and in this, it differs from double-phase elliptic equations). In fact, one of the main ideas to prove the Harnack inequality for evolutionary non-linear operators of p -growth is to use a parabolic transformation to map solutions to a similar equation that has an exponential dependence on time only on the non-homogeneous terms. In this way, it is possible to apply the techniques of the elliptic case to prove the results that can be brought back to the original equation. However, this idea can prove particularly difficult in some cases (for example, the anisotropic p -laplacian). The main problem arises from the fact that the new geometry may have different behavior with respect to the different directions of diffusion. Taking as an example the work of [7], one can see that the definition of the intrinsic cube K has the benefit of having the volume independent of M (and so, independent of the function if M ≈ ‖ u ‖ L ∞ ). However, it has the downside of having singular or degenerate behavior depending on the range of p i . So when M goes to infinity, the set stretches or vanishes along the respective coordinates. This problem may be fixed by choosing a different scaling as in [8]. In fact, one may consider a similar set, such that the geometry associated with it degenerates in only one way. This can be done by choosing the intrinsic cube and the corresponding intrinsic cylinder as

when assuming p i > 2 for all ∀ 1 ≤ i ≤ N . However, with this choice, we see that K ρ ( M ) is no longer a volume-preserving set, but its volume depends on M , and from p i s, one can see that the set vanishes as soon as M vanishes.

The idea of the scaling of [11] is that, by considering the first s variables as the ones of time, one does not have to use an exponential shift and the volume of the set does not come into play. Thus, for equation (1.7), the strategy is to view the polydiscs of two parameters θ , ρ > 0 such as

for a point x 0 ∈ R N denoted by x 0 = ( x 0 ′ , x 0 ″ ) where x 0 ′ ∈ R s and x 0 ″ ∈ R N − s . We will say Q θ , ρ is an intrinsic polydisc when θ depends on the solution u itself, with a relation

We will call the first s variables the nondegenerate variables and the last ( N − s ) the singular variables. By adopting this method from the theory of singular parabolic equations, we obtain an intrinsic geometry that degenerates monotonically with the scalar M , which allows us to accommodate the degeneracy into play.

1.4 Structure of the article

In Section 2, we establish the functional framework, provide definitions and geometrical setting, and outline the necessary tools for our main results. Section 3 is dedicated to presenting our two principal Liouville-type results, each addressing solutions bounded on one side and both sides.

Notation:

• When considering a measurable subset Ω of R N , we represent its Lebesgue measure by ∣ Ω ∣ . To say that Ω is an open bounded set, we employ the notation Ω ⊂ ⊂ R N .

• For a measurable function u , we define the essential infimum and supremum as inf u and sup u , respectively, in the set of consideration. When u : Ω → R and a ∈ R , we omit the domain when considering sub or super level sets, denoted by [ u ≤ a ] = { x ∈ E : u ( x ) ≤ a } . If u is defined on some open set Ω ⊂ R N , we denote the distributional derivatives as ∂ i u = ∂ u ⁄ ∂ x i .

• We adopt the convention that a positive constant γ , which relies solely on the given data, denoted as γ = γ ( N , 2 , p , C 1 , C 2 , C ) and may fluctuate throughout calculations.

• We introduce the following quantities, which will be clearly defined as long as the function u remains bounded:

  μ + ( λ , k ) = sup Q λ , k ( x 0 ) u , μ − ( λ , k ) = inf Q λ , k ( x 0 ) u , ω ( λ , k ) = μ + ( λ , k ) − μ − ( λ , k ) , μ + = sup R N u , μ − = inf R N u and ω = μ + − μ − .