Interval Oscillation Theorems for the Weighted -Sub-Laplacian Equation in the Heisenberg Group
In this paper, we derive a Riccati-type inequality in the Heisenberg group . Based on it, some oscillation criteria are established for the weighted -sub-Laplacian equations in . Our results generalize the oscillation theorems for -sub-Laplacian equations in to ones in .
In this paper, we consider the nonlinear degenerate elliptic equation in the Heisenberg group : where , , is an outer region in , denotes the Heisenberg gradient (see (19)), and and are to be specified later.
In the qualitative theory of nonlinear partial differential equations, one of the important problems is to determine whether or not solutions of the equations are oscillatory. For the second-order linear ordinary differential equation,
A classical result of the oscillation is the famous Fite-Wintner theorem which states that if , then the solutions of (2) are oscillatory (see ). In , Kamenev studied the oscillatory behavior of the solutions of (2) under the assumption that .
Soon after, Fite-Wintner’s theorem and Kamenev’s theorem were extended to various forms of second-order differential equations. In , by using the Riccati-type transformation, Noussair and Swanson extended Fite-Wintner’s theorem to the equation:
Usami  established Fite-Wintner-type theorem to the quasilinear elliptic equation in divergence form:
It knows that the -Laplacian equations play a critical role in physical phenomena. We refer the readers to Díaz  for detailed references on physical background of the -Laplacian equations. In this paper, we derive several oscillation criteria for the weighted -sub-Laplacian equation in . One of the difficulties is that there does not exist a good divergence formula in as in . In this paper, we overcome this difficulty.
Before stating our main results, we introduce some notations and notions. For positive constants , we denote
where denotes the norm in (see (22)). A domain is called the outer region in if there exists a positive constant such that . Let us restrict our attention to the nontrivial solution of (1), that is, to the solution satisfying
Now, we give a definition.
Definition 1. Let
We call that belongs to the function class , if and there exist and such that
for and for all
In this paper, we always assume that the following conditions are satisfied.
The coefficient matrix is a real symmetric positive definite matrix function (i.e., it is the ellipticity condition in ) with , the smallest (necessarily positive) eigenvalue of is denoted by , and there exists a function such that where .
For convenience, denote where denotes the sphere in with the center and the radius and denotes the -dimensional Hausdorff measure in (see ): where is the homogeneous dimension of and ( is a constant) denotes the area of unit sphere in .
One of the main results is the following.
Theorem 2. Assume that for any , there exist and such that Then, (1) is oscillatory.
Denote and take in (10), where is a constant. The following is a Kamenev-type oscillation criterion.
Theorem 3. Assume . If for each , there exist such that Then, (1) is oscillatory.
The paper is organized as follows. In Section 2, we collect some well-known results for the Heisenberg group and introduce two lemmas. Section 3 is devoted to the proofs of the Riccati-type inequality. The proofs of Theorems 2 and 3 are given in Section 4.
The Heisenberg group is (or ) endowed with the group law defined by where . The group plays the important roles as in conformal geometry, geometry of several complexes, and harmonic analysis (e.g., see Folland and Stein in ).
The left invariant vector fields on are of the form
The family satisfies Hörmander’s rank condition (see ). The Heisenberg gradient of a smooth function is defined by
The divergence of a smooth vector value function on is defined by
For , the usual divergence on is
The norm for is
With the norm, the distance between two points and in is defined by where denotes the inverse of with respect to , that is, .
The sphere of radius centered at the origin of is the set: and the open ball of radius centered at is the set:
Now, we first introduce two well-known lemmas.
Lemma 4 (refer to ). If and are nonnegative constants and , then
Lemma 5 (the divergence formula in ). Let be a bounded domain in with boundary and denote the unit outward normal to . For any vector field , we have where is the identity matrix of .
The following lemma plays a critical role in proving the Riccati-type inequality. The proof is similar to Theorem 2.3 in , and we omit it.
Lemma 6. For any vector field and , we have
3. A Riccati-Type Inequality
In this section, we establish a Riccati-type inequality and then prove two lemmas.
Lemma 7 (type inequality). Suppose that is the nontrivial solution of (1) with for . Let Then,
Proof. It easily knows that and similarly, Hence, Combining (1) and (32), we have that is, By integrating (39) over , it follows Using (28) and (10), it implies For in (32), we have It yields In view of , we have By (30) and (33), we have By combining (41), (44), (45), and (11), it yields The proof is complete.
Using Lemma 7, we have the following.
Proof. Changing to in (34), multiplying (34) by , and integrating from to , we have in view of and that Take Then, by (27), we get By combining (48), it shows By letting and dividing both sides by , it follows (47).
4. Proofs of the Main Results
The following lemma is useful for proving Theorem 2.
Proof. Suppose that the statement is incorrect; without loss generality, we may assume that there exists a solution of (1) such that From Lemmas 8 and 9, it implies that (47) and (52) hold. Using them, we have which contradicts to (12). Thus, the claim is true.
Now, we give the following.
Proof of Theorem 1. Take the sequence
By assumptions, we see that for each , there exist such that and (12) holds. In view to Lemma 10, we conclude that every nontrivial solution of (1) has at least one zero in . By noting , it follows that every solution has arbitrarily large zeros. Hence, (1) is oscillatory.
As an immediate consequence of Theorem 2, the following result is true.
Proof of Theorem 2. From Definition 1 and the definitions of and , we get Thus, it follows Noting , we see By combining (63) and (15), it yields which implies (56). Similarly, (57) holds by combining (63) and (16). From Corollary 11, (1) is oscillatory.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research is supported by the National Natural Science Foundation of China (Grant No. 11771354).
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