An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group

Abstract

The aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$. We first derive a critical inequality in $H^{n}$. Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$.

1 Introduction

In this paper, we give an oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$

$$ {\nabla _{H}} \cdot \bigl(A(z,t){\nabla _{H}}v(z,t) \bigr)+{B^{T}}(z,t){\nabla _{H}}v(z,t)+C(z,t)f \bigl(v(z,t) \bigr)=0 $$

(1.1)

and the corresponding homogeneous ordinary differential equation

$$ \bigl(p(r)u'(r) \bigr)'+ b(r)u'(r)+c(r)u(r)=0, $$

(1.2)

where $(z,t) \in \Omega$, Ω is the outer region in $H^{n}$, ${\nabla _{H}}$ denotes the Heisenberg gradient (see (2.1)), $r= \vert (z,t) \vert _{H}$, $\vert (z,t) \vert _{H}$ denotes the norm in $H^{n}$ (see (2.4)), and $A(z,t)$, $B(z,t)$, $C(z,t)$, $f(v)$, $p(r)$, $b(r)$, and $c(r)$ are to be specified later.

The oscillation result obtained in this paper implies that if Eq. (1.2) is oscillatory. Then the judgement of oscillation for Eq. (1.1) can be transformed into comparing the relationship between the corresponding coefficients in Eqs. (1.1) and (1.2).

The oscillation properties for ordinary differential equations in the real line R have been investigated by many authors with different methods (see [1, 5, 6, 11, 12, 16] and references therein). A well-known result was shown by Picone [11], in which an identity (now it is called the Picone identity) between

$$ \bigl(p(t)y'(t) \bigr)'+q(t)y(t)=0 $$

(1.3)

and

$$ \bigl(P(t)x'(t) \bigr)'+Q(t)x(t)=0 $$

(1.4)

was established, where $p(t),P(t) \in {C^{1}}([a,\infty ),R)$, $q(t),Q(t)\in C([a,\infty ),R)$, and a is a positive constant. Then the Sturm comparison principle for Eqs. (1.3) and (1.4) was derived. Finally, the oscillation theorem for Eq. (1.4) followed.

The oscillation criteria for linear or nonlinear partial differential equations have also been concerned extensively. Noussair [10] concluded some oscillation criteria for semilinear elliptic inequalities by the Riccati transformation. Xu [14] formulated oscillation theorems for elliptic equations with damping by using the way similar to [10]. For more results, we refer the readers to [8, 9, 13] and references therein. Recently, Zhuang [15] obtained oscillation criteria for second-order nonlinear elliptic differential equations.

Motivated by the ideas in [10, 11], and [15], in this paper, we deduce an oscillation criterion for (1.1) in $H^{n}$. The difficulty is that there is no good divergence formula in $H^{n}$ as in $R^{n}$. Fortunately, we find a good estimate (see Lemma 2.3) to help us build the criterion.

Before stating our main result, we introduce some notations and notions. For positive constants a, $a_{1}$, $a_{2}$, we denote

$$\begin{aligned}& G[a_{1},a_{2}]= \bigl\{ (z,t) \in H^{n}: a_{1} \le \bigl\vert (z,t) \bigr\vert _{H}\le {a_{2}} \bigr\} , \\& G[a,+\infty )= \bigl\{ (z,t) \in H^{n}: \bigl\vert (z,t) \bigr\vert _{H} \ge a \bigr\} . \end{aligned}$$

We say that Ω is an outer region in $H^{n}$ if there exists a positive constant $a_{0}$ such that $G[a_{0}, +\infty ) \subset \Omega$. Let us restrict our attention to the nontrivial solutions $v(z,t)$ of Eq. (1.1), that is, to the solutions $v(z,t)$ of (1.1) satisfying

$$ \sup \bigl\{ \bigl\vert v(z,t) \bigr\vert :(z,t) \in \Omega \bigr\} >0. $$

A nontrivial solution to Eq. (1.1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

Hereafter, we always assume that the following conditions are satisfied:

$(C_{1})$:

the coefficient matrix $A(z,t)=(a_{ij}(z,t))_{2n \times 2n}$ is real symmetric positive definite with $a_{ij} \in C_{\mathrm{loc}}^{{2}}(\Omega , R)$, the largest (necessarily positive) eigenvalue of $A(z,t)$ is denoted by $\lambda _{\max }(z,t)$, and there exists a function $\lambda (r) \in {C^{1}}(R^{+}, R^{+})$ such that

$$ \lambda (r) \ge \max_{\vert (z,t) \vert _{H}=r} {\lambda _{ \max }}(z,t), \quad r > {a_{0}}; $$

$(C_{2})$:

${B^{T}}(z,t)=(b_{i}( z,t))_{1 \times 2n}$, ${b_{i}}(z,t)\in C_{\mathrm{loc}}^{ {2}}(\Omega , R)$;

$(C_{3})$:

$C(z,t) \in C_{\mathrm{loc}}^{{1}}(\Omega , R)$;

$(C_{4})$:

$f\in C^{1}(R,R)$, $vf(v)>0$ and $f'(v)\ge k>0$ for $v \ne 0$ and a constant k;

$(C_{5})$:

$p(r) \in {C^{1}}([a_{0},\infty ),(0,\infty ))$, $b(r), c(r) \in C([a_{0}, \infty ),R)$.

We denote

$$ Q(r)= \int _{S_{H}(o,r)} { \biggl(C(z,t)-\frac{1}{4k} \bigl(B^{T}A^{-1}B \bigr) (z,t)- \frac{1}{2k}{\nabla _{H}} \cdot B(z,t) \biggr)} \,dH, $$

(1.5)

where ${S_{H}}(o,r)$ denotes the sphere in $H^{n}$ with center $o=(0,0)$ and radius r, and dH denotes the 2n-dimensional Hausdorff measure in $R^{2n+1}$ (see [3]);

$$ g(r)=\frac{{\wp {\alpha _{\wp }}\lambda (r)}}{{k{r^{-2n-1}}}}, $$

(1.6)

where $\wp =2n+2$ is the homogeneous dimension of $H^{n}$, and $\wp \alpha _{\wp }$ ($\alpha _{\wp }$ is a constant) is the area of unit sphere ${S_{H}}(o,1)$ in $H^{n}$.

Now we state our main result.

Theorem 1.1

Assume that there exists $a>0$ such that for any $r \in [a, +\infty )$,

$$\begin{aligned}& p(r)> \bigl(4r^{2}+1 \bigr)g(r), \\& Q(r)\ge \frac{b^{2}(r)}{4(p(r)-(4r^{2}+1)c(r))}+c(r), \end{aligned}$$

and the equality is not true everywhere in any subinterval of $[a, + \infty )$. Then Eq. (1.1) is oscillatory when Eq. (1.2) is oscillatory.

The paper is organized as follows. In Sect. 2, we collect some well-known results for the Heisenberg group and prove a good estimate. Section 3 is devoted to proofs of the Picone-type differential inequality and the Sturm-type comparison principle. The proof of Theorem 1.1 is presented in Sect. 4.

2 Preliminaries

The Heisenberg group $H^{n}$ is $R^{2n+1}$ (or ${C^{n}} \times R$) endowed with the group law ∘ defined by

$$ \bar{\xi }\circ \xi = \Biggl( {x + \bar{x},y + \bar{y},t + \bar{t} + 2 \sum _{i = 1}^{n} { ( {{x_{i}} {{ \bar{y}}_{i}} - {y_{i}} {{ \bar{x}}_{i}}} )} } \Biggr), $$

where $\xi = ( {{x_{1}},{x_{2}}, \ldots ,{x_{n}},{y_{1}},{y_{2}}, \ldots ,{y_{n}},t} ): = ( {x,y,t} ): = ( {z,t} ) \in {R^{2n}} \times R$, $\bar{\xi }= ( {\bar{z},\bar{t}} )$. The group $H^{n}$ plays important roles as $R^{n}$ in conformal geometry, several complex geometry, and harmonic analysis (e.g., see Folland and Stein [2]).

The left-invariant vector fields on $H^{n}$ are of the form

$$ {X_{j}} = \frac{\partial }{{\partial {x_{j}}}} + 2{y_{j}} \frac{\partial }{{\partial t}},\quad\quad {Y_{j}} = \frac{\partial }{{\partial {y_{j}}}} - 2{x_{j}} \frac{\partial }{{\partial t}}, \quad j = 1,2, \ldots ,n,\quad\quad T = \frac{\partial }{{\partial t}}. $$

The family $\{ {{X_{1}}, \ldots ,{X_{n}},{Y_{1}}, \ldots ,{Y_{n}}} \}$ satisfies Hörmander’s rank condition (see [4]). The Heisenberg gradient of a smooth function u is defined by

$$ {\nabla _{H}}u = ( {{X_{1}}u, \ldots ,{X_{n}}u,{Y_{1}}u, \ldots ,{Y_{n}}u} ). $$

(2.1)

The divergence of a smooth vector value function $F = ( {{F_{1}}, \ldots ,{F_{2n}}} )$ on $H^{n}$ is defined by

$$ {\nabla _{H}} \cdot F = {X_{1}} {F_{1}} + \cdots + {X_{n}} {F_{n}} + {Y_{1}} {F_{n + 1}} + \cdots + {Y_{n}} {F_{2n}}. $$

(2.2)

For $F = ( {{F_{1}}, \ldots ,{F_{2n + 1}}} )$, the usual divergence divF on $R^{2n+1}$ is

$$ \operatorname{div} F = \frac{{\partial {F_{1}}}}{{\partial {x_{1}}}} + \cdots + \frac{{\partial {F_{n}}}}{{\partial {x_{n}}}} + \frac{{\partial {F_{n + 1}}}}{{\partial {y_{1}}}} + \cdots + \frac{{\partial {F_{2n}}}}{{\partial {y_{n}}}} + \frac{{\partial {F_{2n + 1}}}}{{\partial t}}. $$

(2.3)

The norm ${ \vert \xi \vert _{H}}$ for $\xi \in {H^{n}}$ is

$$ { \vert \xi \vert _{H}} = { \bigl( {{{ \bigl( {{x^{2}} + {y^{2}}} \bigr)}^{2}} + {t^{2}}} \bigr)^{\frac{1}{4}}}. $$

(2.4)

With this norm, the distance between two points ξ and η in $H^{n}$ is defined by

$$ {d_{H}} ( {\xi ,\eta } ) = { \bigl\vert {{\eta ^{ - 1}} \circ \xi } \bigr\vert _{H}}, $$

where $\eta ^{-1}$ denotes the inverse of η with respect to ∘, that is $\eta ^{-1}=-\eta$.

The sphere of radius $r > 0$ centered at the origin $o = (0,0)$ of $H^{n}$ is the set

$$ {S_{H}}(o,r) = \bigl\{ {\xi \in {H^{n}}:{d_{H}} ( {\xi ,o} ) = r} \bigr\} , $$

and the open ball of radius $r>0$ centered at o is the set

$$ {B_{H}}(o,r) = \bigl\{ {\xi \in {H^{n}}:{d_{H}} ( {\xi ,o} ) < r} \bigr\} . $$

From [3] we know that the area of ${S_{H}}(o,r)$ is

$$ \bigl\vert {{S_{H}}(o,r)} \bigr\vert = \wp {\alpha _{\wp }} {r^{2n + 1}}, $$

where ${\alpha _{\wp }}$ is the volume of ${B_{H}}(o,1)$.

For simplicity, we denote ${B_{H}}(o,r)$ and ${S_{H}}(o,r)$ by $B_{r}$ and $S_{r}$, respectively.

The value of ${\alpha _{\wp }}$ can be concretely computed. In fact, let

$$ z = \rho z' \quad \text{for } \bigl\vert {z'} \bigr\vert = 1. $$

Then

$$ \begin{aligned} {\alpha _{\wp }} &= \bigl\vert {{B_{H}}(0,1)} \bigr\vert = \int _{{{ \vert z \vert }^{4}} + {t^{2}} < 1} {{{ }}\,dz\,dt} = \frac{{2{\pi ^{n}}}}{{\Gamma ( n )}} \int _{{{ \vert \rho \vert }^{4}} + {t^{2}} < 1} {{{ }} {\rho ^{2n - 1}}\,d\rho \,dt} \\ &= \frac{{{\pi ^{n}}}}{{\Gamma ( n )}} \int _{{r^{2}} + {t^{2}} < 1,r > 0} {{{ }} {{{r}}^{n - 1}}\,dr\,dt} = \frac{{2{\pi ^{n}}}}{{\Gamma ( n )}} \int _{0}^{1} { \int _{0}^{ \frac{\pi }{2}} {{s^{n}} {{\sin }^{n - 1}}\theta } } \,ds\,d\theta \\ &= \frac{{2{\pi ^{n}}}}{{ ( {n + 1} )\Gamma ( n )}} \int _{0}^{\frac{\pi }{2}} {{{\sin }^{n - 1}}\theta } \,d \theta = \frac{{{\pi ^{n}}}}{{ ( {n + 1} )\Gamma ( n )}}B \biggl( {\frac{n}{2},\frac{1}{2}} \biggr) \\ &= \frac{{{\pi ^{n + \frac{1}{2}}}}}{{ ( {n + 1} )\Gamma ( n )}} \frac{{\Gamma ( {\frac{n}{2}} )}}{{\Gamma ( {\frac{n}{2} + 1} )}}. \end{aligned} $$

Now we first introduce two well-known lemmas.

Lemma 2.1

(Lagrange’s identity [7])

For any two sequences $\{ {{a_{i}}} \}$ and $\{ {{b_{i}}} \}$, we have

$$ \Biggl( {\sum_{i = 1}^{n} {{a_{i}}^{2}} } \Biggr) \Biggl( { \sum _{i = 1}^{n} {{b_{i}}^{2}} } \Biggr) = { \Biggl( {\sum_{i = 1}^{n} {{a_{i}} {b_{i}}} } \Biggr)^{2}} + {\sum _{1 \le i < j \le n} { ( {{a_{i}} {b_{j}} - {a_{j}} {b_{i}}} )} ^{2}}. $$

(2.5)

Lemma 2.2

(The divergence formula in $H^{n}$ [2])

Let $\Omega _{0}$ be a bounded domain in $H^{n}$ with $C^{1}$ boundary $\partial {\Omega _{0}}$, and let ν denote the unit outward normal to $\partial {\Omega _{0}}$. For any ${C^{1}(\Omega _{0})}$ vector field $V=(V_{1},\ldots,V_{2n})$, we have

$$ \int _{{\Omega _{0}}} {{\nabla _{H}} \cdot V} \,dz\,dt = \int _{\partial { \Omega _{0}}} {MV} \cdot \nu \,dH, $$

(2.6)

where

$$M={\left(\begin{array}{cc}{I}_n& 0\\ 0& I_n\\ 2y& -2x\end{array}\right)}_{(2n+1)\times 2n},$$

and $I_{n}$ is the identity matrix of $R^{n}$.

Based on these lemmas, we get the following critical estimate.

Lemma 2.3

For any ${C^{1}(B_{r})}$ vector field $V=(V_{1},\ldots,V_{2n})$, we have

$$ \Vert MV\cdot \nu \Vert ^{2}_{L^{1}(S_{r})}\le \wp {\alpha _{\wp }} {r^{2n+1}} \bigl(4r^{2}+1 \bigr) \Vert V \Vert ^{2}_{L^{2}(S_{r})}. $$

(2.7)

Proof

A direct calculation shows that

$$ MV = \bigl( {{V_{1}},{V_{2}}, \ldots ,{V_{2n}},2 ( {{y_{1}} {V_{1}} + \cdots + {y_{n}} {V_{n}}} ) - 2 ( {{x_{1}} {V_{n + 1}} + \cdots + {x_{n}} {V_{2n}}} )} \bigr). $$

Then by (2.5) we have

$$\begin{aligned} { \vert {MV} \vert ^{2}} - { \vert V \vert ^{2}} &= 4{ \bigl( { ( {{y_{1}} {V_{1}} + \cdots + {y_{n}} {V_{n}}} ) - ( {{x_{1}} {V_{n + 1}} + \cdots + {x_{n}} {V_{2n}}} )} \bigr)^{2}} \\ &\le 4 \bigl( {{y_{1}}^{2} + \cdots + {y_{n}}^{2} + {{ ( { - {x_{1}}} )}^{2}} + \cdots + {{ ( { - {x_{n}}} )}^{2}}} \bigr) \bigl( {{V_{1}}^{2} + \cdots + {V_{n}}^{2}} \bigr) \\ &\le 4 \vert \xi \vert _{H}^{2}{ \vert V \vert ^{2}} , \end{aligned}$$

that is,

$$ { \vert {MV} \vert ^{2}} \le \bigl( {4 \vert \xi \vert _{H}^{2} + 1} \bigr){ \vert V \vert ^{2}}. $$

(2.8)

By (2.6) and Hölder’s inequality we have

$$ \Vert MV\cdot \nu \Vert ^{2}_{L^{1}(S_{r})}\le \wp {\alpha _{\wp }}r^{2n+1} \Vert MV \Vert ^{2}_{L^{2}(S_{r})}. $$

Combining (2.8), we obtain (2.7). □

3 Picone-type differential inequality and Sturm-type comparison principle

In this section, we derive a Picone-type differential inequality (Lemma 3.1), and then we prove a Sturm-type comparison principle (Lemma 3.2).

Lemma 3.1

(Picone-type differential inequality)

Let $v(z,t)$ and $u(r)$ be the solutions of Eqs. (1.1) and (1.2) respectively, and denote

$$ W(z,t)=\frac{1}{f(v)}(A\nabla _{H} v) (z,t)+\frac{1}{2k}B(z,t). $$

If there exists $a>0$ such that

$$ v(z,t) \ne 0 \quad \textit{for } (z,t) \in G[a, +\infty ) $$

and

$$ p(r) \ne \bigl( {4{r^{2}} + 1} \bigr)g(r) \quad \textit{for } r \in [ {a, + \infty } ), $$

then

$$ \begin{aligned} & \biggl( p(r)u'(r)u(r)-u^{2}(r) \int _{S_{r}} {MW(z,t)\cdot \nu \,dH} \biggr)^{\prime } \\ &\quad \ge \biggl({\frac{u(r)}{\sqrt{(4r^{2}+1)g(r)}} \int _{S_{r}} {MW(z,t) \cdot \nu }\,dH - u'(r )\sqrt{ \bigl(4r^{2}+1 \bigr)g(r)} \biggr)}^{2} \\ & \quad\quad {} + \bigl(p(r)- \bigl(4r^{2}+1 \bigr)g(r) \bigr) \biggl(u'(r)- \frac{b(r)u(r)}{2(p(r)-(4r^{2}+1)g(r))} \biggr)^{2} \\ & \quad\quad{} + \biggl(Q(r)-\frac{b^{2}(r)}{4(p(r)-(4r^{2}+1)g(r))}-c(r) \biggr){u^{2}(r)}. \end{aligned} $$

(3.1)

Proof

Without loss of generality, we assume that

$$ v(z,t) > 0 \quad \text{for } ( {z,t} ) \in G[ {a, + \infty } ). $$

In view of Eq. (1.1), we get

$$ \begin{aligned} &\nabla _{H} \cdot W(z,t) \\ &\quad = \nabla _{H}\cdot \biggl(\frac{1}{f(v)}(A \nabla _{H}v) (z,t)+ \frac{1}{2k}B(z,t) \biggr) \\ &\quad =\nabla _{H} \biggl(\frac{1}{f(v)} \biggr) \cdot A\nabla _{H}v+ \frac{1}{f(v)}\nabla _{H} \cdot (A\nabla _{H}v )+ \frac{1}{2k}{\nabla _{H}} \cdot B \\ &\quad = \biggl(\frac{1}{f(v)} \biggr)^{\prime }{\nabla _{H}}v \cdot A{\nabla _{H}}v - \frac{1}{f(v)} \bigl(B^{T}\nabla _{H}v+Cf(v) \bigr)+\frac{1}{2k}{ \nabla _{H}} \cdot B \\ &\quad =-\frac{f'(v)}{f^{2}(v)}{\nabla _{H}}v\cdot A{\nabla _{H}}v- \frac{1}{f(v)}{B^{T}} {\nabla _{H}}v -C+ \frac{1}{2k}{\nabla _{H}} \cdot B \\ &\quad \le -k \biggl(W-\frac{1}{2k}B \biggr)^{T}A^{-1} \biggl(W-\frac{1}{2k}B \biggr)-B^{T}A^{-1} \biggl(W- \frac{1}{2k}B \biggr)-C+\frac{1}{2k}{ \nabla _{H}} \cdot B \\ &\quad =-k{W^{T}}A^{-1}W+\frac{1}{4k}{B^{T}}A^{-1}B-C+ \frac{1}{2k}{\nabla _{H}} \cdot B. \end{aligned} $$

(3.2)

Now we calculate the left-hand side of (3.1). By combining (3.2) and (2.6) it follows that

$$\begin{aligned} & \biggl(p(r)u'(r)u(r)-u^{2}(r) \int _{S_{r}} {MW(z,t)\cdot \nu } \,dH \biggr)^{\prime } \\ &\quad = pu^{\prime 2}+ \bigl(pu' \bigr)^{\prime }u - 2uu' \int _{S_{r}} {MW \cdot \nu } \,dH- u^{2} \int _{\partial S_{r}} {MW\cdot \nu } \,ds \\ &\quad = pu^{\prime 2}-buu'-cu^{2}- 2uu' \int _{S_{r}} {MW \cdot \nu }\,dH -u^{2} \int _{S_{r}}{ \nabla _{H}}\cdot W\,dH \\ &\quad \ge pu^{\prime 2}-buu'-cu^{2}-2uu' \int _{S_{r}} {MW\cdot \nu } \,dH \\ & \quad\quad{} -u^{2} \int _{S_{r}} { \biggl(-kW^{T}A^{-1}W+ \frac{1}{4k}B^{T}A^{-1}B-C+ \frac{1}{2k}{\nabla _{H}} \cdot B \biggr)} \,dH \\ &\quad = pu^{\prime 2}-buu'-cu^{2}-2uu' \int _{S_{r}}{MW \cdot \nu }\,dH+ku^{2} \int _{S_{r}} {W^{T}A^{-1}W} \,dH+{u^{2}}Q(r). \end{aligned}$$

By $(C_{1})$ and (2.7) this implies that

$$ \int _{S_{r}} {W^{T}A^{-1}W} \,dH \ge \lambda ^{-1} \int _{S_{r}} {W^{2}}\,dH \ge \frac{1}{\lambda \wp {\alpha _{\wp }}r^{2n+1}(4r^{2}+1)} \biggl( \int _{S_{r}}{MW \cdot \nu \,dH} \biggr)^{2}. $$

(3.3)

Recalling $g(r)$ in (1.6), we have

$$\begin{aligned} & \biggl(p(r)u'(r)u(r)-u^{2}(r) \int _{S_{r}} {MW\cdot \nu \,dH} \biggr)' \\ &\quad \ge pu^{\prime 2}-buu'-cu^{2}-2uu' \int _{S_{r}} {MW\cdot \nu }\,dH+ \frac{u^{2}}{(4r^{2}+1)g} \biggl( \int _{S_{r}} {MW\cdot \nu } \,dH \biggr)^{2}+u^{2}Q \\ &\quad = \bigl(p- \bigl(4r^{2}+1 \bigr)g \bigr) \biggl(u'- \frac{bu}{2(p-(4r^{2}+1)g)} \biggr)^{2}+ \biggl(Q-\frac{b^{2}}{4(p-(4r^{2}+1)g)}-c \biggr)u^{2} \\ & \quad\quad{} + \biggl(\frac{u}{\sqrt{(4r^{2}+1)g}} \int _{S_{r}}{MW\cdot \nu }\,dH-u' \sqrt{ \bigl(4r^{2}+1 \bigr)g} \biggr)^{2}. \end{aligned}$$

 □

Lemma 3.2

(Sturm-type comparison principle)

Let $u(r)$ be a nontrivial solution of Eq. (1.2), and let $a_{1}$ and $a_{2}$ be two adjacent zeros of $u(r)$. If

$$ p(r) \ne \bigl(4r^{2}+1 \bigr)g(r) $$

and

$$ \begin{aligned} & \int _{a_{1}}^{a_{2}} \biggl(Q(r)- \frac{b^{2}(r)}{4(p(r)-(4r^{2}+1)g(r))}-c(r) \biggr){u^{2}(r)}\,dr \\ & \quad{} + \int _{a_{1}}^{a_{2}} \bigl(p(r)- \bigl(4r^{2}+1 \bigr)g(r) \bigr) \biggl(u'(r)- \frac{b(r)u(r)}{2(p(r)-(4r^{2}+1)g(r))} \biggr)^{2}\,dr > 0, \end{aligned} $$

(3.4)

then any solution $v(z,t)$ of Eq. (1.1) has at least one zero on $G [ {{a_{1}},{a_{2}}} ]$.

Proof

We use the contradiction and suppose that there exists a nontrivial solution $v(z,t)$ of Eq. (1.1) such that

$$ v(z,t)>0 \quad \text{on } G[a_{1},a_{2}]. $$

Integrating (3.1) in Lemma 3.1 from $a_{1}$ to $a_{2}$ gives

$$\begin{aligned} & \int _{a_{1}}^{a_{2}} \biggl(pu'u-u^{2} \int _{S_{r}}{MW \cdot \nu } \,dH \biggr)' \,dr \\ &\quad \ge \int _{a_{1}}^{a_{2}} \biggl(\frac{u}{\sqrt{(4r^{2}+1)g}} \int _{S_{r}}{MW \cdot \nu }\,dH-u'\sqrt{ \bigl(4r^{2}+1 \bigr)g} \biggr)^{2} \\ & \quad\quad{} + \int _{a_{1}}^{a_{2}} \biggl(Q-\frac{b^{2}}{4(p-(4r^{2}+1)g)}-c \biggr)u^{2} \\ & \quad\quad{} + \bigl(p- \bigl(4r^{2}+1 \bigr)g \bigr) \biggl(u'- \frac{bu}{2(p-(4r^{2}+1)g)} \biggr)^{2}. \end{aligned}$$

Noting that the left-hand side is 0, this yields

$$\begin{aligned} \int _{a_{1}}^{a_{2}} \biggl(Q-\frac{b^{2}}{4(p-(4r^{2}+1)g)}-c \biggr)u^{2}+ \bigl(p- \bigl(4r^{2}+1 \bigr)g \bigr) \biggl(u'- \frac{bu}{2(p-(4r^{2}+1)g)} \biggr)^{2}\le 0, \end{aligned}$$

which contradicts to (3.4). The proof is completed. □

An immediate consequence of Lemma 3.2 is the following:

Corollary 3.3

Let $u(r)$ be a nontrivial solution of Eq. (1.2), and let $a_{1}$ and $a_{2}$ be two adjacent zeros of $u(r)$. If there exists $a>0$ such that for any $r\in [a,+ \infty )$,

$$\begin{aligned}& p(r)> \bigl(4r^{2}+1 \bigr)g(r), \\& Q(r)\ge \frac{b^{2}(r)}{4(p(r)-(4r^{2}+1)g(r))}+c(r), \end{aligned}$$

and the equality is not always true in any subinterval of $[a,+\infty )$, then any solution $v(z,t)$ of Eq. (1.1) has at least one zero on $G[a_{1},a_{2}]$.

4 Proof of Theorem 1.1

Proof of Theorem 1.1

Let $u(r)$ be an oscillatory solution of Eq. (1.2), and let $r_{i}$ be the sequence of zeros of $u(r)$ satisfying

$$ r_{0}\le r_{1}< r_{2}< \cdots < r_{i} \cdots ,\quad\quad \lim_{i \to \infty }{r_{i}}=\infty . $$

For every ${G_{i}}=\{(z,t)\in H^{n}:{r_{i}}< \vert (z,t) \vert _{H}<r_{i+1}\}$, by Corollary 3.3 it follows that every solution $v(z,t)$ of Eq. (1.1) has at least one zero on $\bar{G_{i}}$, which shows that $v(z,t)$ is oscillatory. From the arbitrariness of $v(z,t)$ we claim that Eq. (1.1) is oscillatory. □