# A generalized nonlinear Picone identity for the p-biharmonic operator and its applications

## Abstract

A generalized nonlinear Picone identity for the p-biharmonic operator is established in this paper. As applications, a Sturmian comparison principle to the p-biharmonic equation with singular term, a Liouville’s theorem to the p-biharmonic system, and a generalized Hardy–Rellich type inequality are obtained.

## Introduction and results

In 1971, Dunninger  established a Picone identity

\begin{aligned} &\operatorname{div} \biggl[ {u\nabla ( {a\Delta u} ) - a\Delta u \nabla u - \frac{{{u^{2}}}}{v}\nabla ( {A\Delta v} ) + A\Delta v \nabla \biggl( { \frac{{{u^{2}}}}{v}} \biggr)} \biggr] \\ &\quad = - \frac{{{u^{2}}}}{v}\Delta ( {A\Delta v} ) + u \Delta ( {a\Delta u} ) + ( {A - a} ){ ( {\Delta u} )^{2}} \\ &\qquad {} - A{ \biggl( {\Delta u - \frac{u}{v}\Delta v} \biggr)^{2}} + A\frac{ {2\Delta v}}{v}{ \biggl( {\nabla u - \frac{u}{v} \nabla v} \biggr)^{2}}, \end{aligned}
(1.1)

where u, v, $$a\Delta u$$, $$A\Delta v$$ are twice continuously differentiable functions with $$v \ne 0$$ and a and A are positive weights. In , the integral form of (1.1) was used to study qualitative results for the fourth order elliptic system

\begin{aligned}& {\Delta \bigl( {a ( x )\Delta u} \bigr) - c ( x )u = 0,} \\& {\Delta \bigl( {A ( x )\Delta v} \bigr) - C ( x )v = 0.} \end{aligned}

A Sturmian comparison principle, an integral inequality of Wirtinger type, and lower bound for eigenvalue were obtained. Jaroš  extended (1.1) to the case where $$\Delta ( {a ( x )\Delta u} )$$ and $$\Delta ( {A ( x )\Delta v} )$$ were replaced by the weighted p-biharmonic operators $$\Delta ( {a ( x ) {{ | {\Delta u} |}^{p - 2}}\Delta u} )$$ and $$\Delta ( {A ( x ){{ \vert {\Delta v} \vert } ^{p - 2}}\Delta v} )$$, respectively, and showed some results similar to  for the fourth order elliptic system

\begin{aligned}& \Delta \bigl( {a ( x ){{ \vert {\Delta u} \vert }^{p - 2}} \Delta u} \bigr) - c ( x ){ \vert u \vert ^{p - 2}}u = 0, \\& \Delta \bigl( {A ( x ){{ \vert {\Delta v} \vert }^{p - 2}}\Delta v} \bigr) - C ( x ){ \vert v \vert ^{p - 2}}v = 0. \end{aligned}

With some simplifications in (1.1), recently, Dwivedi and Tyagi  have obtained the following linear Picone identity (see Theorem 1.1) for the biharmonic operator $${\Delta ^{2}}u = \Delta ( {\Delta u} )$$ and gave several remarks on the qualitative questions such as Morse index and Hardy–Rellich type inequality.

### Theorem 1.1

()

Let u and v be differentiable functions in $$\varOmega \subset \mathbb{R}^{n}$$ ($$n\geq 3$$) such that $$u \geq 0$$, $$v > 0$$, and $$- \Delta v > 0$$

\begin{aligned}& L ( {u,v} ) = { \biggl( {\Delta u - \frac{u}{v}\Delta v} \biggr) ^{2}} - \frac{{2\Delta v}}{v}{ \biggl( {\nabla u - \frac{u}{v}\nabla v} \biggr)^{2}}, \\& R ( {u,v} ) = { \vert {\Delta u} \vert ^{2}} - \Delta \biggl( { \frac{{{u^{2}}}}{v}} \biggr)\Delta v. \end{aligned}

Then $$R(u,v) = L(u,v)$$. Moreover, $$L ( {u,v} ) \geq 0$$, and $$L ( {u,v} ) = 0$$ if and only if $$u = \alpha v$$ for $$\alpha \in \mathbb{R}$$.

It is noteworthy that Dwivedi and Tyagi  established a Caccioppoli-type inequality by an application of Theorem 1.1. Moreover, Dwivedi and Tyagi  extended the result of Theorem 1.1 on Heisenberg group and obtained its applications.

Recently, Dwivedi  has extended the linear Picone identity in Theorem 1.1. He obtained the following linear Picone identity (see Theorem 1.2) for the p-biharmonic operator: $$\Delta _{p}^{2}u = \Delta ( {{{ \vert {\Delta u} \vert } ^{p - 2}}\Delta u} )$$, $$p > 1$$.

### Theorem 1.2

()

Let u and v be differentiable functions in $$\varOmega \subset \mathbb{R}^{n}$$ ($$n\geq 3$$) such that $$u \geq 0$$, $$v > 0$$, and $$- \Delta v > 0$$. Denote

\begin{aligned}& L ( {u,v} ) = { \vert {\Delta u} \vert ^{p}} + \frac{ { ( {p - 1} ){u^{p}}}}{{{v^{p}}}}{ \vert {\Delta v} \vert ^{p}} - \frac{{p{u^{p - 1}}}}{{{v^{p - 1}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v\Delta u \\& \hphantom{L ( {u,v} ) =}{} - \frac{{p ( {p - 1} ){u^{p - 2}}}}{{{v^{p - 1}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v{ \biggl( {\nabla u - \frac{u}{v}\nabla v} \biggr)^{2}}, \\& R ( {u,v} ) = { \vert {\Delta u} \vert ^{p}} - \Delta \biggl( { \frac{{{u^{p}}}}{{{v^{p - 1}}}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}}\Delta v. \end{aligned}

Then $$R(u,v) = L(u,v)$$. Moreover, $$L ( {u,v} ) \geq 0$$, and $$L ( {u,v} ) = 0$$ if and only if $$u = \alpha v$$ for $$\alpha \in \mathbb{R}$$.

Dwivedi and Tyagi  established a nonlinear Picone identity (see Theorem 1.3) for the biharmonic operator and also discussed some qualitative results for biharmonic equation (system).

### Theorem 1.3

()

Let u and v be differentiable functions in $$\varOmega \subset \mathbb{R}^{n}$$ ($$n\geq 3$$) such that $$u \geq 0$$, $$v > 0$$, and $$- \Delta v > 0$$. Suppose that $$f: \mathbb{R} \to ( {0,\infty } )$$ is a $${C^{2}}$$ function such that $${f'} ( y ) \geq 1$$ and $${{f''}} ( y ) \leq 0$$, $$\forall 0 < y \in \mathbb{R}$$. Denote

\begin{aligned}& L ( {u,v} ) = { \vert {\Delta u} \vert ^{2}} - \frac{ {{{ \vert {\Delta u} \vert }^{2}}}}{{{{f'}}(v)}} + { \biggl( {\frac{ {\Delta u}}{{\sqrt{{{f'}}(v)} }} - \frac{u}{{f(v)}} \sqrt{{{f'}}(v)} \Delta v} \biggr)^{2}} \\& \hphantom{L ( {u,v} ) =}{} - \frac{{2 \Delta v}}{{f(v)}}{ \biggl( {\nabla u - \frac{{u{{f'}}(v)}}{{f(v)}} \nabla v} \biggr)^{2}} + \frac{{{u^{2}}{{f''}}(v)}}{{f(v)}}{ \vert {\nabla v} \vert ^{2}}\Delta v, \\& R ( {u,v} ) = { \vert {\Delta u} \vert ^{2}} - \Delta \biggl( { \frac{{{u^{2}}}}{{f(v)}}} \biggr)\Delta v. \end{aligned}

Then $$R(u,v) = L(u,v)$$. Moreover, $$L ( {u,v} ) \geq 0$$, and $$L ( {u,v} ) = 0$$ if and only if $$u = cv + d$$ for $$c,d \in \mathbb{R}$$.

From the biharmonic operator to the p-biharmonic operator, Dwivedi  developed a nonlinear Picone identity of Dwivedi and Tyagi  in the following Theorem 1.4 and obtained some qualitative results for p-biharmonic equation (system).

### Theorem 1.4

()

Let u and v be differentiable functions in $$\varOmega \subset \mathbb{R}^{n}$$ ($$n\geq 3$$) such that $$u \geq 0$$, $$v > 0$$, and $$- \Delta v > 0$$. Suppose that $$f:\mathbb{R} \to ( {0,\infty } )$$ is a $${C^{2}}$$ function such that $${{f'}} ( y ) \geq ( {p - 1} ){ [ {f ( y )} ]^{\frac{{p - 2}}{{p - 1}}}}$$, $$p > 1$$, and $${{f''}} ( y ) \leq 0$$, $$\forall 0 < y \in \mathbb{R}$$. Denote

\begin{aligned}& L ( {u,v} ) = { \vert {\Delta u} \vert ^{p}} + \frac{ {{{f'}}(v){u^{p}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p}} - \frac{{p{u^{p - 1}}}}{{f(v)}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v\Delta u \\& \hphantom{L ( {u,v} ) =}{} + \frac{{{u^{p}}{{f''}}(v){{ \vert {\nabla v} \vert }^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v + \frac{{{u^{p}}{{f''}}(v){{ \vert {\nabla v} \vert }^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v, \\& R ( {u,v} ) = { \vert {\Delta u} \vert ^{p}} - \Delta \biggl( { \frac{{{u^{p}}}}{{f(v)}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}}\Delta v. \end{aligned}

Then $$R(u,v) = L(u,v)$$. Moreover, $$L ( {u,v} ) \geq 0$$, and $$L ( {u,v} ) = 0$$ if and only if $$u = cv + d$$ for $$c,d \in \mathbb{R}$$.

The purpose of this paper is to present a generalized nonlinear Picone identity for the p-biharmonic operator, which extends the results of Dwivedi and Tyagi  and Dwivedi . As applications, a Sturmian comparison principle to the p-biharmonic equation with singular term, a Liouville’s theorem to the p-biharmonic system, and a generalized Hardy–Rellich type inequality are obtained. Our main result is described as follows.

### Theorem 1.5

Let u and v be differentiable functions in $$\varOmega \subset \mathbb{R}^{n}$$ ($$n\geq 3$$) such that $$u \geq 0$$, $$v > 0$$, and $$- \Delta v > 0$$. Suppose that $$f:\mathbb{R} \to ( {0,\infty } )$$ and $$g:\mathbb{R} \to ( {0, \infty } )$$ are $${C^{2}}$$ functions with

$$\textstyle\begin{cases} g(u)>0,\qquad {g'}(u)>0,\qquad {g''}(u) > 0, \quad {u > 0},& \textit{if } x\in \varOmega , \\ {g(u)=0,\qquad {g'}(u)=0,\qquad {g''}(u) = 0}, \quad {u = 0}, & \textit{if } x\in \partial \varOmega , \end{cases}$$

and $$f(v)>0$$, $${f'}(v) > 1$$, $${f''}(v) \leq 0$$ in Ω such that f and g satisfy

\begin{aligned} \frac{{g(u){f'}(v)}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p}} \geq ( {p - 1} ){ \biggl[ {\frac{ {{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr] ^{\frac{p}{{p - 1}}}}- \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v \end{aligned}
(1.2)

and

\begin{aligned} \sqrt{2{g''}(u)g(u)} \geq {g'}(u), \end{aligned}
(1.3)

respectively. Denote

\begin{aligned} L(u,v) ={}& { \vert {\Delta u} \vert ^{p}} - \biggl( {\frac{{{g''}(u) {{ \vert {\nabla u} \vert }^{2}}}}{{f(v)}} + \frac{{{g'}(u)\Delta u}}{{f(v)}}} \\ &{} - \frac{{2{g'}(u){f'}(v)\nabla u \cdot \nabla v}}{{{{ [ {f(v)} ]}^{2}}}} - \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}} - \frac{{g(u){f'}(v)\Delta v}}{{{{ [ {f(v)} ]}^{2}}}} \\ &{} + \frac{{2g(u){{ [ {{f'}(v)} ]} ^{2}}{{ \vert {\nabla v} \vert }^{2}}}}{{{{ [ {f(v)} ]} ^{3}}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}}\Delta v \end{aligned}
(1.4)

and

\begin{aligned} R(u,v) = { \vert {\Delta u} \vert ^{p}} - \Delta \biggl( {\frac{ {g(u)}}{{f(v)}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}} \Delta v, \end{aligned}
(1.5)

respectively. Then $$R(u,v) = L(u,v)$$. Moreover, $$L ( {u,v} ) \geq 0$$, and $$L ( {u,v} ) = 0$$ if and only if

\begin{aligned}& u = cv,\quad c \in \mathbb{R}, \end{aligned}
(1.6)
\begin{aligned}& { \vert {\Delta u} \vert ^{p}} = { \biggl[ { \frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr]^{\frac{p}{{p - 1}}}}, \end{aligned}
(1.7)
\begin{aligned}& \frac{{g(u){f'}(v)}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p}} = ( {p - 1} ){ \biggl[ {\frac{ {{g'}(u){{ \vert {\nabla v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr] ^{\frac{p}{{p - 1}}}} - \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v, \end{aligned}
(1.8)
\begin{aligned}& \sqrt{{g''}(u)} \nabla u = \frac{{\sqrt{2g(u)} {f'}(v)\nabla v}}{ {f(v)}} \quad \textit{and}\quad \sqrt{2{g''}(u)g(u)} = {g'}(u). \end{aligned}
(1.9)

### Remark 1.6

If $$p = 2$$, $$g(u) = {u^{2}}$$ and $$f(v)=v$$ in (1.4) and (1.5), which is the result of Dwivedi and Tyagi  (see Theorem 1.1).

### Remark 1.7

If $$p =2$$, $$g(u) = {u^{2}}$$ and $${f'} ( v ) \geq 1$$ and $${f''} ( v ) \leq 0$$, $$\forall 0 < v \in \mathbb{R}$$ in (1.4) and (1.5), which is the result of Dwivedi and Tyagi  (see Theorem 1.3).

### Remark 1.8

If $$p > 2$$, $$g(u) = {u^{p}}$$ and $$f(v) = {v^{p-1}}$$ in (1.4) and (1.5), which is the result of Dwivedi  (see Theorem 1.2).

### Remark 1.9

If $$p > 2$$, $$g(u) = {u^{p}}$$ and $${f'} ( v ) \geq ( {p - 1} ){ [ {f ( v )} ]^{\frac{{p - 2}}{ {p - 1}}}}$$, $$p > 1$$ and $${f''} ( v ) \leq 0$$, $$\forall 0 < v \in \mathbb{R}$$ in (1.4) and (1.5), which is the result of Dwivedi  (see Theorem 1.4).

We give the proof of Theorem 1.5 in the following.

### Proof

We first prove that $$R(u,v) = L(u,v)$$ by expanding $$R(u,v)$$:

\begin{aligned} R(u,v) ={}& { \vert {\Delta u} \vert ^{p}} - \Delta \biggl( { \frac{ {g(u)}}{{f(v)}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}}\Delta v \\ ={}& { \vert {\Delta u} \vert ^{p}} - \biggl( {\frac{{{g''}(u){{ \vert {\nabla u} \vert }^{2}}}}{{f(v)}} + \frac{ {{g'}(u)\Delta u}}{{f(v)}} - \frac{{2{g'}(u){f'}(v)\nabla u \cdot \nabla v}}{{{{ [ {f(v)} ]}^{2}}}} - \frac{{g(u){f''}(v) {{ \vert {\nabla v} \vert }^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}} \\ & {} - \frac{{g(u){f'}(v)\Delta v}}{{{{ [ {f(v)} ]}^{2}}}} + \frac{{2g(u){{ [ {{f'}(v)} ]} ^{2}}{{ \vert {\nabla v} \vert }^{2}}}}{{{{ [ {f(v)} ]} ^{3}}}} \biggr){ \vert {\Delta v} \vert ^{p - 2}}\Delta v \\ ={}& { \vert {\Delta u} \vert ^{p}} - \frac{{{g'}(u)\Delta u}}{ {f(v)}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v + \frac{{g(u) {f'}(v)}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p}} \\ & {} - \frac{{{{ \vert {\Delta v} \vert }^{p - 2}}\Delta v}}{ {f(v)}} \biggl( {{g''}(u){{ \vert {\nabla u} \vert }^{2}} - \frac{ {2{g'}(u){f'}(v)\nabla u \cdot \nabla v}}{{f(v)}} + \frac{{2g(u){{ [ {{f'}(v)} ]}^{2}}{{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}} \biggr) \\ &{} + \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert }^{2}}}}{ {{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}} \Delta v \\ ={}& L(u,v). \end{aligned}

Next we verify $$L(u,v) \geq 0$$, we can rewrite $$L(u,v)$$ as

\begin{aligned} L(u,v) =& p \biggl( {\frac{1}{p}{{ \vert {\Delta u} \vert }^{p}} + \frac{ {p - 1}}{p}{{ \biggl[ {\frac{{{g'}(u){{ \vert {\Delta v} \vert } ^{p - 1}}}}{{pf(v)}}} \biggr]}^{\frac{p}{{p - 1}}}}} \biggr) - \frac{ {{g'}(u) \vert {\Delta u} \vert }}{{f(v)}}{ \vert {\Delta v} \vert ^{p - 1}} \\ &{} + \frac{{g(u){f'}(v)}}{{{{ [ {f(v)} ]}^{2}}}} { \vert {\Delta v} \vert ^{p}} - ( {p - 1} ){ \biggl[ {\frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr] ^{\frac{p}{{p - 1}}}} + \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v \\ &{} + \frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 2}}}}{{f(v)}} \bigl( { \vert {\Delta u} \vert \vert {\Delta v} \vert - \Delta u\Delta v} \bigr) \\ &{} - \frac{{{{ \vert {\Delta v} \vert }^{p - 2}}\Delta v}}{ {f(v)}} ( {{{ \biggl( {\sqrt{{g''}(u)} \nabla u - \frac{{\sqrt{2g(u)} {f'}(v)\nabla v}}{{f(v)}}} \biggr)}^{2}}} \\ &{} + { \frac{{2 ( {\sqrt{2{g''}(u)g(u)} - {g'}(u)} ) {f'}(v)\nabla u \cdot \nabla v}}{{f(v)}}} \\ : =&F_{1} + F_{2}+F_{3}+F_{4}, \end{aligned}
(1.10)

where

\begin{aligned}& F_{1} = p \biggl( {\frac{1}{p}{{ \vert {\Delta u} \vert }^{p}} + \frac{ {p - 1}}{p}{{ \biggl[ {\frac{{{g'}(u){{ \vert {\Delta v} \vert } ^{p - 1}}}}{{pf(v)}}} \biggr]}^{\frac{p}{{p - 1}}}}} \biggr) - \frac{ {{g'}(u) \vert {\Delta u} \vert }}{{f(v)}}{ \vert {\Delta v} \vert ^{p - 1}}, \\& F_{2}= \frac{{g(u){f'}(v)}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p}} - ( {p - 1} ){ \biggl[ {\frac{ {{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr] ^{\frac{p}{{p - 1}}}} + \frac{{g(u){f''}(v){{ \vert {\nabla v} \vert } ^{2}}}}{{{{ [ {f(v)} ]}^{2}}}}{ \vert {\Delta v} \vert ^{p - 2}}\Delta v, \\& F_{3}=\frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 2}}}}{ {f(v)}} \bigl( { \vert {\Delta u} \vert \vert {\Delta v} \vert - \Delta u\Delta v} \bigr), \\& F_{4}= - \frac{{{{ \vert {\Delta v} \vert }^{p - 2}} \Delta v}}{{f(v)}} \biggl( {{{ \biggl( { \sqrt{{g''}(u)} \nabla u - \frac{ {\sqrt{2g(u)} {f'}(v)\nabla v}}{{f(v)}}} \biggr)}^{2}}} \\& \hphantom{F_{4}=} {} + \frac{{2 ( {\sqrt{2{g''}(u)g(u)} - {g'}(u)} ){f'}(v)\nabla u \cdot \nabla v}}{{f(v)}} \biggr). \end{aligned}

We now recall Young’s inequality

\begin{aligned} {a_{0}} {b_{0}} \geq \frac{{{a_{0}}^{p}}}{p} + \frac{{{b_{0}}^{q}}}{q}, \end{aligned}
(1.11)

where $${a_{0}} \geq 0$$, $${b_{0}} \geq 0$$, $$p > 1$$, $$q > 1$$, and $$\frac{1}{p} + \frac{1}{q} = 1$$, the equality holds if and only if $${a_{0}}^{p} = {b_{0}}^{q} = {b_{0}}^{\frac{p}{{p - 1}}}$$. Setting $${a_{0}} = \vert {\Delta u} \vert$$, $${b_{0}} = \frac{{{g'}(u){{ \vert {\Delta v} \vert } ^{p - 1}}}}{{pf(v)}}$$ in (1.11), we obtain

$$\frac{{{g'}(u) \vert {\Delta u} \vert }}{{f(v)}}{ \vert {\Delta v} \vert ^{p - 1}} \leq p \biggl( { \frac{1}{p}{{ \vert {\Delta u} \vert } ^{p}} - \frac{{p - 1}}{p}{{ \biggl[ {\frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{pf(v)}}} \biggr]}^{\frac{p}{{p - 1}}}}} \biggr),$$

which implies $$F_{1} \geq 0$$. Clearly $$F_{2} \geq 0$$ by (1.2). Since $$\vert {\Delta u} \vert \vert {\Delta v} \vert - \Delta u\Delta v \geq 0$$, the equality holds if and only if $$u = cv$$, $$c \in \mathbb{R}$$, and combining with $$\frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 2}}}}{{f(v)}} \geq 0$$, we obtain $$F_{3} \geq 0$$. By $$- \Delta v > 0$$, $$f(v) > 0$$, and (1.3), we have $$F_{4} \geq 0$$. Hence $$L(u,v) \geq 0$$ from (1.10).

We now verify $$L(u,v) = 0$$ by (1.6)–(1.9). It follows from (1.6) that there exists a positive constant c such that $$u=cv$$, namely we have

$$\vert {\Delta v} \vert \vert {\Delta u} \vert - \Delta v \cdot \Delta u = c \vert {\Delta v} \vert \vert {\Delta v} \vert - c\Delta v \cdot \Delta v = c{ \vert {\Delta v} \vert ^{2}} - c { \vert {\Delta v} \vert ^{2}} = 0,$$

which implies $$F_{3}= 0$$. By $${ \vert {\Delta u} \vert ^{p}} = { [ {\frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{ {pf(v)}}} ]^{\frac{p}{{p - 1}}}}$$ in (1.7), we obtain

\begin{aligned} \frac{{{g'}(u){{ \vert {\Delta v} \vert }^{p - 1}}}}{{f(v)}} = p { \vert {\Delta u} \vert ^{p - 1}}. \end{aligned}
(1.12)

It follows from (1.12) that

\begin{aligned} I &= p \biggl( {\frac{1}{p}{{ \vert {\Delta u} \vert }^{p}} + \frac{ {p - 1}}{p}{{ \biggl[ {\frac{{{g'}(u){{ \vert {\Delta v} \vert } ^{p - 1}}}}{{pf(v)}}} \biggr]}^{\frac{p}{{p - 1}}}}} \biggr) - \frac{ {{g}(u) \vert {\Delta u} \vert {{ \vert {\Delta v} \vert } ^{p - 1}}}}{{f(v)}} \\ &= p \biggl( {\frac{1}{p}{{ \vert {\Delta u} \vert }^{p}} + \frac{ {p - 1}}{p}{{ \vert {\Delta u} \vert }^{p}}} \biggr) - \vert {\Delta u} \vert p{ \vert {\Delta u} \vert ^{p - 1}} \\ & = { \vert {\Delta u} \vert ^{p}} + ( {p - 1} ) { \vert {\Delta u} \vert ^{p}} - p{ \vert {\Delta u} \vert ^{p}} \\ & = 0. \end{aligned}

We can prove $$F_{2}= 0$$ by (1.8). A direct calculation shows

$${ \biggl( {\sqrt{{g''}(u)} \nabla u - \frac{{\sqrt{2g(u)} {f'}(v) \nabla v}}{{f(v)}}} \biggr)^{2}} = 0$$

by $$\sqrt{{g''}(u)} \nabla u = \frac{{\sqrt{2g(u)} {f'}(v)\nabla v}}{{f(v)}}$$ in (1.9), we can also show

$$\frac{{2 ( {\sqrt{2{g''}(u)g(u)} - {g^{\prime }}(u)} ){f'}(v) \nabla u \cdot \nabla v}}{{f(v)}} = 0$$

by $$\sqrt{2{g''}(u)g(u)} = {g'}(u)$$ in (1.9), hence $$F_{4}{\text{ = 0}}$$ by (1.9). Summing up these, it follows $$L(u,v) = F_{1}+F_{2}+F_{3} + F_{4} = 0$$. Hence we can conclude that $$L(u,v) = 0$$ if and only if (1.6)–(1.9) hold. In fact, if $$u = {\text{0}}$$, it clearly follows. If $$u \ne {\text{0}}$$, the conclusion holds from the above process of proof. □

## Applications

Throughout this section, we always assume that f and g are $${C^{2}} ( \varOmega )$$ functions and satisfy the conditions in Theorem 1.5, unless otherwise stated, and give applications for the generalized nonlinear Picone identity. We first show a Sturmian comparison principle to the p-biharmonic equation with singular term by Theorem 1.5 as follows.

### Proposition 2.1

Let $${k_{1}}(x)$$ and $${k_{2}}(x)$$ be two continuous weighted functions with $${k_{1}}(x) < {k_{2}}(x)$$. Assume that there exists a positive solution satisfying

\begin{aligned} \textstyle\begin{cases} { \Delta _{p}^{2}u = \frac{{{k_{1}}(x)g(u)}}{u},} & {x \in \varOmega ,} \\ {g(u) > 0,\quad u > 0,} & {x \in \varOmega ,} \\ {g(u) = 0,\quad u = 0,} & {x \in \partial \varOmega .} \end{cases}\displaystyle \end{aligned}
(2.1)

Then any nontrivial solution v of the following p-biharmonic equation

\begin{aligned} \Delta _{p}^{2}v = k_{2}(x)f(v), \quad x \in \varOmega , \end{aligned}
(2.2)

must change sign.

### Proof

Suppose that v of (2.2) does not change sign. Without loss of generality, we assume that $$v > 0$$ in Ω. By (2.1), (2.2), and Theorem 1.5, we have

\begin{aligned} 0 &\leq \int _{\varOmega }{L(u,v)\,dx} = \int _{\varOmega }{R(u,v)\,dx} \\ &= \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx - \int _{\varOmega }{\Delta \biggl( {\frac{{g(u)}}{{f(v)}}} \biggr){{ \vert {\Delta v} \vert }^{p - 2}}\Delta v}\,dx \\ & = \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx - \int _{\varOmega }{\frac{{g(u)}}{{f(v)}}\Delta _{p}^{2}v} \,dx \\ &= \int _{\varOmega }{{k_{1}}(x)g(u)}\,dx - \int _{\varOmega }{{k_{2}}(x)g(u)}\,dx \\ & = \int _{\varOmega }{ \bigl( {{k_{1}}(x) - {k_{2}}(x)} \bigr)g(u)}\,dx \\ & < 0, \end{aligned}

which is a contradiction. This accomplishes the proof. □

We next show a Liouville’s theorem for the p-biharmonic system by Theorem 1.5 as follows.

### Proposition 2.2

Let $$(u,v) \in [ {{W^{2,p}} ( \varOmega ) \cap W_{0} ^{1,p} ( \varOmega )} ] \times [ {{W^{2,p}} ( \varOmega ) \cap W_{0}^{1,p} ( \varOmega )} ]$$ be a pair of weak solutions to the p-biharmonic system

\begin{aligned} \textstyle\begin{cases} { \Delta _{p}^{2}u = f(v),} & x \in \varOmega , \\ { \Delta _{p}^{2}v = \frac{{{{ [ {f(v)} ]}^{2}}u}}{{g(u)}},} & x \in \varOmega , \\ {g(u) > 0,\qquad f(v) > 0,\qquad u > 0,\qquad v > 0,} & x \in \varOmega , \\ {g(u) = 0,\qquad f(v) = 0,\qquad u = 0,\qquad v = 0,} & x \in \partial \varOmega . \end{cases}\displaystyle \end{aligned}
(2.3)

Then $$u = cv$$ in Ω, where c is a constant.

### Proof

For any test functions $${\phi _{1}},{\phi _{2}} \in {W^{2,p}} ( \varOmega ) \cap W_{0}^{1,p} ( \varOmega )$$, it follows from (2.3) that

\begin{aligned}& \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p - 2}}\Delta u\Delta {\phi _{1}}\,dx} = \int _{\varOmega }{f(v){\phi _{1}}\,dx}, \end{aligned}
(2.4)
\begin{aligned}& \int _{\varOmega }{{{ \vert {\Delta v} \vert }^{p - 2}}\Delta v\Delta {\phi _{2}}\,dx} = \int _{\varOmega }{\frac{{{{ [ {f(v)} ]}^{2}}u}}{ {g(u)}}{\phi _{2}}\,dx}. \end{aligned}
(2.5)

Taking $${\phi _{1}} = u$$ and $${\phi _{2}} = \frac{{g(u)}}{{f(v)}}$$ in (2.4) and (2.5), respectively, we obtain

$$\int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}\,dx} = \int _{\varOmega }{f(v)u\,dx} = \int _{\varOmega }{\Delta \biggl( {\frac{{g(u)}}{{f(v)}}} \biggr) {{ \vert {\Delta v} \vert }^{p - 2}}\Delta v\,dx},$$

which implies

$$\int _{\varOmega }{L(u,v)\,dx} = \int _{\varOmega }{R(u,v)\,dx} = \int _{\varOmega } {{{ \vert {\Delta u} \vert }^{p}}\,dx} - \int _{\varOmega }{\Delta \biggl( {\frac{{g(u)}}{{f(v)}}} \biggr){{ \vert {\Delta v} \vert }^{p - 2}} \Delta v\,dx = 0},$$

hence the conclusion follows by an application of Theorem 1.5. □

Finally, we obtain a generalized Hardy–Rellich type inequality by Theorem 1.5.

### Proposition 2.3

Suppose that a function $$0 < v \in {C^{2}}(\varOmega )$$ with $$-\Delta v > 0$$ in Ω, and it satisfies

\begin{aligned} \Delta _{p}^{2}v \geq \lambda k(x)f(v),\quad x\in \varOmega , \end{aligned}
(2.6)

where $$\lambda > 0$$ is a constant, $$k(x)$$ is a positive continuous function. Then there holds

\begin{aligned} \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx \geq \lambda \int _{\varOmega }{k(x)g(u)}\,dx \end{aligned}
(2.7)

for any $$0 \leq u \in C_{0}^{2}(\varOmega )$$.

### Proof

It follows from (2.6) and Theorem 1.5 that

\begin{aligned} 0 &\leq \int _{\varOmega }{L(u,v)\,dx} = \int _{\varOmega }{R(u,v)\,dx} \\ &= \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx - \int _{\varOmega }{\Delta \biggl( {\frac{{g(u)}}{{f(v)}}} \biggr){{ \vert {\Delta v} \vert }^{p - 2}}\Delta v}\,dx \\ & = \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx - \int _{\varOmega }{\frac{{g(u)}}{{f(v)}}\Delta _{p}^{2}v} \,dx \\ & \leq \int _{\varOmega }{{{ \vert {\Delta u} \vert }^{p}}}\,dx - \int _{\varOmega }{\lambda k(x)g(u)}\,dx, \end{aligned}

which implies (2.7). □

## References

1. 1.

Dunninger, D.R.: A Picone integral identity for a class of fourth order elliptic differential inequalities. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 50, 630–641 (1971)

2. 2.

Dwivedi, G.: Picone’s Identity for p-biharmonic operator and Its Applications. arXiv:1503.05535

3. 3.

Dwivedi, G., Tyagi, J.: Remarks on the qualitative questions for biharmonic operators. Taiwan. J. Math. 19(6), 1743–1758 (2015)

4. 4.

Dwivedi, G., Tyagi, J.: A note on the Caccioppoli inequality for biharmonic operators. Mediterr. J. Math. 13(4), 1823–1828 (2016)

5. 5.

Dwivedi, G., Tyagi, J.: Picone’s identity for biharmonic operators on Heisenberg group and its applications. Nonlinear Differ. Equ. Appl. 23(2), 1–26 (2016)

6. 6.

Jaroš, J.: Picone’s identity for the p-biharmonic operator with applications. Electron. J. Differ. Equ. 2011, 122 (2011)

## Funding

This work is supported by the National Natural Science Foundation of China (11701453, 11701322).

## Author information

Authors

### Contributions

The author read and approved the final manuscript.

### Corresponding author

Correspondence to Tingfu Feng.

## Ethics declarations

### Competing interests

The author declares that they have no competing interests.
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