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.

1 Introduction and results

In 1971, Dunninger [1] 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 [1], 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 [6] 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 [1] 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 [3] 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

([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 $- \Delta v > 0$

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

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 [4] established a Caccioppoli-type inequality by an application of Theorem 1.1. Moreover, Dwivedi and Tyagi [5] extended the result of Theorem 1.1 on Heisenberg group and obtained its applications.

Recently, Dwivedi [2] 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

([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 [3] 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

([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 [2] developed a nonlinear Picone identity of Dwivedi and Tyagi [3] in the following Theorem 1.4 and obtained some qualitative results for p-biharmonic equation (system).

Theorem 1.4

([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 $- \Delta v > 0$. 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 [3] and Dwivedi [2]. 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 [3] (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 [3] (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 [2] (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 [2] (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. □

2 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). □