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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 [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. □

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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  MATH  Google Scholar 

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

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Feng, T. A generalized nonlinear Picone identity for the p-biharmonic operator and its applications. J Inequal Appl 2019, 56 (2019). https://doi.org/10.1186/s13660-019-2008-8

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MSC

  • 26D10
  • 26D15

Keywords

  • p-biharmonic operator
  • Generalized nonlinear Picone identity
  • Sturmian comparison principle
  • Liouville’s theorem
  • Hardy–Rellich type inequality