In 1910, Picone [1] considered the homogeneous linear second order differential system

$ \begin{align} \left\{ {\begin{array}{*{20}{c}} {{{( {{a_1}(x)u'} )}^\prime } + {b_1}(x)u = 0, } \\ {{{( {{a_2}(x)v'} )}^\prime } + {b_2}(x)v = 0, } \\ \end{array} } \right. \end{align} $

(1.1)

where $ {a_1}(x) $, $ {a_2}(x) $, $ u $ and $ v $ are differentiable functions to $ x $, sign $ {{}^{\prime} } $ denotes $ \frac{d}{dx} $, he established the following identity: for $ v(x) \ne 0 $,

$ \begin{align} {( {\frac{u} {v}( {{a_1}u'v - {a_2}uv'} )} )^\prime } = ( {{b_2} - {b_1}} ){u^2} + ( {{a_1} - {a_2}} ){(u')^{^2}} + {a_2}{( {u' - v'\frac{u} {v}} )^2}. \end{align} $

(1.2)

As applications, a Sturmian comparison principle and the oscilation theory of solutions of (1.1) were obtained by (1.2). Moreover, the extension of (1.2) to the multidimensional case (Laplace operator $ \Delta u $) is the following identity: for differentiable functions $ v > 0 $ and $ u \geqslant 0 $,

$ \begin{align} {( {\nabla u - \frac{u} {v}\nabla v} )^2} = {| {\nabla u} |^2} + \frac{{{u^2}}} {{{v^2}}}{| {\nabla v} |^2} - 2\frac{u} {v}\nabla v \cdot \nabla u = {| {\nabla u} |^2} - \nabla ( {\frac{{{u^2}}} {v}} )\nabla v\geq 0. \end{align} $

(1.3)

Dunninger [2] and Allegretto and Huang [3] independently extended (1.3) to $ p $-Laplace operator $ {\Delta _p}u = $ div$ ( {{{| {\nabla u} |}^{p - 2}}\nabla u} ) $ with $ p > 1 $, respectively, namely, for differentiable functions $ v > 0 $ and $ u \geqslant 0 $,

$ \begin{align*} \label{eq1.4} {| {\nabla u} |^p} + (p - 1)\frac{{{u^p}}} {{{v^p}}}{| {\nabla v} |^p} - p\frac{{{u^{p - 1}}}} {{{v^{p - 1}}}}{| {\nabla v} |^{p - 2}}\nabla v \cdot \nabla u = {| {\nabla u} |^p} - \nabla ( {\frac{{{u^p}}} {{{v^{p - 1}}}}} ){| {\nabla v} |^{p - 2}}\nabla v\geq 0. \end{align*} $

As well known, There is a lot of literatures on the study of linear Picone identity, which show Picone identity is an important tool in the analysis of partial differential equations, we refer to [4-8] and related references. However, there is very little literatures on the study of nonlinear Picone identity. We first briefly review the research development of nonlinear Picone identity for Laplace operator and $ p $-Laplace operator, respectively.

Recently, a nonlinear Picone identity for Laplace operator was presented by Tyagi [9] as follows.

Theorem 1.1[9] Let $ v > 0 $ and $ u \geqslant 0 $ be two differentiable functions in the domain $ \Omega \subset {\mathbb{R}^n}( {n \geqslant 3} ) $. Denote

$ \begin{array}{l} L(u, v) = \alpha |\nabla u{|^2} - \frac{{|\nabla u{|^2}}}{{f'(v)}} + {(\frac{{u\sqrt {f'(v)} \nabla v}}{{f(v)}} - \frac{{\nabla u}}{{\sqrt {f'(v)} }})^2}, \\ R(u, v) = \alpha |\nabla u{|^2} - \nabla (\frac{{{u^2}}}{{f(v)}})\nabla v, \end{array} $

where $ f(y) > 0, 0 < y \in R, $ and $ f'(y) \geqslant \frac{1} {\alpha } $ for some $ \alpha > 0 $. Then $ R(u, v) = L(u, v) $. Moreover, $ L(u, v) \geqslant 0 $, and $ L(u, v) = 0 $ a.e. in $ \Omega $ if and only if $ u = {c_1}v + {c_2} $ a.e. in $ \Omega $ for some constants $ {c_1}, {c_2} $.

Bal [10] extended the nonlinear Picone identity of Tyagi [9] from Laplace operator to $ p $-Laplace operator as follows. Moreover, Bal [11] also obtained the nonlinear Picone identity of [10] to $ p $-sub-Laplace operator in Heisenberg group, which is an extension of the result in Euclidean space.

Theorem 1.2[10] Let $ v > 0 $ and $ u \geqslant 0 $ be two differentiable functions in the domain $ \Omega \subset {\mathbb{R}^n} ( {n \geqslant 3} ) $. Denote

$ \begin{array}{l} L(u, v) = |\nabla u{|^p} - \frac{{p{u^{p - 1}}|\nabla v{|^{p - 2}}\nabla v \cdot \nabla u}}{{f(v)}} + \frac{{{u^p}f'(v)|\nabla v{|^p}}}{{{{[f(v)]}^2}}}, \\ R(u, v) = |\nabla u{|^p} - \nabla (\frac{{{u^p}}}{{f(v)}})|\nabla v{|^{p - 2}}\nabla v, {\rm{ }} \end{array} $

where $ f(y) > 0, 0 < y \in R, $ and $ f'(y) \geqslant ( {p - 1} ) [ {f{{(y)}^{\frac{{p - 2}} {{p - 1}}}}} ], p > 1 $. Then $ R(u, v) = L(u, v) $. Moreover, $ L(u, v) \geqslant 0 $, and $ L(u, v) = 0 $ a.e. in $ \Omega $ if and only if $ u = {c_1}v + {c_2} $ a.e. in $ \Omega $ for some constants $ {c_1}, {c_2} $.

Let us recall our previous works in Feng and Cui [12], in the following, we establish a linear Picone identity to anisotropic Laplace operator $ \sum\limits_{i = 1}^n {\frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial u}} {{\partial {x_i}}}} )} , {p_i} > 1 $.

Theorem 1.3[12] Let $ v > 0 $ and $ u \geqslant 0 $ be two differentiable functions in the domain $ \Omega \subset {\mathbb{R}^n} ( {n \geqslant 3} ) $. Denote

$ \begin{array}{l} L(u, v) = \sum\limits_{i = 1}^n {|\frac{{\partial u}}{{\partial {x_i}}}{|^{{p_i}}}} - \sum\limits_{i = 1}^n {{p_i}\frac{{{u^{{p_i} - 1}}}}{{{v^{{p_i} - 1}}}}|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i} - 2}}\frac{{\partial v}}{{\partial {x_i}}}\frac{{\partial u}}{{\partial {x_i}}}} + \sum\limits_{i = 1}^n {({p_i} - 1)\frac{{{u^{{p_i}}}}}{{{v^{{p_i}}}}}|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i}}}} , \\ R(u, v) = \sum\limits_{i = 1}^n {|\frac{{\partial u}}{{\partial {x_i}}}{|^{{p_i}}}} - \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}(\frac{{{u^{{p_i}}}}}{{{v^{{p_i} - 1}}}})|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i} - 2}}\frac{{\partial v}}{{\partial {x_i}}}} , \end{array} $

where $ {p_i} > 1 $ $ ({i = 1, \cdots , n} ) $. Then $ R(u, v) = L(u, v) $. Moreover, $ L(u, v) \geqslant 0 $, and $ L(u, v) = 0 $ a.e. in $ \Omega $ if and only if $ u = {c_1}v + {c_2} $ a.e. in $ \Omega $ for some constants $ {c_1}, {c_2} $.

Based on our previous works in [12], in this paper, our aim is to obtain a nonlinear Picone identity to anisotropic Laplace operator and give its applications. Including that a Sturmian comparison principle to an anisotropic elliptic equation, a Liouville's theorem to an anisotropic elliptic system and a generalized anisotropic Hardy type inequality are obtained. Our main result is the following.

Theorem 1.4 Let $ v > 0 $ and $ u \geqslant 0 $ be two differentiable functions in the domain $ \Omega \subset {\mathbb{R}^n} ( {n \geqslant 3} ) $. Denote

$ \begin{array}{l} L(u, v) = \sum\limits_{i = 1}^n {|\frac{{\partial u}}{{\partial {x_i}}}{|^{{p_i}}}} - \sum\limits_{i = 1}^n {{p_i}\frac{{{u^{{p_i} - 1}}}}{{f(v)}}|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i} - 2}}\frac{{\partial v}}{{\partial {x_i}}}} \frac{{\partial u}}{{\partial {x_i}}} + \sum\limits_{i = 1}^n {\frac{{{u^{{p_i}}}f'(v)}}{{{{[f(v)]}^2}}}|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i}}}} , \\ R(u, v) = \sum\limits_{i = 1}^n {|\frac{{\partial u}}{{\partial {x_i}}}{|^{{p_i}}}} - \sum\limits_{i = 1}^n {\frac{\partial }{{\partial {x_i}}}(\frac{{{u^{{p_i}}}}}{{f(v)}})|\frac{{\partial v}}{{\partial {x_i}}}{|^{{p_i} - 2}}\frac{{\partial v}}{{\partial {x_i}}}} , {\rm{ }} \end{array} $

where $ f(v) > 0 $ and $ f'(v) \geqslant ( {{p_i} - 1} ){ [ {f(v)} ]^{\frac{{{p_i} - 2}} {{{p_i} - 1}}}}, {p_i} > 1 $. Then $ R(u, v) = L(u, v) $. Moreover, $ L(u, v) \geqslant 0 $, and $ L(u, v) = 0 $ a.e. in $ \Omega $ if and only if $ u = {c}v $ a.e. in $ \Omega $ for a constant $ c $.

Remark 1.5 We mention that if $ {p_i} = 2 $, then Theorem 1.4 is the result in Tyagi [9] with $ \alpha {\text{ = }}1 $; if $ {p_i} = 2 $ and $ f(v) = v $, then Theorem 1.4 is the result in Picone [1]; if $ {p_i} > 2 $ and $ f(v) = {v^{{p_i} - 1}} $, then Theorem 1.4 is the result in Feng and Cui [12]; if $ {p_i} = p > 2 $, then Theorem 1.4 is the result in Feng and Yu [13]; if $ {p_i} = p > 2 $ and $ f(v) = {v^{p - 1}} $, then Theorem 1.4 is the result in Jaroš [8] with $ p{\text{ = }}r $.

This paper is organized as follows. The proof of Theorem 1.4 is given in Section 2; Section 3 is devoted to applications of Theorem 1.4.

Proof of Theorem 1.4 We first prove $ R(u, v) = L(u, v) $. Expanding $ R(u, v) $ by a direct calculation,

$ \begin{align*} R(u, v) & = \sum\limits_{i = 1}^n {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} - \sum\limits_{i = 1}^n {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} \nonumber\\ & = \sum\limits_{i = 1}^n {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} - \sum\limits_{i = 1}^n {\frac{{{p_i}{u^{{p_i} - 1}}\frac{{\partial u}} {{\partial {x_i}}}f(v) - {u^{{p_i}}}f'(v)\frac{{\partial v}} {{\partial {x_i}}}}} {{{{ [ {f(v)} ]}^2}}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} \nonumber\\ & = \sum\limits_{i = 1}^n {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} - \sum\limits_{i = 1}^n {{p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} \frac{{\partial u}} {{\partial {x_i}}} + \sum\limits_{i = 1}^n {\frac{{{u^{{p_i}}}f'(v)}} {{{{ [ {f(v)} ]}^2}}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i}}}} \nonumber\\ & = L(u, v). \end{align*} $

We next prove $ L(u, v) \geqslant 0 $. We can rewrite $ L(u, v) $ as

$ \begin{align} L(u, v) & = \sum\limits_{i = 1}^n ({{p_i} [ {\frac{1} {{{p_i}}}{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}} + \frac{{{p_i} - 1}} {{{p_i}}}{{ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )}^{\frac{{{p_i}}} {{{p_i} - 1}}}}} ]} - {p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i} - 1}} | {\frac{{\partial u}} {{\partial {x_i}}}} | ) \\ &\quad + \sum\limits_{i = 1}^n ({ - ( {{p_i} - 1} ){{ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )}^{\frac{{{p_i}}} {{{p_i} - 1}}}}} + \frac{{{u^{{p_i}}}f'(v)}} {{{{ [ {f(v)} ]}^2}}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i}}} ) \\ &\quad + \sum\limits_{i = 1}^n {{p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}} ( { | {\frac{{\partial v}} {{\partial {x_i}}}} | | {\frac{{\partial u}} {{\partial {x_i}}}} | - \frac{{\partial v}} {{\partial {x_i}}}\frac{{\partial u}} {{\partial {x_i}}}} ) \\ &: = I + II + III, \end{align} $

(2.1)

where

$ \begin{align*} I & = \sum\limits_{i = 1}^n ({{p_i} [ {\frac{1} {{{p_i}}}{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}} + \frac{{{p_i} - 1}} {{{p_i}}}{{ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )}^{\frac{{{p_i}}} {{{p_i} - 1}}}}} ]} - {p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i} - 1}} | {\frac{{\partial u}} {{\partial {x_i}}}} | ), \\ II& = \sum\limits_{i = 1}^n ({ - ( {{p_i} - 1} ){{ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )}^{\frac{{{p_i}}} {{{p_i} - 1}}}}} + \frac{{{u^{{p_i}}}f'(v)}} {{{{ [ {f(v)} ]}^2}}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i}}} ), \\ III & = \sum\limits_{i = 1}^n {{p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}} ( { | {\frac{{\partial v}} {{\partial {x_i}}}} | | {\frac{{\partial u}} {{\partial {x_i}}}} | - \frac{{\partial v}} {{\partial {x_i}}}\frac{{\partial u}} {{\partial {x_i}}}} ). \end{align*} $

Let us recall Young's inequality

$ \begin{align} ab \leqslant \frac{{{a^{{p_i}}}}} {{{p_i}}} + \frac{{{b^{{q_i}}}}} {{{q_i}}} \end{align} $

(2.2)

for $ a \geqslant 0, b \geqslant 0 $, where $ {p_i} > 1, {q_i} > 1 $ and $ \frac{1} {{{p_i}}} + \frac{1} {{{q_i}}} = 1 $, and the equality holds if and only if $ a = {b^{\frac{1} {{{p_i} - 1}}}} $. Taking $ a = | {\frac{{\partial u}} {{\partial {x_i}}}} | $ and $ b = \frac{{{u^{{p_i} - 1}}}} {{f(v)}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i} - 1}} $ in (2.2), it yields

$ \begin{align} {p_i}\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{p - 1}} | {\frac{{\partial u}} {{\partial {x_i}}}} | \leqslant {p_i} [ {\frac{1} {{{p_i}}}{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}} + \frac{{{p_i} - 1}} {{{p_i}}}{{ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )}^{\frac{{{p_i}}} {{{p_i} - 1}}}}} ], \end{align} $

(2.3)

and thus $ I \geqslant 0 $ by (2.3). Since $ f(v) > 0 $ and $ f'(v) \geqslant ( {{p_i} - 1} ){ [ {f(v)} ]^{\frac{{{p_i} - 2}} {{{p_i} - 1}}}} $, we obtain

$ \begin{align} &\quad - ( {{p_i} - 1} ){ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )^{\frac{{{p_i}}} {{{p_i} - 1}}}} + \frac{{{u^{{p_i}}}f'(v)}} {{{{ [ {f(v)} ]}^2}}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i}}} \\ &\geqslant - ( {{p_i} - 1} ){ ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )^{\frac{{{p_i}}} {{{p_i} - 1}}}} + \frac{{{u^{{p_i}}} ( {{p_i} - 1} ){{ [ {f(v)} ]}^{\frac{{{p_i} - 2}} {{{p_i} - 1}}}}}} {{{{ [ {f(v)} ]}^2}}}{ | {\frac{{\partial v}} {{\partial {x_i}}}} |^{{p_i}}} = 0, \end{align} $

(2.4)

then $ II \geqslant 0 $ by (2.4). Let us recall Hölder's inequality

$ \begin{align} {a_0}{b_0} \leqslant { ( {{a_0}^2} )^{\frac{1} {2}}}{ ( {{b_0}^2} )^{\frac{1} {2}}} \end{align} $

(2.5)

for $ {a_0} \geqslant 0, {b_0} \geqslant 0 $, and the equality holds if and only if $ {a_0} = c{b_0} $, $ c $ is a constant. Taking $ {a_0} = \frac{{\partial v}} {{\partial {x_i}}} $ and $ {b_0} = \frac{{\partial u}} {{\partial {x_i}}} $ in (2.5), it yields

$ \begin{align} | {\frac{{\partial v}} {{\partial {x_i}}}} | | {\frac{{\partial u}} {{\partial {x_i}}}} | - \frac{{\partial v}} {{\partial {x_i}}}\frac{{\partial u}} {{\partial {x_i}}} \geqslant 0, \end{align} $

(2.6)

it follows from $ f(v) > 0, u \geqslant 0 $ and (2.6) that $ III \geqslant 0 $. Hence $ L(u, v){\kern 1pt} {\kern 1pt} \geqslant 0 $ by (2.1).

The proof process above also shows that $ L(u, v){\kern 1pt} {\kern 1pt} = 0 $ a.e. in $ \Omega $ if and only if

$ \begin{align} | {\frac{{\partial u}} {{\partial {x_i}}}} | = { ( {\frac{{{u^{{p_i} - 1}}}} {{f(v)}}{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 1}}} )^{\frac{1} {{{p_i} - 1}}}}, f(v) > 0, \end{align} $

(2.7)

$ \begin{align} f'(v) = ( {{p_i} - 1} ){ [ {f(v)} ]^{\frac{{{p_i} - 2}} {{{p_i} - 1}}}}, f(v) > 0, \end{align} $

(2.8)

$ \begin{align} \frac{{\partial v}} {{\partial {x_i}}}\frac{{\partial u}} {{\partial {x_i}}} = | {\frac{{\partial v}} {{\partial {x_i}}}} | | {\frac{{\partial u}} {{\partial {x_i}}}} | \end{align} $

(2.9)

a.e. in $ \Omega $. It follows from (2.7), (2.8) and (2.9) that $ u = cv $ a.e. in $ \Omega $ for a constant $ c $. Hence $ L(u, v) = 0 $ a.e. in $ \Omega $ if and only if $ u = cv $ a.e. in $ \Omega $. The proof of Theorem 1.4 is ended.

Throughout this section, we always assume that $ v > 0, f(v) > 0 $ in $ \Omega $, and $ f'(v) \geqslant ( {{p_i} - 1} ) [ {f{{(v)}^{\frac{{{p_i} - 2}} {{{p_i} - 1}}}}} ] $ for $ {p_i} > 1, i = 1, \cdots , n. $

Let us state anisotropic Sobolev space $ W_0^{1, ( {{p_i}} )}(\Omega ) $ needed in this paper, see Adams [14], Lu [15] and Troisi [16] et al. Let $ \Omega \subset {\mathbb{R}^n} ( {n \geqslant 3} ) $ be a bounded domain, $ {p_i} > 1{\kern 1pt} {\kern 1pt}, i = 1, 2, \cdots , n $. We define anisotropic Sobolev space $ W_0^{1, ( {{p_i}} )}(\Omega ) $ by

$ W_0^{1, ( {{p_i}} )}(\Omega ) = \{ {u \in W_0^{1, 1}(\Omega ):\frac{{\partial u}} {{\partial {x_i}}} \in {L^{{p_i}}}(\Omega ), i = 1, \cdots , n} \} $

with the norm $ { \| u \|_{W_0^{1, ( {{p_i}} )}(\Omega )}} = \sum\limits_{i = 1}^n {{{ ( {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}dx} } )}^{\frac{1} {{{p_i}}}}}}. $ It is well known that $ W_0^{1, {p_i}}(\Omega ) $ is a separable reflexive Banach space.

We give three examples of applications of Theorem 1.4. The first is a Liouville's theorem to an anisotropic elliptic system.

Example 3.1 Let $ g(u, v) $ be an integrable function in $ \Omega $. Assume that $ (u, v) \in W_0^{1, ( {{p_i}} )}(\Omega ) \times W_0^{1, ( {{p_i}} )}(\Omega ) $ is a pair of solution to an anisotropic elliptic system

$ \begin{equation} \left\{ {\begin{array}{*{20}{c}} { \sum\limits_{i = 1}^n {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} = g(u, v), } & {} & {x \in \Omega , } & {} \\ {\sum\limits_{i = 1}^n {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} = g(u, v), } & {} & {x \in \Omega , } & {} \\ {u > 0, v > 0, } & {} & {x \in \Omega , } & {} \\ {u = 0, v = 0, } & {} & {x \in \partial \Omega .} & {} \\ \end{array} } \right. \end{equation} $

(3.1)

Then $ u = cv $ a.e. in $ \Omega $ for some constants $ c $.

Proof It follows from (3.1) that

$ \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}dx} = \int_\Omega {g(u, v)dx} = } \sum\limits_{i = 1}^n {\int_\Omega {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}dx} }, $

which implies

$ \int_\Omega {L(u, v)dx} = \int_\Omega {R(u, v)dx} = \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}dx} } - \sum\limits_{i = 1}^n {\int_\Omega {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}dx} } = 0, $

hence the conclusion is true by Theorem 1.4.

The next is a Sturmian comparison principle to an anisotropic elliptic equation.

Example 3.2 Let $ {f_1}(x) $ and $ {f_2}(x) $ be two continuous functions with $ {f_1}(x) < {f_2}(x) $. Assume that $ u \in W_0^{1, ( {{p_i}} )}(\Omega ) $ satisfies the following Dirichlet problem

$ \begin{equation} \left\{ {\begin{array}{*{20}{c}} { - \sum\limits_{i = 1}^n {\frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial u}} {{\partial {x_i}}}} ) = \sum\limits_{i = 1}^n {{f_1}(x){u^{{p_i} - 1}}} } , } & {} & {x \in \Omega , } \\ {u > 0, } & {} & {x \in \Omega , } \\ {u = 0, } & {} & {x \in \partial \Omega .} \\ \end{array} } \right. \end{equation} $

(3.2)

Then any nontrivial solution $ v $ of the equation

$ \begin{equation} - \sum\limits_{i = 1}^n {\frac{{{u^{{p_i}}}}} {{f(v)}}\frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} ) = \sum\limits_{i = 1}^n {{f_2}(x){u^{{p_i}}}} } , x \in \Omega \end{equation} $

(3.3)

must change sign.

Proof Assume that the solution $ v $ in (3.3) does not change sign and without loss of generality, let $ v>0 $ in $ \Omega $. We have by (3.2) and Theorem 1.4 that

$ \begin{align*} 0 & \leqslant \int_\Omega {L(u, v)dx} = \int_\Omega {R(u, v)dx} \nonumber\\ & = \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} + \sum\limits_{i = 1}^n {\int_\Omega {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}dx} } \nonumber\\ & = \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} - \sum\limits_{i = 1}^n {\int_\Omega {\frac{{{u^{{p_i}}}}} {{f(v)}}\frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} )} dx} \nonumber\\ & = \sum\limits_{i = 1}^n {\int_\Omega { ( {{f_1}(x) - {f_2}(x)} ){u^{{p_i}}}dx} } < 0, \end{align*} $

which is a contradiction. This accomplishes the proof.

The end is a generalized anisotropic Hardy type inequality.

Example 3.3 Assume that there exists a constant $ k > 0 $ and a function $ h(x) $ such that

$ \begin{equation} - \frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} ) \geqslant kh(x)f(v), x \in \Omega \end{equation} $

(3.4)

for every $ i = 1, \cdots , n $. Then,

$ \begin{equation} \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} \geqslant \sum\limits_{i = 1}^n {k\int_\Omega {h(x){u^{{p_i}}}dx} } \end{equation} $

(3.5)

for any $ 0 \leqslant u \in C_0^1(\Omega ) $.

Proof By (3.4) and Theorem 1.4, we have

$ \begin{align*} 0 & \leqslant \int_\Omega {L(u, v)dx} = \int_\Omega {R(u, v)dx} \nonumber\\ & = \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} + \sum\limits_{i = 1}^n {\int_\Omega {\frac{\partial } {{\partial {x_i}}} ( {\frac{{{u^{{p_i}}}}} {{f(v)}}} ){{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}dx} } \nonumber\\ & = \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} - \sum\limits_{i = 1}^n {\int_\Omega {\frac{{{u^{{p_i}}}}} {{f(v)}}\frac{\partial } {{\partial {x_i}}} ( {{{ | {\frac{{\partial v}} {{\partial {x_i}}}} |}^{{p_i} - 2}}\frac{{\partial v}} {{\partial {x_i}}}} )} dx} \nonumber\\ &\leqslant \sum\limits_{i = 1}^n {\int_\Omega {{{ | {\frac{{\partial u}} {{\partial {x_i}}}} |}^{{p_i}}}} dx} - \sum\limits_{i = 1}^n {k\int_\Omega {h(x){u^{{p_i}}}dx} }, \end{align*} $

which implies (3.5).