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.
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1 Introduction and results
In 1971, Dunninger [1] established a Picone identity
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
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
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 \(- \Delta v > 0\)
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
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
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
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
and \(f(v)>0\), \({f'}(v) > 1\), \({f''}(v) \leq 0\) in Ω such that f and g satisfy
and
respectively. Denote
and
respectively. Then \(R(u,v) = L(u,v)\). Moreover, \(L ( {u,v} ) \geq 0\), and \(L ( {u,v} ) = 0\) if and only if
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 \(R(u,v)\):
Next we verify \(L(u,v) \geq 0\), we can rewrite \(L(u,v)\) as
where
We now recall Young’s inequality
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
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
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
It follows from (1.12) that
We can prove \(F_{2}= 0\) by (1.8). A direct calculation shows
by \(\sqrt{{g''}(u)} \nabla u = \frac{{\sqrt{2g(u)} {f'}(v)\nabla v}}{{f(v)}}\) in (1.9), we can also show
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
Then any nontrivial solution v of the following p-biharmonic equation
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
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
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
Taking \({\phi _{1}} = u\) and \({\phi _{2}} = \frac{{g(u)}}{{f(v)}}\) in (2.4) and (2.5), respectively, we obtain
which implies
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
where \(\lambda > 0\) is a constant, \(k(x)\) is a positive continuous function. Then there holds
for any \(0 \leq u \in C_{0}^{2}(\varOmega )\).
Proof
It follows from (2.6) and Theorem 1.5 that
which implies (2.7). □
References
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Dwivedi, G.: Picone’s Identity for p-biharmonic operator and Its Applications. arXiv:1503.05535
Dwivedi, G., Tyagi, J.: Remarks on the qualitative questions for biharmonic operators. Taiwan. J. Math. 19(6), 1743–1758 (2015)
Dwivedi, G., Tyagi, J.: A note on the Caccioppoli inequality for biharmonic operators. Mediterr. J. Math. 13(4), 1823–1828 (2016)
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)
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).
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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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DOI: https://doi.org/10.1186/s13660-019-2008-8