Recently, there has been an active research on the biharmonic equation with a p-Laplacian term
as well as the evolutionary biharmonic equations with a p-Laplacian term
where , α and β are real constants.
Equation (1.1) is the stationary state of the Equation (1.2) while the traveling wave solution for (1.3) satisfies an equation of the form (1.1) as was shown in Strauss  . For the analysis and applications of (1.1), (1.2), and (1.3), see, for example,  -  . In this article, we shall use the Morawetz multiplier    to show that there are no nontrivial solutions of certain decay order for (1.1) and a system of coupled biharmonic equations with p-Laplacian terms,
where , , and are real-valued func- tions, , are all real constants.
As usual, , ∇u denotes the gradient of u, denotes the divergence of u, and . Also the subscript denotes the partial derivative, thus . We also use the notation and . denotes . is the space of functions whose partial derivatives of order up to and including k are continuously differentiable.
Define eight sets of functions , , and which we will use in this article:
where and are the antiderivative of with respect to u and with respect to v, respectively, such that and .
Remark 1. A function u is said to be of decay order (h, k) if and only if .
All the functions are assumed to be real-valued.
2. A Biharmonic Equation with a P-Laplacian Term
We consider the equation (1.1) in this section. Multiplying both sides of the equation (1.1) by the Morawetz multiplier
where Y depends on ζ and u as well as their partial derivatives up to and including the third order and , and
Note that we use the Einstein summation notation in the expressions for and .
Let u be a solution of (1.1) such that u ϵ . Assume .
(a) If and , then .
(b) If and , then .
Let . Integrating both sides of (2.1) in and using the Divergence theorem, we get
Let . We get
where A, B, C, P are defined as in (2.2)?(2.5).
The above equation (2.6) can be written as
Since u is assumed to be of decay order and ,
, after substituting ζ by r.
To prove the assertion (a), since β ≤ 0, we have
Since , and , we have
Thus . Since
Assertion (b) follows with a similar argument from (2.7).
Remark 2. As an example for , let . Then
For u to be in , we need
This would be satisfied if
The above condition (2.8) would be satisfied if u is of decay order .
As for u to be in , since
u would be in if
Thus, if , then u is in .
Therefore, if u is of decay order and , u satisfies the assumptions of Theorem 1 on u.
Remark 3. A similar conclusion can be obtained for where .
3. A System of Biharmonic Equations with p-Laplacian Terms
We consider the system (1.4.a) and (1.4.b) in this section. Let and be the antiderivatives of with respect to u and with respect to v, respectively, such that and .
Assume also . Multiplying both sides of (1.4.a) by and both sides of (1.4.b) by , then adding them up, we get
where Y depends on ζ, u, and v as well as their partial derivatives up to and including the third order, , , , , , , and . Here we assume . Z is similar to Section 2 with appropriate modification to allow terms containing , , , , and .
Let u and v be solutions of the system (1.4.a) and (1.4.b) with . Assume , and . Let . Assume and .
Assume further that and .
(a) If and , then and .
(b) If , and , then and .
Let . Following the same steps as in Theorem 1, we get
since and .
Since and , and .
A partial result of this work was presented in the 2016 International Workshop on Geometric Analysis & Subelliptic PDEs, May 24-26, 2016, Taipei, and NCTS International Workshop on Harmonic Analysis and Geometric Analysis, May 23?25, 2017, Taipei. The author wishes to thank the meeting organizer, Dr. Der- Chen Chang, for the hospitality and the support during the workshop.
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