Nonexistence of Nontrivial Solutions with Decay Order for a Biharmonic P-Laplacian Equation and System ()
1. Introduction
Recently, there has been an active research on the biharmonic equation with a p-Laplacian term
(1.1)
as well as the evolutionary biharmonic equations with a p-Laplacian term
(1.2)
and
(1.3)
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 [1] . For the analysis and applications of (1.1), (1.2), and (1.3), see, for example, [2] - [42] . In this article, we shall use the Morawetz multiplier [43] [44] [45] 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,
(1.4a)
(1.4b)
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:
Math_32#
and
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
and
, we get
(2.1)
where Y depends on ζ and u as well as their partial derivatives up to and including the third order and
, and
where
(2.2)
(2.3)
(2.4)
and
(2.5)
where
Note that we use the Einstein summation notation in the expressions for
and
.
Theorem 1
Let u be a
solution of (1.1) such that u ϵ
. Assume
.
(a) If
and
, then
.
(b) If
and
, then
.
Proof:
Let
. Integrating both sides of (2.1) in
and using the Divergence theorem, we get
Let
. We get
Thus
(2.6)
where A, B, C, P are defined as in (2.2)?(2.5).
The above equation (2.6) can be written as
Let
.
Since u is assumed to be of decay order
and
,
, after substituting ζ by r.
Thus
(2.7)
To prove the assertion (a), since β ≤ 0, we have
Thus
Since
, and
, we have
Thus
. Since
Assertion (b) follows with a similar argument from (2.7).
Remark 2. As an example for
, let
. Then
where
.
Assume
.
For u to be in
, we need
that is,
This would be satisfied if
(2.8)
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
.
Theorem 2
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
.
Proof:
Let
. Following the same steps as in Theorem 1, we get
since
and
.
Thus
Since
and
,
and
.
Acknowledgements
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.