_{1}

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We use the Morawetz multiplier to show that there are no nontrivial solutions of certain decay order for a biharmonic equation with a p-Laplacian term and a system of coupled biharmonic equations with p-Laplacian terms in the entire Euclidean space.

Recently, there has been an active research on the biharmonic equation with a p-Laplacian term

Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) = 0. (1.1)

as well as the evolutionary biharmonic equations with a p-Laplacian term

∂ u / ∂ t = Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) (1.2)

and

∂ 2 u / ∂ t 2 = Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) (1.3)

where x ∈ R n , n ≥ 2 , p > 1 , 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 [

Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) + [ a ( x ) u 2 + b ( x ) v 2 ] u = 0 , (1.4a)

Δ 2 v + σ ∇ ⋅ ( | ∇ u | p − 2 ∇ v ) + μ ⋅ v + g ( x , v ) + [ c ( x ) u 2 + d ( x ) v 2 ] v = 0 , (1.4b)

where u = u ( x ) , v = v ( x ) , a ( x ) , b ( x ) , c ( x ) and d ( x ) are real-valued func- tions, x ∈ R n , n ≥ 2 , α , β , σ , μ are all real constants.

As usual, x = ( x 1 , x 2 , ⋯ , x n ) , ∇u denotes the gradient of u, ∇ ⋅ u denotes the divergence of u, and r = | x | . Also the subscript denotes the partial derivative, thus u s = ∂ u / ∂ s . We also use the notation u r = ∂ u / ∂ r = ( x / r ) ⋅ ∇ u and ∂ j = ∂ / ∂ x j . F r ( x , s ) denotes ∂ F ( x , s ) / ∂ r = ( x / r ) ⋅ ∇ x F ( x , s ) . C k ( R n ) is the space of functions whose partial derivatives of order up to and including k are continuously differentiable.

Define eight sets of functions A m ( R n ) , B m ( R n ) , D h , k ( R n ) , E m ( R n ) , F n ( R n ) , G n ( R n ) , N F ( R n ) and N G ( R n ) which we will use in this article:

A m ( R n ) = { a | a ∈ C 1 ( R n ) , sup | x | ≥ ρ ( | x | − m | a ( x ) | ) < ∞ forsome ρ > 0 and m > 0 , and r a r ( x ) − ( n − 2 ) a ( x ) ≤ 0 }

B m ( R n ) = { b | b ∈ C 1 ( R n ) , sup | x | ≥ ρ ( | x | − m | b ( x ) | ) < ∞ for some ρ > 0 and m > 0 , and 2 r b r ( x ) − ( n − 3 ) b ( x ) ≤ 0 }

D h , k ( R n ) = { u | u ∈ C k ( R n ) , lim R → ∞ ( sup | x | ≤ R ( | x α | | D β u ( x ) | ) ) = 0 , forallmulti-indices α and β ∈ N 0 n suchthat | α | ≤ h and | β | ≤ k , where α = ( α 1 , α 2 , ⋯ , α n ) , β = ( β 1 , β 2 , ⋯ , β n ) , | α | = α 1 + α 2 + ⋯ + α n and | β | = β 1 + β 2 + ⋯ + β n } ,

E m ( R n ) = { a | a ∈ C 1 ( R n ) , sup | x | ≥ ρ ( | x | − m | a ( x ) | ) < ∞ for some ρ > 0 and m > 0 } ,

F n ( R n ) = { u | lim R → ∞ ( R n sup | x | = R | F ( x , u ( x ) ) | ) = 0 } , Math_32#

N F ( R n ) = { u | ∫ R n [ n F ( x , u ( x ) ) + r F r ( x , u ( x ) ) − ( ( n − 1 ) / 2 ) f ( x , u ( x ) ) u ( x ) ] d x ≤ 0 } ,

and

N G ( R n ) = { v | ∫ R n [ n G ( x , v ( x ) ) + r G r ( x , v ( x ) ) − ( ( n − 1 ) / 2 ) g ( x , v ( x ) ) v ( x ) ] d x ≤ 0 } ,

where F ( x , u ) and G ( x , v ) are the antiderivative of f ( x , u ) with respect to u and g ( x , v ) with respect to v, respectively, such that F ( x , 0 ) = 0 and G ( x , 0 ) = 0 .

Remark 1. A function u is said to be of decay order (h, k) if and only if u ∈ D h , k ( R n ) .

All the functions are assumed to be real-valued.

We consider the equation (1.1) in this section. Multiplying both sides of the equation (1.1) by the Morawetz multiplier ζ ( u r + ( ( n – 1 ) u / ( 2 r ) ) ) , where ζ ∈ C 3 ( R n ) and ζ ( x ) = ζ ( | x | ) = ζ ( r ) , we get

0 = [ Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) ] ζ ( u r + ( ( n − 1 ) u / ( 2 r ) ) ) = ∇ ⋅ Y + Z , (2.1)

where Y depends on ζ and u as well as their partial derivatives up to and including the third order and F ( x , u ) , and

Z = ( 3 ζ ′ / 2 ) ( Δ u ) 2 + A ( u r ) 2 + B ( | ∇ u | 2 − | u r | 2 ) + C u 2 + ( ζ − r ζ ′ ) P + α { ( ( ζ / r ) – ζ ′ ) | ∇ u | p − 2 [ ( u r ) 2 + ( ( n − 1 ) / ( 2 r ) ) u r u ] + [ ( ζ ′ / p ) – ( ζ / ( 2 p r ) ) ( ( n + 1 ) p − 2 ( n − 1 ) ) ] | ∇ u | p } − β { ( 1 / 2 ) ζ r | ∇ u | 2 + ( ∇ u ⋅ ∇ ζ ) u r − ζ r | u r | 2 + ( ( 1 / r ) ζ − ζ r ) ( | ∇ u | 2 − | u r | 2 ) + ( ( n – 1 ) / ( 2 r ) ) [ ( 1 / r ) ζ r − ( 1 / 2 ) ( Δ ζ ) + ( ( n − 3 ) / ( 2 r 2 ) ) ζ ] u 2 } – ( ζ r + ( ( n − 1 ) / r ) ζ ) F ( x , u ) + ( ( n − 1 ) / ( 2 r ) ) ζ f ( x , u ) u − ζ F r ( x , u )

where

A = − 7 ζ ‴ / 2 − ( n − 1 ) ( n − 3 ) ( ζ ′ − ( ζ / r ) ) / ( 2 r 2 ) , (2.2)

B = − 3 ζ ‴ / 2 + ( n − 5 ) ζ ″ / r − ( n 2 + 2 n − 19 ) ( ζ ′ − ζ / r ) / ( 2 r 2 ) , (2.3)

C = ( ( n – 1 ) / 2 ) [ ζ ' ' ' ' / ( 2 r ) + ( n − 3 ) ζ ‴ / ( r 2 ) + ( n − 3 ) ( n − 7 ) ζ ″ / ( 2 r 3 ) – 3 ( n − 3 ) ( n − 5 ) ( ζ ′ − ζ / r ) / ( 2 r 4 ) ] , (2.4)

and

P = ( 2 / r ) [ ∑ ( S i j u ) 2 − ∑ ( ∑ ( x j / r ) S i j u ) 2 ] ≥ 0 , (2.5)

where

S i j u = ( x i / r 3 ) ∑ [ x k ( x k ∂ j – x j ∂ k ) u r ] + ∂ j ∑ [ ( x k / r 2 ) ( x k ∂ i – x i ∂ k ) u ] .

Note that we use the Einstein summation notation in the expressions for P and S i j u .

Theorem 1

Let u be a c 4 solution of (1.1) such that u ϵ D n , 3 ( R n ) ∩ F n ( R n ) ∩ N F ( R n ) . Assume β ≤ 0 .

(a) If p ≥ ( 2 n ) / ( n + 1 ) and α ≤ 0 , then u ≡ 0 .

(b) If 1 < p ≤ ( 2 n ) / ( n + 1 ) and α ≥ 0 , then u ≡ 0 .

Proof:

Let R > 0 . Integrating both sides of (2.1) in | x | ≤ R and using the Divergence theorem, we get

∫ | x | = R Y ⋅ ( x / R ) d s + ∫ | x | ≤ R Z d x = 0.

Let R → ∞ . We get

lim R → ∞ ∫ | x | = R Y ⋅ ( x / R ) d s + ∫ R n Z d x = 0.

Thus

0 = lim R → ∞ ∫ | x | = R Y ⋅ ( x / R ) d s + ∫ R n { ( 3 ζ ′ / 2 ) ( Δ u ) 2 + A ( u r ) 2 + B ( | ∇ u | 2 − | u r | 2 ) + C u 2 + ( ζ − r ζ ′ ) P + α { ( ( ζ / r ) − ζ ′ ) | ∇ u | p − 2 [ ( u r ) 2 + ( ( n − 1 ) / ( 2 r ) ) u r u ] + [ ( ζ ′ / p ) − ( ζ / ( 2 p r ) ) ( ( n + 1 ) p − 2 ( n − 1 ) ) ] | ∇ u | p } − β { ( 1 / 2 ) ζ r | ∇ u | 2 + ( ∇ u ⋅ ∇ ζ ) u r − ζ r | u r | 2 + ( ( 1 / r ) ζ − ζ r ) ( | ∇ u | 2 − | u r | 2 ) + ( ( n – 1 ) / ( 2 r ) ) [ ( 1 / r ) ζ r − ( 1 / 2 ) ( Δ ζ ) + ( ( n − 3 ) / ( 2 r 2 ) ) ζ ] u 2 } – ( ζ r + ( ( n − 1 ) / r ) ζ ) F ( x , u ) + ( ( n − 1 ) / ( 2 r ) ) ζ f ( x , u ) u − ζ F r ( x , u ) } d x , (2.6)

where A, B, C, P are defined as in (2.2)?(2.5).

The above equation (2.6) can be written as

0 = lim R → ∞ ∫ | x | = R Y ⋅ ( x / R ) d s + ∫ R n { ( 3 ζ ′ / 2 ) ( Δ u ) 2 + A ( u r ) 2 + B ( | ∇ u | 2 − | u r | 2 ) + C u 2 + ( ζ − r ζ ′ ) P + α { ( ( ζ / r ) − ζ ′ ) | ∇ u | p − 2 [ ( u r ) 2 + ( ( n − 1 ) / ( 2 r ) ) u r u ] + [ ( ζ ′ / p ) − ( ζ / ( 2 p r ) ) ( ( n + 1 ) p − 2 ( n − 1 ) ) ] | ∇ u | p } − β { ( 1 / 2 ) ζ r | ∇ u | 2 + ( ∇ u ⋅ ∇ ζ ) u r − ζ r | u r | 2 + ( ( 1 / r ) ζ − ζ r ) ( | ∇ u | 2 − | u r | 2 ) + ( ( n − 1 ) / ( 2 r ) ) [ ( 1 / r ) ζ r − ( 1 / 2 ) ( Δ ζ ) + ( ( n − 3 ) / ( 2 r 2 ) ) ζ ] u 2 } d x = ∫ R n ( ζ r + ( ( n − 1 ) / r ) ζ ) F ( x , u ) − ( ( n − 1 ) / ( 2 r ) ) ζ f ( x , u ) u + ζ F r ( x , u ) } d x ,

Let ζ ( x ) = ζ ( | x | ) = ζ ( r ) = r .

Since u is assumed to be of decay order ( n , 3 ) and u ∈ F n ( R n ) ,

lim R → ∞ ∫ | x | = R Y ⋅ ( x / R ) d s = 0 , after substituting ζ by r.

Thus

∫ R n { ( 3 / 2 ) ( Δ u ) 2 + α [ ( 1 / p ) − ( 1 / ( 2 p ) ) ( ( n + 1 ) p − 2 ( n − 1 ) ) ] | ∇ u | p − ( β / 2 ) | ∇ u | 2 } d x = ∫ R n [ n F ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u + r F r ( x , u ) ] d x . (2.7)

To prove the assertion (a), since β ≤ 0, we have

∫ R n { ( 3 / 2 ) ( Δ u ) 2 + α [ ( 1 / p ) − ( 1 / ( 2 p ) ) ( ( n + 1 ) p − 2 ( n − 1 ) ) ] | ∇ u | p } d x ≤ ∫ R n [ n F ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u + r F r ( x , u ) ] d x

Thus

∫ R n { ( 3 / 2 ) ( Δ u ) 2 + α [ ( 2 n − ( n + 1 ) p ) / ( 2 p ) ] | ∇ u | p } d x ≤ ∫ R n [ n F ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u + r F r ( x , u ) ] d x

Since α ≤ 0 , p ≥ ( 2 n ) / ( n + 1 ) , and u ∈ N F ( R n ) , we have

0 ≤ ∫ R n ( 3 / 2 ) ( Δ u ) 2 d x ≤ ∫ R n [ n F ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u + r F r ( x , u ) ] d x ≤ 0 ,

Thus ∫ R n ( 3 / 2 ) ( Δ u ) 2 d x = 0 . Since u ∈ D n , 3 ( R n ) , u ≡ 0

Assertion (b) follows with a similar argument from (2.7).

Remark 2. As an example for f ( x , u ) , let f ( x , u ) = q ( x ) | u | s − 1 u , s ≥ 1 . Then

F ( x , u ) = ( 1 / ( s + 1 ) ) q ( x ) | u | s + 1 , where q ( x ) = q ( | x | ) = q ( r ) .

Assume q ∈ E m ( R n ) .

For u to be in F n ( R n ) , we need

lim R → ∞ ( R n sup | x | = R | F ( x , u ( x ) ) | ) = 0 ,

that is,

lim R → ∞ ( R n sup | x | = R ( q ( | x | ) | u ( x ) | s + 1 ) ) = 0 ,

This would be satisfied if

lim R → ∞ ( R n sup | x | = R ( | x | m | u ( x ) | s + 1 ) ) = 0. (2.8)

The above condition (2.8) would be satisfied if u is of decay order ( n + m , 0 ) .

As for u to be in N F ( R n ) , since

n F ( x , u ) + r F r ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u = [ ( n / ( s + 1 ) ) q + ( r / ( s + 1 ) ) q r − ( ( n − 1 ) / 2 ) q ] | u | s + 1 ,

u would be in N F ( R n ) if ( n / ( s + 1 ) ) q + ( r / ( s + 1 ) ) q r − ( ( n − 1 ) / 2 ) q ≤ 0.

Thus, if r q r ≤ ( ( n s − n − s − 1 ) / 2 ) q , then u is in N F ( R n ) .

Therefore, if u is of decay order ( n + m , 1 ) and r q r ≤ ( ( n s − n − s − 1 ) / 2 ) q , u satisfies the assumptions of Theorem 1 on u.

Remark 3. A similar conclusion can be obtained for f ( x , u ) = q 1 ( x ) | u | a − 1 u + q 2 ( x ) | u | b − 1 u , where a > b ≥ 1 .

We consider the system (1.4.a) and (1.4.b) in this section. Let F ( x , u ) and G ( x , v ) be the antiderivatives of f ( x , u ) with respect to u and g ( x , v ) with respect to v, respectively, such that F ( x , 0 ) = 0 and G ( x , 0 ) = 0 .

Assume also b ( x ) = c ( x ) . Multiplying both sides of (1.4.a) by ζ ( u r + ( ( n − 1 ) u / ( 2 r ) ) ) and both sides of (1.4.b) by ζ ( v r + ( ( n − 1 ) v / ( 2 r ) ) ) , then adding them up, we get

0 = ( Δ 2 u + α ∇ ⋅ ( | ∇ u | p − 2 ∇ u ) + β ⋅ u + f ( x , u ) + [ a ( x ) u 2 + b ( x ) v 2 ] u ) ζ ( u r + ( ( n − 1 ) u / ( 2 r ) ) ) + ( Δ 2 v + σ ∇ ⋅ ( | ∇ v | p − 2 ∇ v ) + μ ⋅ v + g ( x , v ) + [ c ( x ) u 2 + d ( x ) v 2 ] v ) ζ ( v r + ( ( n − 1 ) v / ( 2 r ) ) ) = ∇ · Y + Z ,

where Y depends on ζ, u, and v as well as their partial derivatives up to and including the third order, F ( x , u ) , G ( x , v ) , α , β , σ , μ , a ( x ) , b ( x ) , c ( x ) , and d ( x ) . Here we assume b ( x ) = c ( x ) . Z is similar to Section 2 with appropriate modification to allow terms containing v , σ , μ , a ( x ) , b ( x ) , c ( x ) , and d ( x ) .

Theorem 2

Let u and v be C 4 solutions of the system (1.4.a) and (1.4.b) with b ( x ) = c ( x ) . Assume a ∈ A k , 2 ( R n ) , b ∈ B m , 2 ( R n ) and d ∈ A h , 2 ( R n ) . Let s = max { k , m , h , n } . Assume β ≤ 0 and μ ≤ 0 .

Assume further that u ∈ D s , 3 ( R n ) ∩ F n ( R n ) ∩ N F ( R n ) and v ∈ D s , 3 ( R n ) ∩ G n ( R n ) ∩ N G ( R n ) .

(a) If p ≥ ( 2 n ) / ( n + 1 ) , α ≤ 0 and σ ≤ 0 , then u ≡ 0 and v ≡ 0 .

(b) If 1 < p ≤ ( 2 n ) / ( n + 1 ) , α ≥ 0 , and σ ≥ 0 , then u ≡ 0 and v ≡ 0 .

Proof:

Let ζ ( x ) = ζ ( | x | ) = ζ ( r ) = r . Following the same steps as in Theorem 1, we get

0 ≤ ∫ R n ( 3 / 2 ) ( ( Δ u ) 2 + ( Δ v ) 2 ) d x ≤ ∫ R n { [ n F ( x , u ) − ( ( n − 1 ) / 2 ) f ( x , u ) u + r F r ( x , u ) ] + [ n G ( x , v ) − ( ( n − 1 ) / 2 ) g ( x , v ) v + r G r ( x , v ) ] + ( 1 / 4 ) [ r a r ( x ) − ( n − 2 ) a ( x ) ] u 4 + ( 1 / 4 ) [ r d r ( x ) − ( n − 2 ) d ( x ) ] v 4 + ( 1 / 2 ) [ r b r ( x ) − ( ( n − 3 ) / 2 ) b ( x ) ] u 2 v 2 } d x ≤ 0 ,

since a ∈ A k ( R n ) , b ∈ B m ( R n ) , d ∈ A h ( R n ) , u ∈ D s , 3 ( R n ) ∩ F n ( R n ) ∩ N F ( R n ) and v ∈ D s , 3 ( R n ) ∩ G n ( R n ) ∩ N G ( R n ) .

Thus

∫ R n ( 3 / 2 ) ( ( Δ u ) 2 + ( Δ v ) 2 ) d x = 0.

Since u and v ∈ D s , 3 ( R n ) , u ≡ 0 and v ≡ 0 .

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.

Lin, J.-E. (2017) Nonexistence of Nontrivial Solutions with Decay Order for a Biharmonic P-Laplacian Equation and System. Applied Mathematics, 8, 920-928. https://doi.org/10.4236/am.2017.87072