CUBO A Mathematical Journal
Vol.14, No¯ 02, (175–182). June 2012

On a Condition for the Nonexistence of W-Solutions of
Nonlinear High-Order Equations with L1-Data

Alexander A. Kovalevsky

Institute of Applied Mathematics and Mechanics,

Rosa Luxemburg St. 74, 83114 Donetsk, Ukraine

email: alexkvl@iamm.ac.donetsk.ua

and

Francesco Nicolosi

Department of Mathematics and Informatics,

University of Catania, 95125 Catania, Italy

email: fnicolosi@dmi.unict.it

ABSTRACT

In a bounded open set of Rn we consider the Dirichlet problem for nonlinear 2m-order

equations in divergence form with L1-right-hand sides. It is supposed that 2 6 m < n,

and the coefficients of the equations admit the growth of rate p−1 > 0 with respect to

the derivatives of order m of unknown function. We establish that under the condition

p 6 2 − m/n for some L1-data the corresponding Dirichlet problem does not have

W-solutions.

RESUMEN

En un conjunto abierto y acotado de Rn consideramos el problema de Dirichlet para

ecuaciones no lineales de orden 2m en la forma divergente con lados L1-right-hand.

Se supone que 2 6 m < n, y los coeficientes de las ecuaciones admiten el radio de

crecimiento p−1 > 0 con respecto a las derivadas de orden m de la función desconocida.

Establecemos que bajo la condición p 6 2 − m/n para algn L1- data el problema de

Dirichlet correspondiente no tiene W-soluciones.

Keywords and Phrases: Nonlinear high-order equations in divergence form, L1-data, Dirichlet

problem, W-solution, nonexistence of W-solutions.

2010 AMS Mathematics Subject Classification: 35G30, 35J40, 35J60.



176 Alexander A. Kovalevsky and Francesco Nicolosi CUBO
14, 2 (2012)

1 Introduction

It is known that nonlinear elliptic second-order equations in divergence form whose principal coeffi-

cients grow with respect to the gradient of unknown function u as |∇u|p−1 for some L1-right-hand

sides may not have weak solutions if the exponent p is sufficiently close to 1. The fact of the

nonexistence of weak solutions was observed in [1] by giving the following example: if Ω is an open

set of Rn with n > 2 and 1 < p 6 2 − 1/n, then there exists a function f ∈ L1(Ω) such that the

problem

u ∈ W1,1
loc

(Ω), −∆pu + u = f in D
′
(Ω)

does not have a solution.

The given observation was one of motivations for the development of the theory of entropy

solutions for nonlinear elliptic second-order equations with L1-data [1]. According to the results

of [1], if 1 < p < n, under natural growth, coercivity and strict monotonicity conditions for

coefficients of the equations under consideration an entropy solution exists and is unique for every

L1-right-hand side. Moreover, if p > 2 − 1/n, the entropy solution is a weak solution.

Analogous results on the existence of entropy and weak solutions for nonlinear elliptic high-

order equations with coefficients satisfying a strengthened coercivity condition and L1-right-hand

sides were obtained in [3, 4]. Conditions of the existence of weak solutions for some classes of

degenerate nonlinear elliptic high-order equations with strengthened coercivity and L1-data were

given in [5, 6].

As far as nonlinear elliptic high-order equations with L1-right-hand sides and coefficients

satisfying the natural coercivity condition are concerned, the question on their solvability on the

whole is still open. It seems, for these equations the approaches which work in the cases of second-

order equations with L1-data and high-order equations with strengthened coercivity and L1-data

are not suitable.

On the other hand, the use of the known principle of uniform boundedness (see [2, Chapter

2]) one can consider as a general functional tool for the study of conditions for the nonexistence of

weak solutions of nonlinear arbitrary even order equations in divergence form with L1-data. Using

this principle, in the present article we give such a condition for high-order equations.

2 Main Results

Let m,n ∈ N be numbers such that 2 6 m < n. Let Ω be a bounded open set of Rn.

We shall use the following notation: Λ is the set of all n-dimensional multi-indicies α such

that |α| = m, Rnm is the space of all functions ξ : Λ → R; if u ∈ L
1
loc

(Ω) and the function u has

the weak derivatives Dαu, α ∈ Λ, then ∇mu : Ω → R
n
m is the mapping such that for every x ∈ Ω



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On a Condition for the Nonexistence of W-Solutions ... 177

and for every α ∈ Λ, (∇mu(x))α = D
αu(x).

Let p > 1, c > 0, g ∈ L1/(p−1)(Ω), g > 0 in Ω, and let for every α ∈ Λ, Aα : Ω × R
n
m → R

be a Carathéodory function. We shall assume that for almost every x ∈ Ω and for every ξ ∈ Rnm,
∑

α∈Λ

|Aα(x,ξ)| 6 c
∑

α∈Λ

|ξα|
p−1

+ g(x). (2.1)

For every f ∈ L1(Ω) by (Pf) we denote the following problem:
∑

α∈Λ

(−1)|α|DαAα(x,∇mu) = f in Ω,

Dαu = 0, |α| 6 m − 1, on ∂Ω.

Definition 2.1. Let f ∈ L1(Ω). A W-solution of problem (Pf) is a function u ∈
◦

Wm,1(Ω) such

that

(i) for every α ∈ Λ, Aα(x,∇mu) ∈ L
1(Ω);

(ii) for every ϕ ∈ C∞0 (Ω),
∫

Ω

{
∑

α∈Λ

Aα(x,∇mu)D
αϕ

}

dx =

∫

Ω

fϕdx.

Theorem 2.1. Suppose that

p 6 2 −
m

n
. (2.2)

Then there exists f ∈ L1(Ω) such that problem (Pf) does not have W-solutions.

Proof. Let us assume that for every f ∈ L1(Ω) there exists a W-solution of problem (Pf). This

implies that if f ∈ L1(Ω), then there exists a function uf ∈
◦

Wm,1(Ω) such that for every α ∈ Λ,

Aα(x,∇muf) ∈ L
1(Ω) and

∀ϕ ∈ C∞0 (Ω),

∫

Ω

{
∑

α∈Λ

Aα(x,∇muf)D
αϕ

}

dx =

∫

Ω

fϕdx. (2.3)

Observe that due to the inequality p > 1 and (2.2) we have 0 < 2−p < 1. We set p1 = 1/(2−p).

Clearly, p1 > 1.

Using (2.1) and the inclusion g ∈ L1/(p−1)(Ω), we establish that if f ∈ L1(Ω), then for every

α ∈ Λ, Aα(x,∇muf) ∈ L
p1/(p1−1)(Ω).

For every f ∈ L1(Ω) we define the functional Hf :
◦

Wm,p1(Ω) → R by

〈Hf,ϕ〉 =

∫

Ω

{
∑

α∈Λ

Aα(x,∇muf)D
αϕ

}

dx, ϕ ∈
◦

Wm,p1(Ω).



178 Alexander A. Kovalevsky and Francesco Nicolosi CUBO
14, 2 (2012)

It is easy to see that

∀f ∈ L1(Ω), Hf ∈ (
◦

Wm,p1(Ω))∗. (2.4)

Moreover, taking into account (2.3), for every f ∈ L1(Ω) and for every ϕ ∈ C∞0 (Ω) we get

〈Hf,ϕ〉 =

∫

Ω

fϕdx. (2.5)

From (2.4) and (2.5) it follows that

for every f1,f2 ∈ L
1(Ω), Hf1+f2 = Hf1 + Hf2, (2.6)

for every f ∈ L1(Ω) and for every λ ∈ R, Hλf = λHf. (2.7)

Next, let ϕ ∈
◦

Wm,p1(Ω). We fix a sequence {ϕk} ⊂ C
∞

0 (Ω) such that

‖ϕk − ϕ‖Wm,p1 (Ω) → 0. (2.8)

For every k ∈ N we define the functional Fk : L
1(Ω) → R by

〈Fk,f〉 = |〈Hf,ϕk〉|, f ∈ L
1
(Ω).

Using (2.5), we establish the following fact: if k ∈ N and f1,f2 ∈ L
1(Ω), then

|〈Fk,f1〉 − 〈Fk,f2〉| 6
(

max
Ω

|ϕk|
)

‖f1 − f2‖L1(Ω).

This implies that for every k ∈ N the functional Fk is continuous on L
1(Ω). Moreover, with the

use of (2.6) and (2.7) we obtain the next properties:

(i) for every k ∈ N and for every f1,f2 ∈ L
1(Ω),

〈Fk,f1 + f2〉 6 〈Fk,f1〉 + 〈Fk,f2〉;

(ii) for every k ∈ N, for every f ∈ L1(Ω) and for every λ ∈ R,

〈Fk,λf〉 = |λ|〈Fk,f〉.

Finally, taking into account (2.4) and (2.8), we establish that for every f ∈ L1(Ω) the sequence of

the numbers 〈Fk,f〉 is bounded. This along with the nonnegativity and continuity of the functionals

Fk, properties (i) and (ii) and the principle of uniform boundedness [2, Chapter 2] allows us to

conclude that there exists M > 0 such that for every k ∈ N and for every f ∈ L1(Ω),

〈Fk,f〉 6 M‖f‖L1(Ω).

From the result obtained, using the definition of the functionals Fk and (2.4) and (2.8), we deduce

that

∀f ∈ L1(Ω), |〈Hf,ϕ〉| 6 M‖f‖L1(Ω). (2.9)



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On a Condition for the Nonexistence of W-Solutions ... 179

Now let F : L1(Ω) → R be the functional such that for every f ∈ L1(Ω),

〈F,f〉 = 〈Hf,ϕ〉. (2.10)

Owing to (2.6) and (2.7), the functional F is linear, and by virtue of (2.9) and (2.10), for every

f ∈ L1(Ω), |〈F,f〉| 6 M‖f‖L1(Ω). Therefore, F ∈ (L
1(Ω))∗. Then there exists a function ψ ∈ L∞(Ω)

such that for every f ∈ L1(Ω),

〈F,f〉 =

∫

Ω

ψfdx.

This and (2.10) imply that

∀f ∈ L1(Ω), 〈Hf,ϕ〉 =

∫

Ω

ψfdx. (2.11)

Let us show that ϕ = ψ a. e. in Ω. In fact, let f ∈ L1/(p−1)(Ω). Then, by (2.5) and (2.8),

〈Hf,ϕk〉 →

∫

Ω

fϕdx. (2.12)

On the other hand, from (2.4), (2.8) and (2.11) it follows that

〈Hf,ϕk〉 →

∫

Ω

fψdx.

This and (2.12) imply that ∫

Ω

f(ϕ − ψ)dx = 0.

Hence, taking into account the arbitrariness of the function f in L1/(p−1)(Ω), we obtain that ϕ = ψ

a. e. in Ω. Therefore, ϕ ∈ L∞(Ω).

Thus, we conclude that
◦

Wm,p1(Ω) ⊂ L∞(Ω). (2.13)

However, since, by (2.2), we have mp1 6 n, inclusion (2.13) is not true.

For instance, if p < 2 − m/n, y ∈ Ω, v : Ω → R is a function such that for every x ∈ Ω\{y},

v(x) = ln |x − y|, B is a closed ball in Rn with the center y such that B ⊂ Ω, ψ1 ∈ C
∞

0 (Ω) and

ψ1 = 1 in B, then vψ1 ∈
◦

Wm,p1(Ω)\L∞(Ω).

The contradiction obtained allows us to conclude that there exists a function f ∈ L1(Ω) such

that problem (Pf) does not have W-solutions. This completes the proof of the theorem.

Now we give an analogous result for equations with lower-order terms.

Let c0 > 0, 0 < σ < n/(n − m), σ1 =
n

σ(n−m)
, g0 ∈ L

σ1(Ω), g0 > 0 in Ω, and let

A : Ω × R → R be a Carathéodory function such that for almost every x ∈ Ω and for every s ∈ R,

|A(x,s)| 6 c0|s|
σ
+ g0(x). (2.14)



180 Alexander A. Kovalevsky and Francesco Nicolosi CUBO
14, 2 (2012)

For every f ∈ L1(Ω) by (Pf) we denote the following problem:

∑

α∈Λ

(−1)|α|DαAα(x,∇mu) + A(x,u) = f in Ω,

Dαu = 0, |α| 6 m − 1, on ∂Ω.

Definition 2.2. Let f ∈ L1(Ω). A W-solution of problem (Pf) is a function u ∈
◦

Wm,1(Ω) such

that

(i) for every α ∈ Λ, Aα(x,∇mu) ∈ L
1(Ω);

(ii) A(x,u) ∈ L1(Ω);

(iii) for every ϕ ∈ C∞0 (Ω),

∫

Ω

{
∑

α∈Λ

Aα(x,∇mu)D
αϕ + A(x,u)ϕ

}

dx =

∫

Ω

fϕdx.

Theorem 2.2. Suppose that condition (2.2) is satisfied. Then there exists f ∈ L1(Ω) such that

problem (Pf) does not have W-solutions.

Proof. Let us assume that for every f ∈ L1(Ω) there exists a W-solution of problem (Pf).

This implies that if f ∈ L1(Ω), then there exists a function uf ∈
◦

Wm,1(Ω) such that for every

α ∈ Λ, Aα(x,∇muf) ∈ L
1(Ω), A(x,uf) ∈ L

1(Ω) and

∀ϕ ∈ C∞0 (Ω),

∫

Ω

{
∑

α∈Λ

Aα(x,∇muf)D
αϕ + A(x,uf)ϕ

}

dx =

∫

Ω

fϕdx. (2.15)

We set p1 = 1/(2 − p). As in the proof of the previous theorem, we have: if f ∈ L
1(Ω), then

for every α ∈ Λ, Aα(x,∇muf) ∈ L
p1/(p1−1)(Ω). Moreover, taking into account that, by Sobolev

embedding theorem,
◦

Wm,1(Ω) ⊂ Ln/(n−m)(Ω) and using the inclusion g0 ∈ L
σ1(Ω) and (2.14),

we obtain that for every f ∈ L1(Ω), A(x,uf) ∈ L
σ1(Ω).

Next, we define

V =
◦

Wm,p1(Ω) ∩ Lσ1/(σ1−1)(Ω).

The set V is a Banach space with the norm

‖u‖V = ‖u‖Wm,p1 (Ω) + ‖u‖Lσ1/(σ1−1)(Ω).

For every f ∈ L1(Ω) we define the functional Gf : V → R by

〈Gf,ϕ〉 =

∫

Ω

{
∑

α∈Λ

Aα(x,∇muf)D
αϕ + A(x,uf)ϕ

}

dx, ϕ ∈ V.



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On a Condition for the Nonexistence of W-Solutions ... 181

Clearly,

∀f ∈ L1(Ω), Gf ∈ V
∗. (2.16)

Furthermore, taking into account (2.15), for every f ∈ L1(Ω) and for every ϕ ∈ C∞0 (Ω) we get

〈Gf,ϕ〉 =

∫

Ω

fϕdx. (2.17)

We denote by V1 the closure of C
∞

0 (Ω) in V. Using (2.16) and (2.17) and arguing by analogy

with the proof of Theorem 2.1, we establish that

V1 ⊂ L
∞

(Ω). (2.18)

However, since, by condition (2.2), we have mp1 6 n, inclusion (2.18) is not true.

For instance, if p < 2−m/n and v and ψ1 are the functions described at the end of the proof

of Theorem 2.1, then vψ1 ∈ V1\L
∞(Ω).

The contradiction obtained allows us to conclude that there exists f ∈ L1(Ω) such that problem

(Pf) does not have W-solutions. This completes the proof of the theorem.

3 Remarks

Simple examples of the functions Aα and A satisfying inequalities (2.1) and (2.14) are as

follows:

(i) Aα(x,ξ) = |ξα|
p−1 or Aα(x,ξ) = |ξα|

p−1signξα if α ∈ Λ;

(ii) A(x,s) = as or A(x,s) = a|s|σ or A(x,s) = a|s|σsigns, where a ∈ R and 0 < σ <

n/(n − m).

Finally, observe that if 1 < p < n/m, Aα(x,ξ) = |ξα|
p−1signξα, α ∈ Λ, A(x,s) = a|s|

σsigns,

where a > 0 and σ ∈ (0,n/(n − m)), then for every f ∈ Lnp/(n−mp)(Ω) problem (Pf) has a W-

solution u ∈
◦

Wm,p(Ω). This fact simply follows from the known results of the theory of monotone

operators (see for instance [7]). However, if 1 < p 6 2− m
n
, according to Theorem 2.2, there exists

f ∈ L1(Ω) such that problem (Pf) does not have W-solutions.

Received: March 2011. Revised: December 2011.

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