EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 3, 2020, 579-586
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On solvability of p- harmonic type equations in grand
Sobolev spaces

Alik M. Najafov1,2,∗, Sain T. Alekberli3

1 Azerbaijan University of Architecture and Construction, Baku, Azerbaijan
2 Institute of Mathematics and Mechanics, National Academy of Science of Azerbaijan,
Baku, Azerbaijan
3 Baku Engineering University, Baku, Azerbaijan

Abstract. In this paper with the help of variational method existence and uniqueness of solution
of p- harmonic type equations in grand Sobolev spaces is studied.
2020 Mathematics Subject Classifications: 35A01,35A02, 35A15, 35D30
Key Words and Phrases: p- harmonic type equations, grand Sobolev space, variational method,
Dirichlet problem

1. Introduction and preliminary notes

It is well known that the existence and uniqueness of Dirichlet problem for p-harmonic
equations

div
(
|∇u|p−2∇u

)
= divf, (1)

u|∂G = 0 (2)

in Sobolev and grand Sobolev spaces were studied, e.g., in [1, 2] see also [4–7, 10–13].
Namely, in these papers the different problems for p-harmonic equations were considered.
Similar and various problems of partial differential equations in grand Sobolev, Besov
and Morrey type spaces were studied in [8, 9, 14–16, 18–23] and others. Most of these
papers were used the variational methods. Evidently, in the above-mentioned papers only
p-harmonic equations (1) was considered.

In this paper we consider Dirichlet problem for p-harmonic type equation has a form

div
(
|∇u|p−q∇u

)
= divf, (3)

u|∂G = ϕ|∂G , (4)

where 1 < p < ∞; 2 ≤ q < ∞; ϕ ∈ W 1
p)(G), f ∈ L(p−ε)′(G), (p− ε)′ =

p− ε
p− ε− 1

and

G in Rn is a bounded domain.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i3.3762

Email addresses: aliknajafov@gmail.com (A.M. Najafov), sain.elekberli@bk.ru (S.T. Alekberli)

https://www.ejpam.com 579 c© 2020 EJPAM All rights reserved.



A.M. Najafov, S.T. Alekberli / Eur. J. Pure Appl. Math, 13 (3) (2020), 579-586 580

Definition 1. ([6, 17, 23]) Denote by W 1
p)(G) the grand Sobolev space of locally summable

functions u on G having the weak partial derivatives D1
xiu (i = 1, 2, . . . , n) with the finite

norm
‖u‖W 1

p)
(G) = ‖u‖Lp)(G) + ‖∇u‖Lp)(G),

where

‖u‖Lp)(G) = sup
0<ε<p−1

 ε

|G|

∫
G

|u(x)|p−εdx

 1
p−ε

and |G| is the Lebesgue measure of G.

We note that the correct choice of space for problem (3)-(4) is the grand Lebesgue
space (or grand Sobolev space).

In this paper using the variational method an existence and uniqueness of solution
to Dirichlet problem for p− harmonic type equations (3)- (4) in grand Sobolev spaces is
studied.

A weak solution for the problem (3)-(4) on G is a function u (x) ∈W 1
p) (G) , if u−ϕ ∈

◦
W

1

p)(G) such that

n∑
i=1

∫
G
|∇u|p−q uxi ϑxi dx =

n∑
i=1

∫
G

f ϑxi dx, (5)

for every ϑ ∈
◦
W

1

p)(G).

2. Main results

In this section we prove the existence and uniqueness of weak solution (5) for the
problem (3)-(4).

Theorem 1. Let G ⊂ Rn is bounded domain, 1 < p <∞; 2 ≤ q <∞; g, h ∈W 1
p−(q−2)(G),

ϕ ∈W 1
p)(G) and f ∈ L1

(p−ε)′. Then the Dirichlet problem for pharmonic type equation (3)

has a unique weak solutions in W 1
p)(G).

Proof. Since functions g and h ∈W 1
p−(q−2)(G), then we consider the bilinear functional

as the form

F (g, h) =

n∑
i=1

∫
G
|∇g|p−qgxi hxi dx−

n∑
i=1

∫
G
f hxi dx =

= I(g, h)−
n∑
i=1

∫
G
f hxi dx =I(g, h)− (f, h), (6)



A.M. Najafov, S.T. Alekberli / Eur. J. Pure Appl. Math, 13 (3) (2020), 579-586 581

since f ∈ L
(p−ε)′ (G), (p− ε)

′
= p−ε

p−ε−1 . Consequently, we have

|I (g, g)| = |I (g)| =

∣∣∣∣∣
n∑
i=1

∫
G
|∇g|p−q gxi gxi dx

∣∣∣∣∣ ≤
≤

n∑
i=1

∫
G
|∇g|p−q |gxi | |gxi | dx =

n∑
i=1

∫
G
|∇g|p−q|gxi |

2 dx =

=

∫
G
|∇g|p−(q−2) dx <∞,

|I (g)| ≤ ‖∇g‖p−(q−2)Lp−(q−2)(G).

Consequently, for every q − 2 < ε < p− 1 function g ∈W 1
p) (G) and

‖g‖W 1
p)
(G)≤C1 ‖g‖W 1

p−(q−2)
(G),

and, note that
‖∇g‖p−εLp)(G)≤C2 |I (g)| , (7)

where C1 and C2 are constants independents on function g.
The variational problem is stated as follows. Find a function g ∈ W 1

p) (G) such that

which gives the minimum value to the integral F (g) and is unique. The Euler-Lagrange
equation for the variational problem (6) under consideration is the equation (3). With the
help of the inequality (7), we have

|F (g, g)| = |F (g)| =

∣∣∣∣∣∣I (g)−
n∑
i=1

∫
G

f gxi dx

∣∣∣∣∣∣ ≥ |I (g)| −

−

∣∣∣∣∣∣
n∑
i=1

∫
G

f gxi dx

∣∣∣∣∣∣ ≥ |I (g)| −
n∑
i=1

∣∣∣∣∣∣
∫
G

fgxi dx

∣∣∣∣∣∣ ≥ |I (g)| −

∣∣∣∣∣∣
n∑
i=1

∫
G

f gxi dx

∣∣∣∣∣∣ ≥
≥ |I (g)| −

n∑
i=1

∫
G
|f | |gxi |dx ≥ C3 ‖g‖p−εW 1

p)
(G)
− ‖g‖p−εLp)(G)−

−‖f‖L
(p−ε)

′ (G) ‖∇g‖Lp)(G) ≥ C4‖g‖W 1
p)
(G) = M0,

C3 and C4 are constants independent on the function g (x).
This means that F (g) is lower bounded on W 1

p)(G) show that there exists g0 ∈W 1
p) (G)

such that F (g0) = min
g∈W 1

p)
(G)

F (g). Fix some sequence {gm} ∈W 1
p) (G) (m = 1, 2, . . . ) such

that lim
m→∞

F (gm) = r0 . Let σ > 0 choose mσ so for m ≥ mσ and s = 1, 2, . . .

it holds F ( gm+s) < r0 + σ. Then noting that 1
2 (gm+s + gm) ∈ W 1

p)(G) we have



A.M. Najafov, S.T. Alekberli / Eur. J. Pure Appl. Math, 13 (3) (2020), 579-586 582

F
(
gm+s+gm

2

)
≥ r0. By direct calculations we show that I

(
gm+s−gm

2

)
< 4σ, and we

have ‖gm+s + gm‖W 1
p)
(G) ≤ 2

(
ε
C

) 1
p−ε . This means that the sequence {gm} is fundamental

in the spaces W 1
p) (G) , consequently in view of completeness the spaces W 1

p) (G) there exist

a function g0 ∈ W 1
p) (G) such that lim

m→∞
‖gm − g0‖W 1

p)
(G) = 0 . By theorem on trace in

W 1
p (G), ([3, p.143]), we get

W 1
p) (G)→W 1

p−ε (G)→ Lt−ε (Gk) , Gk = G
⋂

Rk, p < t ≤ ∞, 1 ≤ k ≤ n.

So
|F (gm)− F (g0)| ≤ C‖gm − g0‖ W 1

p)
(G)

and hence it follows that r0 = lim
m→∞

F (gm) = F (g0). Show that the function delivering

minimum to the functional F (g) is unique and satisfies equation (3) in the space W 1
p) (G).

Since g ∈W 1
p) (G) and F (g0) = r0, we have

0 ≤ I
(
g − g0

2

)
=

1

2
F (g) +

1

2
F (g0)− F

(
g + g0

2

)
≤ r0

2
+
r0
2
− r0 = 0,

I (g − g0) = 0.

By ‖gm − g0‖W 1
p)
(G) → 0, m→∞, it follows that the function g coincides with g0 as an

element of the space W 1
p) (G) . Again from the theorem on trace in space W 1

p) (G) , we have

‖(gm − g0) |∂G‖Lt−ε(∂G) ≤ C ‖gm − g0‖W 1
p)
(G) → 0, m→∞.

Since
‖gm|∂G − ϕ|∂G ‖Lt−ε(∂G) → 0, m→∞,

therefore
‖g0|∂G − ϕ|∂G ‖Lt−ε(∂G) → 0 m→∞.

Taking into account the condition d
dµ (F (g0 + µω))µ=0 = 0, show that the function

g0 ∈W 1
p) (G), minimizing the integral F (g) satisfies the following equation

I (g0, ω)− (f, ω) = 0. (8)

Now prove that the function g0 ∈ W 1
p) (G) minimizing the integral F (g) is the weak

solution of the problem (3)-(4). By θ (t) we denote some monotonically decreasing function
on the segment 1

2 ≤ t ≤ 1 and having the following properties

θ

(
1

2
+ 0

)
= 1, θ (1− 0) = −1, θ(s)

(
1

2
+ 0

)
= θ(s) (1− 0) = 0, s = 1, 2, . . . .



A.M. Najafov, S.T. Alekberli / Eur. J. Pure Appl. Math, 13 (3) (2020), 579-586 583

The function

γ (t) =

{
θ′ (t) , 1

2 ≤ t ≤ 1,
0, −∞ < t < 1

2 , 1 < t <∞

is infinitely differentiable and finite on the real line. Note that the function γ satisfy
condition

γ(s)
(

1

2
+ 0

)
= γ (1− 0) , (s = 1, 2, . . .) .

Let δ > 0 and let Gδ = {y : ρ (y,Rn\G) > δ} be arbitrary point of the domain G, and
r = ρ (x, x0). There ρ (x, x0) is the Euclidean distance between x and x0, where x ∈ G
and x0 be a fixed point in G. Following Sobolev [24], we introduce the function

ω (x) = γ

(
r

l1

)
− γ

(
r

l2

)
,

for 0 < l1 < l2 < δ. It is obvious that ω(x) is a infinitely differentiable finite function with
a support lying on a annular domain l1

2 < r < l2. Therefore ω ∈ C∞0 (G) and D(s)ω|∂G = 0
for all s = 1, 2, . . . . Then from (8) by definition of the weak derivative it follows that∫

G
K

(
r

l1

)
g (x) dx =

∫
G
K

(
r

l2

)
g (x) dx, (9)

where

K

(
r

li

)
= div

(∣∣∣∣∇γ ( rli
)∣∣∣∣p−q ∇γ ( rli

))
− div f , i = 1, 2.

Note that the function K
(
r
li

)
having all properties of kernel. Namely, the following

properties hold:
1) K is infinitely differentiable function with support in the ball r ≤ li;
2) The function K and all its derivatives on sphere R = h are zero;
3)

1

τn lni

∫
G
K

(
r

li

)
dx = 1,

where

τn =
2π

n
2

Γ
(
n
2

) ∫ 1

0
ξn−1K(ξ) dξ.

Then for the function g0(x) we can constructed Sobolev’s averaging g0,li(x), i = 1, 2
on the ball li (i = 1, 2) with centered at the point x as

g0,li (x) =
1

τn lni

∫
Rn

K

(
|z − x|
li

)
g0 (z) dz, i = 1, 2.

The we can rewrite equality (9) in the form g0,l1 (x) = g0,l2 (x) . Consequently, for l < δ

g0,l (x) = g0 (x) .



REFERENCES 584

Since the average functions g0,li(x), i = 1, 2 are continuous and has continuous derivatives
for any order, then g0 (x) also is a kernel. Integrating by parts in the equality I (g0, ω)−
(f, ω) = 0, whence is the limit case

n∑
i=1

∫
G
ω(x)

∂

∂xi

(
|∇g0|p−q

∂

∂xi
g0 (x)

)
dx =

n∑
i=1

∫
G
ω(x)

∂

∂xi
f(x) dx .

Hence by the arbitrariness of the functions ω(x) it follows that

n∑
i=1

∂

∂xi

(
|∇g0|p−q

∂

∂xi
g0 (x)

)
=

n∑
i=1

∂

∂xi
f(x)

i.e
div
(
|∇g0|p−q∇g0

)
= divf.

Thus, solution of the variational problem (5) from the class W 1
p) (G) is also solution of

Dirichlet problem (3)-(4) and this solution is unique.

3. Conclusion

In conclusion, we note that for a p-harmonic type equation in the grand Sobolev space,
a result is obtained on the existence and uniqueness of a weak solution.

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