# Variational Approaches to Characterize Weak Solutions for Some Problems of Mathematical Physics Equations

Authors: Irina Meghea
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving thep-Laplacian and the p-pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned operators.


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## Body

## 1. Introduction

Obtaining and/or characterization of weak solutions for problems of mathematical physics equations involvingp-Laplacian and p-pseudo-Laplacian is a subject matter previously discussed by the author through several approach methods in [1–6]. New similar results can be found by other authors, for instance, Amiri and Zivari-Rezapour in [7], El Khalil and Ouanan in [8], Rhouma and Sayeb in [9], Yoshida in [10]. The importance of these operators also devolves from their involvement in actual modelings of natural phenomena as thermal transfer by Lanchon-Ducauquis, Tulita, and Meuris in [11] or glacier sliding or flow by Partridge in [12] and Pélissier in [13]. In this paper three methods are proposed following three sequences of results, starting from abstract statements and finishing with their application to find weak solutions of Dirichlet problems for p-Laplacian and for p-pseudo-Laplacian as well. In the first succession of assertions, two results of Ghoussoub from [14] have been generalized replacing the frame of Hilbert space by reflexive strictly convex Banach space and the condition imposed to the goal function to be of C
1-Fréchet class by the weaker condition to have a lower semicontinuous and Gâteaux differentiable function. These two theoretical statements were used, together with other results concerning Dirichlet problems for both cited generalized operators, in finding weak solutions for these problems. The second proposition sequence involves critical points for nondifferentiable functional. In this context, two results of Chang from [15] have been generalized changing the space H
1
(
Ω
) in W
0
1
,
p
Ω and introducing Δ
p and Δ
p
s in some problems formulated for Δ by Costa and Gonçalves in [16]. The last series of assertions starts from Ghoussoub-Maurey linear principle which is used here to characterize weak solutions for some mathematical physics equations. Moreover, the problems were discussed and solved using important findings for these operators obtained by the author in [5, 6] in connection with specific properties of the Sobolev spaces involved.


## 2. Critical Points and Weak Solutions for Elliptic Type Equations

### 2.1. Theoretical Support

In order to introduce the first result, a theoretical support will be given starting with the following.Ekeland Principle (see [1, 17, 18]). Let (
X
,
d
) be a complete metric space and φ
:
X
→
(
-
∞
,
+
∞
] bounded from below, lower semicontinuous, and proper. For any ε
>
0 and u of X with(1)φu≤inf⁡φX+εand for any λ
>
0, there exists v
ε in X such that(2)φvε<φw+ελdvε,w∀w∈X∖vε,φvε≤φu,anddvε,u≤λ.Definition 1.
LetX be a real normed space, β be a bornology on X, and let φ
:
X
→
R. Let c be in R and F a nonempty subset of X. φ verifies thePalais-Smale condition on the level
c
around
F (orrelative to
F), P
S
c
,
F, with respect to β, when ∀
u
n
n
≥
1 a sequence of points in X for which(3)limn→∞φun=c,limn→∞∇βφun=0,limn→∞distun,F=0,this sequence has a convergent subsequence.
To clarify the above notation, letβ be a bornology on X and let f
:
X
→
R
- locally finite in the point a. By definition f is β-differentiable in a, if there exists φ in the dual X
∗ such that for every S in β we have l
i
m
t
→
0
,
h
∈
S
u
f
(
a
+
t
h
)
-
f
(
a
)
/
t
=
φ
(
h
) (uniform limit on S for t
→
0). φ is the β-derivative of f in a, and it is denoted by ∇
β
f
a.Through the minimization onF of a functional (minimization with constraints) global critical points of this may be obtained.As a preliminary, we generalize some results from [14] introducing Banach space instead of Hilbert space and the Gâteaux differentiability instead of C
1-class Fréchet.Proposition 2.
LetX be a real reflexive strictly convex Banach space, let φ
:
X
→
R be lower semicontinuous and Gâteaux differentiable and let F be a closed subset of X such that for every u from F with the metric gradient ∇
φ
(
u
)
≠
0, for sufficiently small r
>
0,(4)u-δ∇φu∇φu∈F,∀δ∈0,r.
Then, ifφ is lower bounded on F, for every v
n
n
≥
1 a minimizing sequence for φ on F, there exists a sequence u
n
n
≥
1 in F such that(5)φ′un≤εn,(6)φun≤φvn∀n,(7)limn→∞un-vn=0,where ε
n
>
0 and ε
n
→
0.Remark 3.
This result is reported in [14] as Lemma  9 in the frame of the Hilbert spaces having the function φ of C
1-class Fréchet, but condition (5) is more complicated due to another condition imposed to the set F.Explanations. Let us introduce the definition of the metric gradient in order to provide other observations relative to this central notion for this statement. In a real normed space X, consider the Gâteaux differentiable functional f
:
X
→
R. Themetric gradientof f is the multiple-valued mapping ∇
f
:
X
→
P
X, ∇
f
x
=
i
-
1
J
∗
f
w
′
x, where J
∗
:
X
∗
→
P
X
∗
∗ is the duality mapping on X
∗ corresponding to the identity and i the canonical injection of X into X
∗
∗
:
i
x
=
x
∗
∗, x
∗
∗
,
x
∗
=
x
∗
,
x
,
∀
x
∗
∈
X
∗. Consequently, for any x
∈
X
:
∇
f
x
=
y
∈
X
:
i
y
∈
J
∗
f
w
′
x
=
{
y
∈
X
:
i
y
,
f
w
′
x
=
f
w
′
x
,
y
=
f
w
′
x
2
,
i
y
=
y = f
w
′
x
}. If X is reflexive, for any x
∈
X, ∇
f
(
x
) is nonempty. X
∗
∗ being strictly convex, J
∗ is single-valued. So, if X is reflexive and strictly convex, then ∇
f
:
X
→
X, ∇
f
x
=
i
-
1
J
∗
f
w
′
x, and the following equalities hold:
(8)fw′x,∇fx=fw′x2,∇fx=fw′x.Go now to the proof of Proposition2.Proof.
Denotec
≔
inf
⁡
φ
F and let n be from N. For ε
n
≔
φ
(
v
n
)
-
c
+
1
/
n, hence ε
n
>
0, we have φ
v
n
<
c
+
ε
n. Apply the enunciatedEkeland principle with λ
=
ε
n, ∃
u
n in F with known properties. Thus we obtain the sequence u
n
n
≥
1 satisfying (6), (7) (
u
n
-
v
n
≤
ε
n
,
ε
n
→
0
), and(9)φv≥φun-εnv-un∀v∈F.
Verify (5). It is sufficient to work under the assumption φ
w
′
u
n
>
0
∀
n. Thus apply the hypothesis made in the statement with respect to F with u
=
u
n and denoting, for δ
∈
0
,
r
,
v
δ
≔
u
n
-
δ
∇
φ
u
n
/
∇
φ
u
n
∈
F, replace v
δ in (9) and find,(10)εnvδ-un≥φun-φvδ,multiply this inequality by 1
/
δ, δ
>
0, and take the limit for δ
→
0
+ in order to keep the sense of the inequality. Remark that l
i
m
δ
→
0
v
δ
=
u
n; l
i
m
δ
→
0
v
δ
-
u
n
/
δ
=
l
i
m
δ
→
0
δ
∇
φ
(
u
n
)
/
∇
φ
(
u
n
)
/
δ
=
1. Consider that the existence of the limit for δ
→
0 implies the existence of the limit for δ
→
0
± together with their equality, l
i
m
δ
→
0
+
φ
(
u
n
)
-
φ
(
v
δ
)
/
δ
=
l
i
m
δ
→
0
+
φ
u
n
-
δ
∇
φ
u
n
/
∇
φ
u
n
-
φ
(
u
n
)
/
-
δ
=
φ
w
′
u
n
∇
φ
u
n
/
∇
φ
u
n = 1
/
∇
φ
u
n
φ
w
′
u
n
,
∇
φ
u
n = 1
/
∇
φ
u
n
φ
w
′
u
n
2
=
φ
w
′
u
n, taking into account the definition of the Gâteaux derivative and the above considerations on the metric gradient, and (5) is also fulfilled.Remark 4.
The Gâteaux derivative from the above statement can be replaced by anyβ-derivative and the result remains the same. In the case of the Fréchet derivative, it must remove the condition “φ lower semicontinuous” from the statement.Notation 1.
φ
:
X
→
R
β-differentiable; c
∈
R
⇒
(11)Kcφ≔x∈X:φx=c,∇βφx=0.Proposition 5.
LetX be a real reflexive strictly convex Banach space and φ
:
X
→
R lower semicontinuous and Gâteaux differentiable and F is a nonempty convex closed subset such that (
I
-
∇
φ
)
(
F
)
⊂
F, I the identity map. If φ is lower bounded on F, then for every v
n
n
≥
1 a minimizing sequence for φ on F, there is a sequence u
n
n
≥
1 in F such that(12)φun≤φvn∀n,limn→∞⁡un-vn=0,limn→∞⁡φw′un=0.
Moreover, ifφ satisfies P
S
c
,
F, where c
=
inf
⁡
φ
F, then(13)F∩Kcφ≠⌀.Proof.
Applying Proposition2, (4) is satisfied indeed: if u
∈
F and φ
w
′
u
≠
0, then, F being convex,(14)u-δ∇φu∇φu=1-δφw′uu+δφw′uI-∇φu∈F.Let u
n
n
≥
1 be the sequence given by the statement. c
≤
φ
u
n
≤
φ
v
n
∀
n, hence φ
u
n
→
c. φ
w
′
u
n
≤
5
ε
n, hence φ
w
′
u
n
→
0, clearly d
i
s
t
u
n
,
F
=
0, and consequently u
n
n
≥
1 has a convergent subsequence u
k
n
n
≥
1, u
k
n
→
u
0
∈
F. This implies φ
w
′
u
k
n
→
φ
w
′
u
0
=
0, and u
0 is a global critical point of φ contained in F.


### 2.2. Weak Solutions

Open set of
C
1
class in
R
N. Use the notations (the norm is that Euclidean from R
N
-
1): R
+
N
=
x
=
x
′
,
x
N
:
x
N
>
0, Q
=
x
=
x
′
,
x
N
:
x
′
<
1
,
x
N
<
1, Q
+
=
Q
∩
R
+
N, Q
0
=
x
=
x
′
,
x
N
:
x
′
<
1
,
x
N
=
0. Let Ω be an open nonempty set in R
N, Ω
≠
R
N
, and ∂
Ω its boundary. By definition, Ω is of C
1
classif ∀
x from ∂
Ω
∃
U a neighbourhood of x in R
N and f
:
Q
→
U bijective such that f
∈
C
1
Q
¯, f
-
1
∈
C
1
U
¯, f
Q
+
=
U
∩
Ω, and f
Q
0
=
U
∩
∂
Ω.Weak solution. Let Ω be an open bounded nonempty set in R
N, N
>
1, f
:
Ω
×
R
N
→
R
, and u
0
∈
W
0
1
,
p
Ω. Consider the problems∗-Δpu=fx,u,x∈Ωu=0on∂Ω,∗∗-Δpsu=fx,u,x∈Ωu=0on∂Ω.Actually u
=
u
0 on ∂
Ω means u
∣
∂
Ω
=
u
0, where γ
:
u
→
u
∣
∂
Ω is thetrace operator, a continuous linear mapping from W
1
,
p
Ω in L
p
∂
Ω, p
∈
[
1
,
+
∞
). We have γ
-
1
0
=
W
0
1
,
p
(
Ω
)
=
C
c
1
Ω
¯
W
1
,
p
Ω and u
∈
W
1
,
p
Ω
∩
C
Ω
¯
⇒
γ
u
=
u
∣
∂
Ω. u
- from X
=
W
0
1
,
p
Ω is by definition aweak solution for ∗ and ∗
∗, respectively, if u
-
=
0 on ∂
Ω and(15)∫Ω∇u-p-2∇u-·∇vdx-∫Ωfx,u-xvdx=0∀v∈W01,pΩ(16)respectively∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx-∫Ωfx,u-xvdx=0∀v∈W01,pΩ.Remark 6.
Here∇
w is the weak gradient, and it is equal to ∂
w
/
∂
x
1
,
…
,
∂
w
/
∂
x
N
,
∂
w
/
∂
x
i the weak derivatives; ∇
w
2
=
∑
i
=
1
N
∂
w
/
∂
x
i
2. X
≔
W
0
1
,
p
Ω is endowed in the first case ∗ with the norm ·
1
,
p; that is, u
1
,
p
=
u
W
0
1
,
p
(
Ω
)
=
u
L
p
Ω
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω, which is equivalent to the norm u
→
u
L
p
Ω
p
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p. For the second case ∗
∗, equip the same vector space with the norm u
→
❙
u
❙
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p, which is equivalent to u
→
u
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω.Nemytskii Operator. Let be R
N, N
≥
1, μ the Lebesgue measure in R
N, Ω open nonempty Lebesgue measurable and M
Ω
≔
u
:
Ω
→
R
∣
u
Lebesgue measurable. By definition f
:
Ω
×
R
→
R is aCarathéodory functionif f
·
,
s is Lebesgue measurable ∀
s
∈
R and f
(
x
,
·
) is continuous ∀
x
∈
Ω
∖
A, μ
(
A
)
=
0. In this case, for every u from M
Ω one may consider the function N
f
:
M
(
Ω
)
→
M
Ω, N
f
u
:
N
f
u
x
=
f
x
,
u
x,Nemytskii operator. Suppose μ
Ω
<
+
∞
. Then u
n
x
→
x
∈
Ω
μ
u
0
x
⇒
N
f
u
n
x
→
x
∈
Ω
μ
N
f
u
0
x. Assume that f satisfies thegrowth condition: f
x
,
s
≤
c
s
p
-
1
+
b
x
,
∀
x
∈
Ω
∖
A with μ
(
A
)
=
0
,
∀
s
∈
R, where c
≥
0, p
>
1, and b
∈
L
q
Ω, q
∈
1
,
+
∞.ThenN
f
L
p
-
1
q
Ω
⊂
L
q
Ω; N
f is continuous q
<
+
∞ and bounded on L
p
-
1
q
(
Ω
). If Ω is bounded and 1
/
p
+
1
/
q
=
1, then N
f
L
p
Ω
⊂
L
q
Ω with N
f continuous; moreover, N
F
L
p
Ω
⊂
L
1
Ω with N
F continuous (see, e.g., [2]), where F
x
,
s
=
∫
0
s
f
x
,
t
d
t, and Φ
:
L
p
Ω
→
R, Φ
u
=
∫
Ω
F
x
,
u
x
d
x is of Fréchet C
1 class and Φ
′
=
N
f [14], so it is also Gâteaux differentiable.Theorem 7.
LetΩ be an open bounded nonempty set in R
N and f
:
Ω
×
R
→
R a Carathéodory function with the growth condition:(17)fx,s≤csp-1+bx,where c
>
0, 2
≤
p
≤
2
N
/
N
-
2when N
≥
3, and 2
≤
p
<
+
∞ when N
=
1
,
2, and where b
∈
L
q
Ω, 1
/
p
+
1
/
q
=
1
.
Then the energy functionalφ
:
W
0
1
,
p
(
Ω
)
→
R,(18)
φ
u
=
1
p
u
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
f
o
r
t
h
e
p
r
o
b
l
e
m
∗
(19)
r
e
s
p
e
c
t
i
v
e
l
y
φ
u
=
1
p
❙
u
❙
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
f
o
r
t
h
e
p
r
o
b
l
e
m
∗
∗
,whereF
(
x
,
s
)
=
∫
0
s
f
(
x
,
t
)
d
t is Gâteaux differentiable onW
0
1
,
p
(
Ω
)
∖
0 and(20)φw′uv=∫Ω∇up-2∇u·∇vdx-∫Ωfx,uxvdx∀u,v∈W01,pΩrespectively(21)φw′uv=∑i=1N∫Ω∂u∂xip-2∂u∂xi∂v∂xidx-∫Ωfx,uxvdx=0∀u,v∈W01,pΩ.Proof.
One may considerφ, in both cases, as the sum of two other functions. The second of these functions being Gâteaux differentiable (see the above consideration), it is sufficient to remark that also the maps u
→
1
/
p
u
1
,
p
pand u
→
1
/
p
❙
u
❙
1
,
p
pare Gâteaux differentiable onW
0
1
,
p
Ω
∖
0 [2, 19] and then φ is Gâteaux differentiable onW
0
1
,
p
Ω
∖
0.Corollary 8.
LetΩand f be as in Theorem 7. Then the weak solutions of ∗ and ∗
∗, respectively, are precisely the critical points of the functionalφ
:
W
0
1
,
p
Ω
→
R:(22)
φ
u
=
1
p
u
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
F
x
,
s
≔
∫
0
s
f
x
,
t
d
t
(23)
r
e
s
p
e
c
t
i
v
e
l
y
φ
u
=
1
p
❙
u
❙
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
F
x
,
s
≔
∫
0
s
f
x
,
t
d
t
.Proof.
Indeed, ifu
- is a weak solution of ∗ and ∗
∗, respectively, then φ
w
′
u
-
v
=
0
∀
v
∈
W
0
1
,
p
Ω((15) and (16), resp., Theorem 7); hence φ
w
′
u
-
=
0. The inverse assertion is obvious.Weak Subsolutions and Weak Supersolutions of
∗
and
∗
∗
. Let Ω be an open bounded set of C
1 class in R
N, N
≥
3, let f
:
Ω
×
R
→
R be a Carathéodory function, and let u
-
∈
W
0
1
,
p
Ω. u
- is aweak subsolutionand aweak supersolution,respectively, of ∗ or ∗
∗ if(24)u-≤0on∂Ωrespectivelyu-≥0on∂Ω,and(25a)∫Ω∇u-p-2∇u-·∇vdx≤∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0,(25b)respectively,∫Ω∇u-p-2∇u-·∇vdx≥∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0,(26a)or∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx≤∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0(26b)respectively∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx≥∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0.Proposition 9.
LetΩbe an open bounded of C
1 class set in R
N, N
≥
3 and f
:
Ω
×
R
→
R a Carathéodory function and u
1, u
2 fromW
0
1
,
p
Ω bounded weak subsolution and weak supersolution of ∗, respectively, with u
1
x
≤
u
2
x a.e. onΩ. Suppose that f verifies (17) and there is ρ
>
0 such that the function g
:
g
x
,
s
=
f
x
,
s
+
ρ
s is strictly increasing in s on inf
⁡
u
1
Ω
,
sup
⁡
u
2
Ω. Then there is a weak solution u
- of ∗ inW
0
1
,
p
Ω with the property(27)u1x≤u-x≤u2xa.e.onΩ.Proof.
Take the equivalent norm onX
=
W
0
1
,
p
(
Ω
):(28)u=ρuLpΩp+∑i=1N∂u∂xiLpΩp1/p.Consider the functional φ
:
W
0
1
,
p
Ω
→
R:(29)φu=1pup-∫ΩGx,uxdx,Gx,s≔∫0sgx,tdt.
φ is Gâteaux differentiable and its critical points are the weak solutions of ∗ (see Corollary 8). φ is also lower bounded, the norm on L
p
Ω actually being of Fréchet C
1 class (see, e.g., [20] or [21]). Use Proposition 5, X
,
· being a reflexive strictly convex Banach space (see, e.g., [2]). Let be(30)
F
≔
u
∈
W
0
1
,
p
Ω
:
u
1
x
≤
u
x
≤
u
2
x
a.e. on
Ω
.
F is closed convex. We also get(31)I-∇φF⊂F.Here∇
φ denotes the metric gradient of φ. Since (
X
,
·
) is reflexive and strictly convex (see, e.g., [2]), thus∇
φ is univaluated and it has the above described properties. Indeed, let u be in F andv
≔
I
-
∇
φ
u. We should prove that v
∈
F. v
=
u
-
∇
φ
u
∈
W
0
1
,
p
Ωand u
1
x
≤
v
x
≤
u
2
x. Since the definition relation of the subsolution for u
1 actually means φ
w
′
u
1
w
≤
0
∀
w in W
0
1
,
p
Ω with w
(
x
)
≥
0 almost everywhere (a.e.) on Ω and that of supersolution for u
2 is φ
w
′
u
2
w
≥
0
∀
w inW
0
1
,
p
Ω verifying w
(
x
)
≥
0 a.e. onΩ, we will prove that v
(
x
)
-
u
1
(
x
)
≥
0 a.e. on Ω and u
2
x
-
v
x
≥
0 a.e. on Ω using the Gâteaux derivatives of φ in u
1 and u
2, respectively. φ
w
′
u
1
(
v
-
u
1
)
=
φ
w
′
(
u
1
)
(
u
1
-
u
)
-
φ
w
′
u
1
∇
φ
u
≤
-
φ
w
′
u
1
∇
φ
u
≤
-
φ
w
′
u
∇
φ
u
=
-
φ
w
′
u
2
≤
0 (take into account that u
1
≤
u, φ
w
′
(
u
1
) is a linear map and some properties of the metric gradient). Also φ
w
′
u
2
u
2
-
v
=
φ
w
′
(
u
2
)
(
u
2
-
v
)
+
φ
w
′
u
2
∇
φ
u
≥
φ
w
′
u
2
∇
φ
u
≥
φ
w
′
u
∇
φ
u
=
φ
w
′
u
2
≥
0. φ is lower bounded on F, φ being actually continuous (for this see, for instance, [2]). Until now, applying Proposition 5, for every v
n
n
≥
1 a minimizing sequence for φ on F, there is a sequence u
n
n
≥
1 in F such that φ
u
n
≤
φ
(
v
n
)
∀
n, l
i
m
n
→
∞
u
n
-
v
n
=
0, l
i
m
n
→
∞
φ
w
′
(
u
n
)
=
0. So l
i
m
n
→
∞
φ
u
n
=
c since c
≔
inf
⁡
φ
F, we have l
i
m
n
→
∞
φ
w
′
u
n
=
0 already, also the last property from P
S
c
,
F condition is verified. Finish the proof applying once again Proposition 5.Example 10.
Consider the problem (Ωopen bounded of C
1 class in R
N, N
≥
3)(32)-Δpu=αxuup-2,u=0on∂Ω,where p
=
2
N
/
N
-
2; α is continuous with 1
≤
α
(
x
)
≤
a
<
+
∞ on Ω. Then u
1
≔
1 is a weak subsolution, u
2
≔
M, M
>
1 sufficiently big, is a weak supersolution, f
x
,
s
≤
a
s
p
-
1 (condition (17)), and s
→
α
(
x
)
s
s
p
-
2
+
s is increasing in s on [
1
,
M
]; consequently, according to Proposition 9, (32) has a weak solutionu
-with 1
≤
u
-
(
x
)
≤
M a.e. onΩ.Proposition 11.
LetΩbe an open bounded of C
1 class set in R
N, N
≥
3 and f
:
Ω
×
R
→
R a Carathéodory function and u
1, u
2 from W
0
1
,
p
Ω bounded weak subsolution and weak supersolution, respectively, of ∗
∗ with u
1
x
≤
u
2
x a.e. onΩ. Suppose that f verifies (17) and there is ρ
>
0 such that the function g
:
g
x
,
s
=
f
x
,
s
+
ρ
s is strictly increasing in s on inf
⁡
u
1
Ω
,
sup
⁡
u
2
Ω. Then there is a weak solution u
- of ∗
∗ inW
0
1
,
p
(
Ω
) with the property(33)u1x≤u-x≤u2xa.e.onΩ.Proof.
Follow step by step the above proof for Proposition9 considering the real reflexive strictly convex Banach space X
=
W
0
1
,
p
Ω endowed with the norm u
→
❙
u
❙
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p or the equivalent norm u
→
u
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω which both are also equivalent to the other two norms used at Proposition 9. The function φ is from (19) having the weak derivative given in Theorem 7. Using similar calculus, obtain similar conclusion.Example 12.
Consider the problem (Ωopen bounded of C
1 class in R
N, N
≥
3):(34)-Δpsu=αxuup-2u=0on∂Ω,where p
=
2
N
/
N
-
2 and α is continuous with 1
≤
α
x
≤
a
<
+
∞ onΩ. Then u
1
≔
1 is a weak subsolution, u
2
≔
M, M
>
1 sufficiently big, is a weak supersolution, f
x
,
s
≤
a
s
p
-
1 (condition (17)), and s
→
α
x
s
s
p
-
2
+
s is increasing in s on [
1
,
M
]; consequently, according to Proposition 11, (34) has a weak solutionu
-with 1
≤
u
-
(
x
)
≤
M a.e. on Ω.


## 2.1. Theoretical Support

In order to introduce the first result, a theoretical support will be given starting with the following.Ekeland Principle (see [1, 17, 18]). Let (
X
,
d
) be a complete metric space and φ
:
X
→
(
-
∞
,
+
∞
] bounded from below, lower semicontinuous, and proper. For any ε
>
0 and u of X with(1)φu≤inf⁡φX+εand for any λ
>
0, there exists v
ε in X such that(2)φvε<φw+ελdvε,w∀w∈X∖vε,φvε≤φu,anddvε,u≤λ.Definition 1.
LetX be a real normed space, β be a bornology on X, and let φ
:
X
→
R. Let c be in R and F a nonempty subset of X. φ verifies thePalais-Smale condition on the level
c
around
F (orrelative to
F), P
S
c
,
F, with respect to β, when ∀
u
n
n
≥
1 a sequence of points in X for which(3)limn→∞φun=c,limn→∞∇βφun=0,limn→∞distun,F=0,this sequence has a convergent subsequence.
To clarify the above notation, letβ be a bornology on X and let f
:
X
→
R
- locally finite in the point a. By definition f is β-differentiable in a, if there exists φ in the dual X
∗ such that for every S in β we have l
i
m
t
→
0
,
h
∈
S
u
f
(
a
+
t
h
)
-
f
(
a
)
/
t
=
φ
(
h
) (uniform limit on S for t
→
0). φ is the β-derivative of f in a, and it is denoted by ∇
β
f
a.Through the minimization onF of a functional (minimization with constraints) global critical points of this may be obtained.As a preliminary, we generalize some results from [14] introducing Banach space instead of Hilbert space and the Gâteaux differentiability instead of C
1-class Fréchet.Proposition 2.
LetX be a real reflexive strictly convex Banach space, let φ
:
X
→
R be lower semicontinuous and Gâteaux differentiable and let F be a closed subset of X such that for every u from F with the metric gradient ∇
φ
(
u
)
≠
0, for sufficiently small r
>
0,(4)u-δ∇φu∇φu∈F,∀δ∈0,r.
Then, ifφ is lower bounded on F, for every v
n
n
≥
1 a minimizing sequence for φ on F, there exists a sequence u
n
n
≥
1 in F such that(5)φ′un≤εn,(6)φun≤φvn∀n,(7)limn→∞un-vn=0,where ε
n
>
0 and ε
n
→
0.Remark 3.
This result is reported in [14] as Lemma  9 in the frame of the Hilbert spaces having the function φ of C
1-class Fréchet, but condition (5) is more complicated due to another condition imposed to the set F.Explanations. Let us introduce the definition of the metric gradient in order to provide other observations relative to this central notion for this statement. In a real normed space X, consider the Gâteaux differentiable functional f
:
X
→
R. Themetric gradientof f is the multiple-valued mapping ∇
f
:
X
→
P
X, ∇
f
x
=
i
-
1
J
∗
f
w
′
x, where J
∗
:
X
∗
→
P
X
∗
∗ is the duality mapping on X
∗ corresponding to the identity and i the canonical injection of X into X
∗
∗
:
i
x
=
x
∗
∗, x
∗
∗
,
x
∗
=
x
∗
,
x
,
∀
x
∗
∈
X
∗. Consequently, for any x
∈
X
:
∇
f
x
=
y
∈
X
:
i
y
∈
J
∗
f
w
′
x
=
{
y
∈
X
:
i
y
,
f
w
′
x
=
f
w
′
x
,
y
=
f
w
′
x
2
,
i
y
=
y = f
w
′
x
}. If X is reflexive, for any x
∈
X, ∇
f
(
x
) is nonempty. X
∗
∗ being strictly convex, J
∗ is single-valued. So, if X is reflexive and strictly convex, then ∇
f
:
X
→
X, ∇
f
x
=
i
-
1
J
∗
f
w
′
x, and the following equalities hold:
(8)fw′x,∇fx=fw′x2,∇fx=fw′x.Go now to the proof of Proposition2.Proof.
Denotec
≔
inf
⁡
φ
F and let n be from N. For ε
n
≔
φ
(
v
n
)
-
c
+
1
/
n, hence ε
n
>
0, we have φ
v
n
<
c
+
ε
n. Apply the enunciatedEkeland principle with λ
=
ε
n, ∃
u
n in F with known properties. Thus we obtain the sequence u
n
n
≥
1 satisfying (6), (7) (
u
n
-
v
n
≤
ε
n
,
ε
n
→
0
), and(9)φv≥φun-εnv-un∀v∈F.
Verify (5). It is sufficient to work under the assumption φ
w
′
u
n
>
0
∀
n. Thus apply the hypothesis made in the statement with respect to F with u
=
u
n and denoting, for δ
∈
0
,
r
,
v
δ
≔
u
n
-
δ
∇
φ
u
n
/
∇
φ
u
n
∈
F, replace v
δ in (9) and find,(10)εnvδ-un≥φun-φvδ,multiply this inequality by 1
/
δ, δ
>
0, and take the limit for δ
→
0
+ in order to keep the sense of the inequality. Remark that l
i
m
δ
→
0
v
δ
=
u
n; l
i
m
δ
→
0
v
δ
-
u
n
/
δ
=
l
i
m
δ
→
0
δ
∇
φ
(
u
n
)
/
∇
φ
(
u
n
)
/
δ
=
1. Consider that the existence of the limit for δ
→
0 implies the existence of the limit for δ
→
0
± together with their equality, l
i
m
δ
→
0
+
φ
(
u
n
)
-
φ
(
v
δ
)
/
δ
=
l
i
m
δ
→
0
+
φ
u
n
-
δ
∇
φ
u
n
/
∇
φ
u
n
-
φ
(
u
n
)
/
-
δ
=
φ
w
′
u
n
∇
φ
u
n
/
∇
φ
u
n = 1
/
∇
φ
u
n
φ
w
′
u
n
,
∇
φ
u
n = 1
/
∇
φ
u
n
φ
w
′
u
n
2
=
φ
w
′
u
n, taking into account the definition of the Gâteaux derivative and the above considerations on the metric gradient, and (5) is also fulfilled.Remark 4.
The Gâteaux derivative from the above statement can be replaced by anyβ-derivative and the result remains the same. In the case of the Fréchet derivative, it must remove the condition “φ lower semicontinuous” from the statement.Notation 1.
φ
:
X
→
R
β-differentiable; c
∈
R
⇒
(11)Kcφ≔x∈X:φx=c,∇βφx=0.Proposition 5.
LetX be a real reflexive strictly convex Banach space and φ
:
X
→
R lower semicontinuous and Gâteaux differentiable and F is a nonempty convex closed subset such that (
I
-
∇
φ
)
(
F
)
⊂
F, I the identity map. If φ is lower bounded on F, then for every v
n
n
≥
1 a minimizing sequence for φ on F, there is a sequence u
n
n
≥
1 in F such that(12)φun≤φvn∀n,limn→∞⁡un-vn=0,limn→∞⁡φw′un=0.
Moreover, ifφ satisfies P
S
c
,
F, where c
=
inf
⁡
φ
F, then(13)F∩Kcφ≠⌀.Proof.
Applying Proposition2, (4) is satisfied indeed: if u
∈
F and φ
w
′
u
≠
0, then, F being convex,(14)u-δ∇φu∇φu=1-δφw′uu+δφw′uI-∇φu∈F.Let u
n
n
≥
1 be the sequence given by the statement. c
≤
φ
u
n
≤
φ
v
n
∀
n, hence φ
u
n
→
c. φ
w
′
u
n
≤
5
ε
n, hence φ
w
′
u
n
→
0, clearly d
i
s
t
u
n
,
F
=
0, and consequently u
n
n
≥
1 has a convergent subsequence u
k
n
n
≥
1, u
k
n
→
u
0
∈
F. This implies φ
w
′
u
k
n
→
φ
w
′
u
0
=
0, and u
0 is a global critical point of φ contained in F.


## 2.2. Weak Solutions

Open set of
C
1
class in
R
N. Use the notations (the norm is that Euclidean from R
N
-
1): R
+
N
=
x
=
x
′
,
x
N
:
x
N
>
0, Q
=
x
=
x
′
,
x
N
:
x
′
<
1
,
x
N
<
1, Q
+
=
Q
∩
R
+
N, Q
0
=
x
=
x
′
,
x
N
:
x
′
<
1
,
x
N
=
0. Let Ω be an open nonempty set in R
N, Ω
≠
R
N
, and ∂
Ω its boundary. By definition, Ω is of C
1
classif ∀
x from ∂
Ω
∃
U a neighbourhood of x in R
N and f
:
Q
→
U bijective such that f
∈
C
1
Q
¯, f
-
1
∈
C
1
U
¯, f
Q
+
=
U
∩
Ω, and f
Q
0
=
U
∩
∂
Ω.Weak solution. Let Ω be an open bounded nonempty set in R
N, N
>
1, f
:
Ω
×
R
N
→
R
, and u
0
∈
W
0
1
,
p
Ω. Consider the problems∗-Δpu=fx,u,x∈Ωu=0on∂Ω,∗∗-Δpsu=fx,u,x∈Ωu=0on∂Ω.Actually u
=
u
0 on ∂
Ω means u
∣
∂
Ω
=
u
0, where γ
:
u
→
u
∣
∂
Ω is thetrace operator, a continuous linear mapping from W
1
,
p
Ω in L
p
∂
Ω, p
∈
[
1
,
+
∞
). We have γ
-
1
0
=
W
0
1
,
p
(
Ω
)
=
C
c
1
Ω
¯
W
1
,
p
Ω and u
∈
W
1
,
p
Ω
∩
C
Ω
¯
⇒
γ
u
=
u
∣
∂
Ω. u
- from X
=
W
0
1
,
p
Ω is by definition aweak solution for ∗ and ∗
∗, respectively, if u
-
=
0 on ∂
Ω and(15)∫Ω∇u-p-2∇u-·∇vdx-∫Ωfx,u-xvdx=0∀v∈W01,pΩ(16)respectively∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx-∫Ωfx,u-xvdx=0∀v∈W01,pΩ.Remark 6.
Here∇
w is the weak gradient, and it is equal to ∂
w
/
∂
x
1
,
…
,
∂
w
/
∂
x
N
,
∂
w
/
∂
x
i the weak derivatives; ∇
w
2
=
∑
i
=
1
N
∂
w
/
∂
x
i
2. X
≔
W
0
1
,
p
Ω is endowed in the first case ∗ with the norm ·
1
,
p; that is, u
1
,
p
=
u
W
0
1
,
p
(
Ω
)
=
u
L
p
Ω
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω, which is equivalent to the norm u
→
u
L
p
Ω
p
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p. For the second case ∗
∗, equip the same vector space with the norm u
→
❙
u
❙
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p, which is equivalent to u
→
u
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω.Nemytskii Operator. Let be R
N, N
≥
1, μ the Lebesgue measure in R
N, Ω open nonempty Lebesgue measurable and M
Ω
≔
u
:
Ω
→
R
∣
u
Lebesgue measurable. By definition f
:
Ω
×
R
→
R is aCarathéodory functionif f
·
,
s is Lebesgue measurable ∀
s
∈
R and f
(
x
,
·
) is continuous ∀
x
∈
Ω
∖
A, μ
(
A
)
=
0. In this case, for every u from M
Ω one may consider the function N
f
:
M
(
Ω
)
→
M
Ω, N
f
u
:
N
f
u
x
=
f
x
,
u
x,Nemytskii operator. Suppose μ
Ω
<
+
∞
. Then u
n
x
→
x
∈
Ω
μ
u
0
x
⇒
N
f
u
n
x
→
x
∈
Ω
μ
N
f
u
0
x. Assume that f satisfies thegrowth condition: f
x
,
s
≤
c
s
p
-
1
+
b
x
,
∀
x
∈
Ω
∖
A with μ
(
A
)
=
0
,
∀
s
∈
R, where c
≥
0, p
>
1, and b
∈
L
q
Ω, q
∈
1
,
+
∞.ThenN
f
L
p
-
1
q
Ω
⊂
L
q
Ω; N
f is continuous q
<
+
∞ and bounded on L
p
-
1
q
(
Ω
). If Ω is bounded and 1
/
p
+
1
/
q
=
1, then N
f
L
p
Ω
⊂
L
q
Ω with N
f continuous; moreover, N
F
L
p
Ω
⊂
L
1
Ω with N
F continuous (see, e.g., [2]), where F
x
,
s
=
∫
0
s
f
x
,
t
d
t, and Φ
:
L
p
Ω
→
R, Φ
u
=
∫
Ω
F
x
,
u
x
d
x is of Fréchet C
1 class and Φ
′
=
N
f [14], so it is also Gâteaux differentiable.Theorem 7.
LetΩ be an open bounded nonempty set in R
N and f
:
Ω
×
R
→
R a Carathéodory function with the growth condition:(17)fx,s≤csp-1+bx,where c
>
0, 2
≤
p
≤
2
N
/
N
-
2when N
≥
3, and 2
≤
p
<
+
∞ when N
=
1
,
2, and where b
∈
L
q
Ω, 1
/
p
+
1
/
q
=
1
.
Then the energy functionalφ
:
W
0
1
,
p
(
Ω
)
→
R,(18)
φ
u
=
1
p
u
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
f
o
r
t
h
e
p
r
o
b
l
e
m
∗
(19)
r
e
s
p
e
c
t
i
v
e
l
y
φ
u
=
1
p
❙
u
❙
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
f
o
r
t
h
e
p
r
o
b
l
e
m
∗
∗
,whereF
(
x
,
s
)
=
∫
0
s
f
(
x
,
t
)
d
t is Gâteaux differentiable onW
0
1
,
p
(
Ω
)
∖
0 and(20)φw′uv=∫Ω∇up-2∇u·∇vdx-∫Ωfx,uxvdx∀u,v∈W01,pΩrespectively(21)φw′uv=∑i=1N∫Ω∂u∂xip-2∂u∂xi∂v∂xidx-∫Ωfx,uxvdx=0∀u,v∈W01,pΩ.Proof.
One may considerφ, in both cases, as the sum of two other functions. The second of these functions being Gâteaux differentiable (see the above consideration), it is sufficient to remark that also the maps u
→
1
/
p
u
1
,
p
pand u
→
1
/
p
❙
u
❙
1
,
p
pare Gâteaux differentiable onW
0
1
,
p
Ω
∖
0 [2, 19] and then φ is Gâteaux differentiable onW
0
1
,
p
Ω
∖
0.Corollary 8.
LetΩand f be as in Theorem 7. Then the weak solutions of ∗ and ∗
∗, respectively, are precisely the critical points of the functionalφ
:
W
0
1
,
p
Ω
→
R:(22)
φ
u
=
1
p
u
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
F
x
,
s
≔
∫
0
s
f
x
,
t
d
t
(23)
r
e
s
p
e
c
t
i
v
e
l
y
φ
u
=
1
p
❙
u
❙
1
,
p
p
-
∫
Ω
F
x
,
u
x
d
x
,
F
x
,
s
≔
∫
0
s
f
x
,
t
d
t
.Proof.
Indeed, ifu
- is a weak solution of ∗ and ∗
∗, respectively, then φ
w
′
u
-
v
=
0
∀
v
∈
W
0
1
,
p
Ω((15) and (16), resp., Theorem 7); hence φ
w
′
u
-
=
0. The inverse assertion is obvious.Weak Subsolutions and Weak Supersolutions of
∗
and
∗
∗
. Let Ω be an open bounded set of C
1 class in R
N, N
≥
3, let f
:
Ω
×
R
→
R be a Carathéodory function, and let u
-
∈
W
0
1
,
p
Ω. u
- is aweak subsolutionand aweak supersolution,respectively, of ∗ or ∗
∗ if(24)u-≤0on∂Ωrespectivelyu-≥0on∂Ω,and(25a)∫Ω∇u-p-2∇u-·∇vdx≤∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0,(25b)respectively,∫Ω∇u-p-2∇u-·∇vdx≥∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0,(26a)or∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx≤∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0(26b)respectively∑i=1N∫Ω∂u-∂xip-2∂u-∂xi∂v∂xidx≥∫Ωfx,u-xvdx∀v∈W01,pΩ,v≥0.Proposition 9.
LetΩbe an open bounded of C
1 class set in R
N, N
≥
3 and f
:
Ω
×
R
→
R a Carathéodory function and u
1, u
2 fromW
0
1
,
p
Ω bounded weak subsolution and weak supersolution of ∗, respectively, with u
1
x
≤
u
2
x a.e. onΩ. Suppose that f verifies (17) and there is ρ
>
0 such that the function g
:
g
x
,
s
=
f
x
,
s
+
ρ
s is strictly increasing in s on inf
⁡
u
1
Ω
,
sup
⁡
u
2
Ω. Then there is a weak solution u
- of ∗ inW
0
1
,
p
Ω with the property(27)u1x≤u-x≤u2xa.e.onΩ.Proof.
Take the equivalent norm onX
=
W
0
1
,
p
(
Ω
):(28)u=ρuLpΩp+∑i=1N∂u∂xiLpΩp1/p.Consider the functional φ
:
W
0
1
,
p
Ω
→
R:(29)φu=1pup-∫ΩGx,uxdx,Gx,s≔∫0sgx,tdt.
φ is Gâteaux differentiable and its critical points are the weak solutions of ∗ (see Corollary 8). φ is also lower bounded, the norm on L
p
Ω actually being of Fréchet C
1 class (see, e.g., [20] or [21]). Use Proposition 5, X
,
· being a reflexive strictly convex Banach space (see, e.g., [2]). Let be(30)
F
≔
u
∈
W
0
1
,
p
Ω
:
u
1
x
≤
u
x
≤
u
2
x
a.e. on
Ω
.
F is closed convex. We also get(31)I-∇φF⊂F.Here∇
φ denotes the metric gradient of φ. Since (
X
,
·
) is reflexive and strictly convex (see, e.g., [2]), thus∇
φ is univaluated and it has the above described properties. Indeed, let u be in F andv
≔
I
-
∇
φ
u. We should prove that v
∈
F. v
=
u
-
∇
φ
u
∈
W
0
1
,
p
Ωand u
1
x
≤
v
x
≤
u
2
x. Since the definition relation of the subsolution for u
1 actually means φ
w
′
u
1
w
≤
0
∀
w in W
0
1
,
p
Ω with w
(
x
)
≥
0 almost everywhere (a.e.) on Ω and that of supersolution for u
2 is φ
w
′
u
2
w
≥
0
∀
w inW
0
1
,
p
Ω verifying w
(
x
)
≥
0 a.e. onΩ, we will prove that v
(
x
)
-
u
1
(
x
)
≥
0 a.e. on Ω and u
2
x
-
v
x
≥
0 a.e. on Ω using the Gâteaux derivatives of φ in u
1 and u
2, respectively. φ
w
′
u
1
(
v
-
u
1
)
=
φ
w
′
(
u
1
)
(
u
1
-
u
)
-
φ
w
′
u
1
∇
φ
u
≤
-
φ
w
′
u
1
∇
φ
u
≤
-
φ
w
′
u
∇
φ
u
=
-
φ
w
′
u
2
≤
0 (take into account that u
1
≤
u, φ
w
′
(
u
1
) is a linear map and some properties of the metric gradient). Also φ
w
′
u
2
u
2
-
v
=
φ
w
′
(
u
2
)
(
u
2
-
v
)
+
φ
w
′
u
2
∇
φ
u
≥
φ
w
′
u
2
∇
φ
u
≥
φ
w
′
u
∇
φ
u
=
φ
w
′
u
2
≥
0. φ is lower bounded on F, φ being actually continuous (for this see, for instance, [2]). Until now, applying Proposition 5, for every v
n
n
≥
1 a minimizing sequence for φ on F, there is a sequence u
n
n
≥
1 in F such that φ
u
n
≤
φ
(
v
n
)
∀
n, l
i
m
n
→
∞
u
n
-
v
n
=
0, l
i
m
n
→
∞
φ
w
′
(
u
n
)
=
0. So l
i
m
n
→
∞
φ
u
n
=
c since c
≔
inf
⁡
φ
F, we have l
i
m
n
→
∞
φ
w
′
u
n
=
0 already, also the last property from P
S
c
,
F condition is verified. Finish the proof applying once again Proposition 5.Example 10.
Consider the problem (Ωopen bounded of C
1 class in R
N, N
≥
3)(32)-Δpu=αxuup-2,u=0on∂Ω,where p
=
2
N
/
N
-
2; α is continuous with 1
≤
α
(
x
)
≤
a
<
+
∞ on Ω. Then u
1
≔
1 is a weak subsolution, u
2
≔
M, M
>
1 sufficiently big, is a weak supersolution, f
x
,
s
≤
a
s
p
-
1 (condition (17)), and s
→
α
(
x
)
s
s
p
-
2
+
s is increasing in s on [
1
,
M
]; consequently, according to Proposition 9, (32) has a weak solutionu
-with 1
≤
u
-
(
x
)
≤
M a.e. onΩ.Proposition 11.
LetΩbe an open bounded of C
1 class set in R
N, N
≥
3 and f
:
Ω
×
R
→
R a Carathéodory function and u
1, u
2 from W
0
1
,
p
Ω bounded weak subsolution and weak supersolution, respectively, of ∗
∗ with u
1
x
≤
u
2
x a.e. onΩ. Suppose that f verifies (17) and there is ρ
>
0 such that the function g
:
g
x
,
s
=
f
x
,
s
+
ρ
s is strictly increasing in s on inf
⁡
u
1
Ω
,
sup
⁡
u
2
Ω. Then there is a weak solution u
- of ∗
∗ inW
0
1
,
p
(
Ω
) with the property(33)u1x≤u-x≤u2xa.e.onΩ.Proof.
Follow step by step the above proof for Proposition9 considering the real reflexive strictly convex Banach space X
=
W
0
1
,
p
Ω endowed with the norm u
→
❙
u
❙
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p or the equivalent norm u
→
u
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω which both are also equivalent to the other two norms used at Proposition 9. The function φ is from (19) having the weak derivative given in Theorem 7. Using similar calculus, obtain similar conclusion.Example 12.
Consider the problem (Ωopen bounded of C
1 class in R
N, N
≥
3):(34)-Δpsu=αxuup-2u=0on∂Ω,where p
=
2
N
/
N
-
2 and α is continuous with 1
≤
α
x
≤
a
<
+
∞ onΩ. Then u
1
≔
1 is a weak subsolution, u
2
≔
M, M
>
1 sufficiently big, is a weak supersolution, f
x
,
s
≤
a
s
p
-
1 (condition (17)), and s
→
α
x
s
s
p
-
2
+
s is increasing in s on [
1
,
M
]; consequently, according to Proposition 11, (34) has a weak solutionu
-with 1
≤
u
-
(
x
)
≤
M a.e. on Ω.


## 3. Critical Points for Nondifferentiable Functionals

The sense of the title actually is “not compulsory differentiable.” Start this section with the following.Definition 13.
x
0 is acritical point (in the sense of Clarke subderivative) for the real function f if 0
∈
∂
f
x
0. In this case f
(
x
0
) is acritical value (in the sense of Clarke subderivative) for f.
To clarify this notion, the Clarke subderivative should be introduced. LetX be a real normed space, E
⊂
X, f
:
E
→
R, x
0
∈
E
°, and v
∈
X. We set f
0
x
0
;
v
≔
l
i
m
¯
x
→
x
0
,
t
→
0
+
f
x
+
t
v
-
f
x
/
t. f
0
x
0
;
v is by definition Clarke derivative (or the generalized directional derivative) of the function f at x
0 in the direction v. The functional ξ from X
∗ is by definition Clarke subderivative (or generalized gradient) of f in x
0 if f
0
x
0
;
v
≥
ξ
(
v
)
∀
v
∈
X. The set of these generalized gradients is designated by ∂
f
x
0.Here it is a generalization atp-Laplacian and p-pseudo-Laplacian of an application from [16] of this concept.LetΩ be a bounded domain of R
N with the smooth boundary ∂
Ω (topological boundary). Consider the nonlinear boundary value problems ∗ and ∗
∗ where f
:
Ω
×
R
→
R is a measurable function withsubcritical growth; that is,Ifx,s≤a+bsσ∀s∈R,x∈Ωa.e.,where a
,
b
>
0, 0
≤
σ
<
N
+
2
/
N
-
2for N
>
2 and σ
∈
[
0
,
+
∞
) for N
=
1 or N
=
2.Set as in [15](35)f_x,t=lim_s→tfx,s,f¯x,t=lims→t-fx,s.SupposeIIf_,f¯:Ω×R⟶Rare measurable with respect tox.We emphasize that I
I is verified in the following two cases:(i)
fis independent of x.
(ii)
f is Baire measurable and s
→
f
x
,
s is decreasing ∀
x
∈
Ω, in which case we have
(36)f¯x,t=max⁡fx,t+,fx,t-,f_x,t=min⁡fx,t+,fx,t-.Definition 14.
u from W
0
1
,
p
Ω
,
p
>
1 is solution of ∗ and ∗
∗, respectively, if u
=
0 on ∂
Ω in the sense of trace and(37)-Δpux∈f_x,ux,f¯x,uxinΩa.e.(38)respectively-Δpsux∈f_x,ux,f¯x,uxinΩa.e.
DefineW
1
,
p
Γ with p
∈
1
,
+
∞, Γ regular differential manifold, e.g., Γ
=
∂
Ω, Ω open of C
1 class with ∂
Ω bounded. In this situation, there exists a unique linear continuous operator γ
:
W
1
,
p
Ω
→
W
1
-
1
/
p
,
p
∂
Ω, the trace, such that γ is surjective and u
∈
W
1
,
p
Ω
∩
C
Ω
-
⇒
γ
u
=
u
∣
∂
Ω. This gives a sense for u
∣
∂
Ω, for any u in W
1
,
p
Ω. Moreover, γ
-
1
0
=
W
0
1
,
p
Ω.
Let beX
≔
W
0
1
,
p
Ω, but in the first case ∗, the norm endowing X is ·
1
,
p, that is, u
1
,
p
=
u
W
0
1
,
p
Ω
=
u
L
p
Ω
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω, which is equivalent to the norm u
→
u
L
p
Ω
p
+
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p. For the second case ∗
∗, equip the same set X by the norm u
→
❙
u
❙
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω
p
1
/
p, which is equivalent to u
→
u
1
,
p
=
∑
i
=
1
N
∂
u
/
∂
x
i
L
p
Ω.Associate to∗ the locally Lipschitz functional Φ
:
X
→
R:(39)Φu=1pu1,pp-∫ΩFx,udx,u∈X,and associate to ∗
∗
(40)
Φ
u
=
1
p
❙
u
❙
1
,
p
p
-
∫
Ω
F
x
,
u
d
x
,
u
∈
X
,where F
(
x
,
s
)
=
∫
0
s
f
x
,
t
d
x. Set(41)
Q
u
≔
1
p
u
1
,
p
p
,
u
∈
X
,
Ψ
1
u
≔
∫
Ω
F
x
,
u
d
x
,
u
∈
X
,
(42)
respectively
Q
u
≔
1
p
❙
u
❙
1
,
p
p
,
u
∈
X
,
Ψ
1
u
≔
∫
Ω
F
x
,
u
d
x
,
u
∈
X
,
F, a map defined on Ω
×
R, taking values in R, is locally Lipschitz (use I). The functional Ψ
:
L
σ
+
1
Ω
→
R, Ψ
u
=
∫
Ω
F
(
x
,
u
)
d
x, is also locally Lipschitz (again I). Using Sobolev embedding X
⊂
L
σ
+
1
Ω, we obtain that Ψ
1
≔
Ψ
∣
X is locally Lipschitz on X, which implies Φ locally Lipschitz on X, and consequently, according to a local extremum result for Lipschitz functions (if x
0 is a point of local extremum for f, then 0
∈
∂
f
x
0), the critical points of Φ for Clarke subderivative can be taken into account. One may state the following.Proposition 15.
SupposeI and I
I are satisfied. Then Ψ is locally Lipschitz on L
σ
+
1 (Ω) and (i)
∂
Ψ
u
⊂
f
_
x
,
u
x
,
f
¯
x
,
u
x in Ω a.e,
(ii)
ifΨ
1
=
Ψ
∣
X, where X
=
W
0
1
,
p
(
Ω
) endowed with the norm ·
1
,
p for the problem ∗ and ❙
·
❙
1
,
p for the problem ∗
∗, respectively, then(43)∂Ψ1u⊂∂Ψu∀u∈X.Proof.
The proof for (i) can be found in [15], Theorem 2.1, which remained here the same, while the problem was solved for the Laplacian with X
=
H
0
1
Ω only. In order to prove (ii), use Theorem 2.2 from [15] observing for both cases (X endowed with each one of those two norms) that X is reflexive and dense in L
σ
+
1
Ω as can be seen, for instance, summarized in [2].Proposition 16.
IfI and I
I are verified, every critical point of Φ is solution for ∗ and ∗
∗, respectively.Proof.
Problem
∗. Let u
0 be a critical point for Φ. We have (44)0∈∂Φu0⊂∂Qu0+∂-Ψ1u0,since Φ
=
39
Q
-
Ψ
1, and apply some rules of subdifferential calculus concerning finite sums. ∂
Q
u
0
=
Q
′
u
0, where Q
′
u
0
v
=
∫
Ω
∇
u
0
p
-
2
·
∇
v
d
x
=
-
Δ
p
u
0
,
v [2].
Using (44) and a specific property of a function f Lipschitz around x
0
(
f
0
x
0
;
v
=
s
u
p
ξ
∈
∂
f
x
0
ξ
v
,
∀
v
∈
X, f
0 the Clarke derivative of f), we find(45)0≤∫Ω∇u0p-2·∇vdx+-Ψ10u0;v.But -
Ψ
1
0
u
0
;
v
=
Ψ
1
0
u
0
;
-
v (a property of the Clarke derivative, see [1]) and thus(46)∫Ω∇u0p-2·∇-vdx≤Ψ0u0;-v∀v∈X;that is,(47)μ0v≔∫Ω∇u0p-2·∇vdx≤Ψ10u0,v∀v∈X,
μ
0
=
-
Δ
p
u
0
∈
∂
Ψ
1
u
0 and, using Proposition 15, -
Δ
p
u
0
∈
∂
Ψ
u
0. Since ∂
Ψ
(
u
0
)
⊂
L
σ
+
1
Ω
∗
=
L
σ
+
1
/
σ
(
Ω
), it results u
0
∈
W
2
,
(
σ
+
1
)
/
σ
(
Ω
) and (37):(48)-Δpu0x∈f_x,u0x,f¯x,u0xinΩa.e.
Problem
∗
∗
. Let u
0 be a critical point for Φ. We have(49)0∈∂Φu0⊂∂Qu0+∂-Ψ1u0,since Φ
=
40
Q
-
Ψ
1, and apply some rules of subdifferential calculus concerning finite sums. ∂
Q
u
0
=
Q
′
u
0, where Q
′
u
0
v
=
∑
i
=
1
N
∫
Ω
∂
u
0
/
∂
x
i
p
-
2
∂
u
0
/
∂
x
i
∂
v
/
∂
x
i
d
x
=
Δ
p
s
u
0
,
v [2].
Using (49) and a mentioned property of a function f Lipschitz around x
0, we find(50)0≤∑i=1N∫Ω∂u0∂xip-2∂u0∂xi∂v∂xidx+-Ψ10u0;v.But -
Ψ
1
0
(
u
0
;
v
)
=
Ψ
1
0
u
0
;
-
v (a property of the Clarke derivative, see [1]) and thus(51)∑i=1N∫Ω∂u0∂xip-2∂u0∂xi∂-v∂xidx≤Ψ0u0;-v∀v∈X;that is, (52)μ0v≔∑i=1N∫Ω∂u0∂xip-2∂u0∂xi∂v∂xidx≤Ψ10u0,v∀v∈X,
μ
0
=
-
Δ
p
s
u
0
∈
∂
Ψ
1
u
0 and, using Proposition 15, -
Δ
p
s
u
0
∈
∂
Ψ
u
0. Since ∂
Ψ
(
u
0
)
⊂
L
σ
+
1
Ω
∗
=
L
(
σ
+
1
)
/
σ
(
Ω
), it results u
0
∈
W
2
,
(
σ
+
1
)
/
σ
(
Ω
) and (38):(53)-Δpsu0x∈f_x,u0x,f¯x,u0xinΩa.e.


## 4. An Application of Ghoussoub-Maurey Linear Principle top-Laplacian and to p-Pseudo-Laplacian

Start with the statement of this generalized perturbed variational principle.Theorem 17 (Ghoussoub-Maurey linear principle).
LetX be reflexive separable space and φ
:
X
→
-
∞
,
+
∞ lower semicontinuous and proper:(I)
Ifφ is bounded from below on the closed bounded nonempty subset C, the set (54)ξ∈X∗:φ+ξstronglyexposesCfrombelow
is ofG
δ type and everywhere dense.
(II)
If, for anyξ from X
∗, φ
+
ξ is bounded from below, the set(55)ξ∈X∗:φ+ξstronglyexposesXfrombelow
is ofG
δ type and everywhere dense.To clarify the involved notions, letX be a real normed space, f
:
X
→
(
-
∞
,
+
∞
], C nonempty subset of X, and x
0
∈
C. f
strongly exposes
C
from below in
x
0, when f
x
0
=
inf
⁡
f
C
<
+
∞ and x
n
∈
C
∀
n
≥
1, f
x
n
→
f
x
0
⇒
x
n
→
x
0. “f strongly exposes C from above in x
0” has a similar definition. Remark that, taking C
=
X in the given definition, we fall on the definition ofstrongly minimum point. And also, a set of G
δ type means a set which is a countable intersection of open sets. A set of F
σ type means a set which is a countable union of closed sets.We imply this theorem in two generalizations of a minimization problem of the form [22](56)
C
f
≔
min
⁡
∫
Ω
1
p
u
p
+
∑
i
=
1
N
∂
u
∂
x
i
p
-
f
u
d
x
:
u
∈
W
0
1
,
p
Ω
,
u
2
∗
=
1
,
(57)
C
f
≔
min
⁡
∫
Ω
1
p
∑
i
=
1
N
∂
u
∂
x
i
p
-
f
u
d
x
:
u
∈
W
0
1
,
p
Ω
,
u
2
∗
=
1
,where Ω is open set of C
1 class in R
N, N
≥
3, f
∈
W
-
1
,
p
′
Ω
=
W
0
1
,
p
Ω
∗, 1
/
p
+
1
/
p
′
=
1, 2
∗
=
2
N
/
N
-
2,  the critical exponent for the Sobolev embedding (for the necessary explanations, here and in the following, see [1, I, § 4, last section]).LetΩ be an open bounded set of C
1 class in R
N, N
≥
3. Consider the problems (∗) and (∗
∗), where f
:
Ω
×
R
→
R is a Carathéodory function with the growth condition(58)fx,s≤csp-1+bx,c>0,2≤p≤2NN-2,b∈Lp′Ω,1p+1p′=1.The functionals φ
:
W
0
1
,
p
(
Ω
)
→
R,(59)φu=∫Ω1pup+∑i=1N∂u∂xip-Fx,uxdx,(60)φu=∫Ω1p∑i=1N∂u∂xip-Fx,uxdx,with F
x
,
s
≔
∫
0
s
f
(
x
,
t
)
d
t, are of C
1-class Fréchet and their critical points are the weak solutions of the problems (∗) and (∗
∗), respectively.Problem (
∗
). Let λ
1 be the first eigenvalue of -
Δ
p inW
0
1
,
p
(
Ω
) with homogeneous boundary condition. We have (see [2, 6.2])(61)λ1=inf⁡u1,ppiu0,pp:u∈W01,pΩ∖0the Rayleigh - Ritz quotient.And now give an answer for (56). Use the norm ·
p on W
0
1
,
p
(
Ω
) (see above). Denote the dual of W
0
1
,
p
Ω
,
·
1
,
p by W
-
1
,
p
′
Ω, where p
′ is the conjugate of p  (i.e., 1
/
p
+
1
/
p
′
=
1).Proposition 18.
Under the above assumptions and in addition the growth condition(62)Fx,s≤c1spp+αxs,with 0
<
c
1
<
λ
1, α
∈
L
q
′
(
Ω
) for some 2
≤
q
≤
2
N
/
N
-
2and f
x
,
-
s
=
-
f
x
,
s, ∀
x from Ω, the following assertions hold: (i)
The set of functionsh from W
-
1
,
p
′
Ω, having the property that the functional φ
h
:
W
0
1
,
p
(
Ω
)
→
R,(63)φhu=1pupp-∫ΩFx,ux+huxdx
has in only one point an attained minimum includes aG
δ set everywhere dense.
(ii)
The set of functionsh from W
-
1
,
p
′
Ω, having the property(64)theproblem-Δpu=fx,u+huinΩu=0on∂Ωhassolutions,
includes aG
δ set everywhere dense.
(iii)
Moreover, ifs
→
f
x
,
s is increasing, then the set of functions h from W
-
1
,
p
′
Ω, having the property(65)theproblem-Δpu=fx,u+huinΩu=0on∂Ωhasauniquesolution,
includes aG
δ set everywhere dense.Remark 19.
This is a generalization top-Laplacian and at W
0
1
,
p
(
Ω
) of Theorem2.13 from [14].Proof.
It is sufficient to justify (i). Consider, for eachh from W
-
1
,
p
′
Ω, the functional ξ
h from W
-
1
,
p
′
Ω:(66)ξhu=-∫Ωhuxdx.One may see that φ
h
=
φ
+
ξ
h (see (59)). Consequently, according to Ghoussoub-Maurey linear principle, I
I, if we show that φ
h is bounded from below for any h from W
-
1
,
p
′
Ω, then (i) is proven. But, taking into account the Sobolev embedding and (62), we have ∀
u
∈
W
0
1
,
p
Ω: (67)φ+ξhu=1pu1,pp-∫ΩFx,uxdx-∫Ωhuxdx≥1pu1,pp-c1∫Ωuxppdx-∫Ωαxuxdx-∫Ωhuxdx≥1pu1,pp-c1λ1pu1,pp-αq′uq-hW-1,p′u1,p≥1p1-c1λ1u1,pp-ru1,p=u1,p1p1-c1λ1u1,pp-r,
r
∈
R, and hence the conclusion since 1
-
c
1
/
λ
1
>
0. To prove some of these inequalities, (68)∫ΩFx,uxdx≤∫Ωuxpp+αxuxdx=1piu0,pp+∫Ωαxuxdx≤1pλ1u1,pp+α0,q′uq≤1pλ1u1,pp+Ku1,p(see q and properties of Sobolev spaces, e.g., [2]) and ∫
Ω
h
u
x
d
x
=
h
,
u
≤
h
W
-
1
,
p
′
u
1
,
p (the norm of the linear continuous map).Problem (
∗
∗
). Let λ
1 be the first eigenvalue of -
Δ
p
s in W
0
1
,
p
(
Ω
) with homogeneous boundary condition. We have (see [2, 7.2])(69)
λ
1
=
inf
⁡
❙
u
❙
1
,
p
p
i
u
0
,
p
p
:
u
∈
W
0
1
,
p
Ω
∖
0
the Rayleigh - Ritz quotient
.And now give an answer for (57). Use the norm ❙
·
❙
p on W
0
1
,
p
(
Ω
) (see above). Denote also the dual of W
0
1
,
p
Ω
,
❙
·
❙
p by W
-
1
,
p
′
Ω, where p
′ is the conjugate of p (i.e., 1
/
p
+
1
/
p
′
=
1).Proposition 20.
Under the above assumptions and in addition the growth condition(70)Fx,s≤c1spp+αxs,with 0
<
c
1
<
λ
1, α
∈
L
q
′
Ω for some 2
≤
q
≤
2
N
/
N
-
2 and f
(
x
,
-
s
)
=
-
f
(
x
,
s
), ∀
x from Ω, the following assertions hold.(i)
The set of functionsh from W
-
1
,
p
′
Ω, having the property that the functional φ
h
:
W
0
1
,
p
(
Ω
)
→
R, (71)
φ
h
u
=
1
p
❙
u
❙
p
p
-
∫
Ω
F
x
,
u
x
+
h
u
x
d
x
has in only one point an attained minimum includes aG
δ set everywhere dense.
(ii)
The set of functionsh from W
-
1
,
p
′
Ω, having the property (72)theproblem-Δpsu=fx,u+huinΩu=0on∂Ωhassolutions,
includes aG
δ set everywhere dense.
(iii)
Moreover, ifs
→
f
x
,
s is increasing, then the set of functions h from W
-
1
,
p
′
Ω, having the property(73)theproblem-Δpsu=fx,u+huinΩu=0on∂Ωhasauniquesolution,
includes aG
δ set everywhere dense.Remark 21.
This is a generalization top-pseudo-Laplacian and at W
0
1
,
p
(
Ω
) of Theorem  2 .13 from [14].Proof.
The proof and the afferent calculus follow step by step those for Proposition18. One replaces there the norm ·
p on W
0
1
,
p
(
Ω
) by the norm ❙
·
❙
p and the inequalities and the considerations remain the same.


## 5. Conclusions

Three ways to obtain and/or characterize weak solutions for two problems of mathematical physics equations involving Dirichlet problems for thep-Laplacian and thep-pseudo-Laplacian are developed.The first sequence of results used the Ekeland variational principle to obtain theoretical propositions which generalize two statements given by Ghoussoub in which the author replaced the real Hilbert space by real reflexive uniformly convex Banach space, and the FréchetC
1-class of the goal function by the conditions imposed to this to be lower semicontinuous and Gâteaux differentiable. It is also worthwhile to underline that the Gâteaux differentiability can be replaced by the property ofβ-differentiability, β being any bornology. These theoretical statements have been used to characterize weak solutions for thep-Laplacian and for thep-pseudo-Laplacian. Some adequate examples were also given.The second succession of statements establishes results for nondifferentiable functionals using Clarke gradient and other specific notions until their insertion in characterization of weak solutions for Dirichlet problems with thep-Laplacian and thep-pseudo-Laplacian, respectively, in W
0
1
,
p
Ω.The last sequence of assertions starts from Ghoussoub-Maurey linear principle which is used in order to solve some minimization problems. Generalizations of a minimization problem for the Laplacian given by Brezis and Nirenberg have been obtained in conjunction with characterization of weak solutions of Dirichlet problems for thep-Laplacian and for thep-pseudo-Laplacian.



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*Source: 2071926-2016-08-25.xml*