# On the Convergence of a Nonlinear Boundary-Value Problem in a Perforated Domain

Authors: Yulia Koroleva
Journal: International Journal of Differential Equations
 (2015)

---

## Abstract

We consider a family with respect to a small parameter of nonlinear boundary-value problems as well as the corresponding spectral problems in a domain perforated periodically along a
part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain.


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## Body

## 1. Introduction

The paper is devoted to study of convergence of nonlinear boundary-value problems in a domain perforated along the boundary. There exist a lot of literatures, where boundary-value problems in perforated domains were studied. We refer to works [1–33]. In these papers and monographs the authors studied different kinds of perforation for linear as well as for nonlinear differential operators. Usually it is considered a family of problems depending on small parameter that characterizes the size of perforation. The main goal of the research is to find a homogenized (limit) model which is close to originally considered problems posed in the perforated domain. The general technique of homogenization method can be found in [23, 24, 27, 28].The present paper will deal with convergence of boundary-value problems in perforated domains for nonlinearp-Laplace operator. Some problems for nonlinear operators were homogenized, for example, in [1, 20, 21, 30–33]. We consider a family of boundary-value problems in n-dimensional domain, n
>
2, which is periodically perforated along the boundary by small sets. It is assumed that the diameter of each set and the distance between them have the same order. In our problem we suppose that the Dirichlet condition holds on the boundary of cavities, while the Naumann boundary condition is fulfilled on the boundary of the domain. We derive the limit (homogenized) problem for the original problems when the small parameter characterizing the size of perforation tends to zero. Moreover, we establish the strong convergence in W
1
,
p of the solutions for the considered problems to the corresponding solution of the limit problem. In addition we have obtained an estimate of the solution in a neighborhood of the eigenvalue of a corresponding spectral problem.One of our goals is to prove the asymptotic behavior for the eigenvalue problem forp-Laplace operator in our perforated domain. Many authors considered spectral problems for p-Laplace operator; see, for example, [33–38]. These papers contain the results on qualitative properties of the p-Laplace spectral problems, convergence of eigenvalue problems, and some estimates for the difference between considered eigenvalues. The applications of our problem do not require the knowledge about the full spectrum of eigenvalues. Therefore we have proved the homogenization theorems only for the first eigenelement of the spectral problem in perforated domain. More precisely, we have proved that the first eigenelement of the spectral problems converges to the corresponding eigenelement of the spectral limit problem. An analogous problem for linear elliptic operators for the two-dimensional domain was considered in [15] and for dimension three in [10].The crucial point in our analysis is the validity of the Friedrichs inequality for functions in perforated domains. We prove this nontrivial result which is of an independent interest. Some papers devoted to this inequality in domains with microinhomogeneous structure are [8–10, 18, 19].


## 2. Preliminaries and the Main Results

LetΩ
⊂
R
n
-
1
×
{
x
n
>
0
}, n
>
2, be a domain with boundary ∂
Ω
=
Γ. We assume that Γ is piece-wise smooth and consists of the parts Γ
i, i
=
1,2
,
3,4
:
Γ
=
Γ
1
∪
Γ
2
∪
Γ
3
∪
Γ
4, where Γ
4
=
-
1
/
2
;
1
/
2
n
-
1
∩
{
x
n
=
0
}, Γ
2, Γ
3 are orthogonal to {
x
n
=
0
} and belong to the planes {
x
1
=
-
1
/
2
} and {
x
1
=
1
/
2
} correspondingly, and Γ
∖
Γ
4 is a smooth surface. In the sequel ε
=
1
/
2
N
+
1 is a small parameter, N
∈
N, N
≫
1.Consider the setB
⊂
R
n belonging to the ball (1)
K
=
ξ
:
∑
i
=
1
n
-
1
ξ
i
2
+
ξ
n
-
1
2
2
<
a
2
,
0
<
a
<
1
2and having a smooth boundary. If one multiplies each coordinate of B with parameter ε and does integer translations of this set along Γ
4, we obtain the set denoted by B
ε
. Let Γ
ε
=
∂
B
ε. Define the perforated domain Ω
ε as Ω
∖
B
ε
¯
. See the illustration for cut of Ω
ε on Figure 1.Figure 1
Structure ofΩ
ε.Denote byW
1
,
p
(
Ω
ε
,
Γ
ε
) a set of functions from W
1
,
p
(
Ω
ε
) with zero trace on Γ
ε. Analogously, by W
1
,
p
(
Ω
,
Γ
4
) we define a set of functions from W
1
,
p
(
Ω
) with zero trace on Γ
4. We also consider the space C
0
∞
(
Ω
¯
,
Γ
4
), which is the set of functions from C
∞
(
Ω
¯
), vanishing in the neighborhood of Γ
4. Analogously, C
0
∞
(
Ω
-
ε
,
Γ
ε
) denotes the set of functions from C
∞
(
Ω
-
ε
), vanishing in a neighborhood of Γ
ε.Remark 1.
One can extend the functionsu
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
) into B
ε by zero. For the extended function we keep the same notation. It is true that u
ε belongs to W
1
,
p
Ω; see [24].Definition 2.
For2
≤
p
<
n define the operator (2)
Δ
u
p
x
≡
∑
j
=
1
n
∂
∂
x
j
∇
u
x
p
-
2
∂
u
x
∂
x
j
.We consider the following spectral problem:(3)
-
Δ
p
u
ε
x
=
λ
ε
u
ε
p
-
2
u
ε
i
n
Ω
ε
,
u
ε
=
0
o
n
Γ
ε
,
∂
u
ε
∂
p
ν
≡
∇
u
p
-
2
∇
u
,
ν
=
0
o
n
∂
Ω
,where ν is the unit outward normal vector to the boundary of Ω
.Definition 3.
One says thatλ
ε is an eigenfunction to problem (3) if there exists u
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
∖
{
0
}, satisfying the integral identity (4)
∫
Ω
ε
∑
j
=
1
n
∇
u
ε
x
p
-
2
∂
u
ε
x
∂
x
j
∂
φ
ε
x
∂
x
j
d
x
=
λ
ε
∫
Ω
ε
u
ε
p
-
2
u
ε
φ
ε
d
xfor every φ
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
. The couple (
u
ε
,
λ
ε
) is called the solution to (3).We will show that the problem(5)
-
Δ
p
u
0
x
=
λ
0
u
0
p
-
2
u
0
i
n
Ω
,
u
0
=
0
o
n
Γ
4
,
∂
u
0
∂
p
ν
≡
∇
u
0
p
-
2
∂
u
0
∂
ν
=
0
o
n
∂
Ω
∖
Γ
¯
4is homogenized (the limit one) for (3).As usual, we understand the solution to this boundary-value problem in the weak sense, that is, iffu
0
∈
W
1
,
p
(
Ω
,
Γ
4
)
∖
{
0
} satisfies (6)
∫
Ω
∑
j
=
1
n
∇
u
0
x
p
-
2
∂
u
0
x
∂
x
j
∂
φ
x
∂
x
j
d
x
=
λ
0
∫
Ω
u
0
p
-
2
u
0
φ
d
xfor every φ
∈
W
1
,
p
(
Ω
,
Γ
4
)
.Moreover, we prove the following results.Theorem 4.
Assume thatF
∈
W
-
1
,
q
(
Ω
), 1
/
p
+
1
/
q
=
1, 2
≤
p
<
∞, q
>
1, and K is an arbitrary compact set belonging to the complex plane C; K does not contain the eigenvalues of problem (5). Then the following statements hold:(1)
There exists a numberε
0
>
0, such that the unique solution to the problem (7)
-
Δ
p
U
ε
x
=
λ
U
ε
p
-
2
U
ε
+
F
in
Ω
ε
,
U
ε
=
0
on
Γ
ε
,
∂
U
ε
∂
p
ν
≡
∇
U
ε
p
-
2
∇
U
ε
,
ν
=
0
on
∂
Ω
does exist for allε
<
ε
0 and for all λ
∈
K
. Moreover, the uniform (in ε and λ) estimate (8)
U
ε
W
1
,
p
≤
C
F
L
q
is valid, whereC does not depend on U
ε and F.
(2)
It yields that(9)
U
ε
-
U
0
W
1
,
p
⟶
0
when
ε
⟶
0
,
whereU
0 is the unique solution of the problem (10)
-
Δ
p
U
0
x
=
λ
U
0
p
-
2
U
0
+
F
in
Ω
,
U
0
=
0
on
Γ
4
,
∂
U
0
∂
p
ν
≡
∇
U
0
p
-
2
∂
U
0
∂
ν
=
0
on
∂
Ω
∖
Γ
¯
4
.Here the solutions to problems (7) and (10) are understood in the weak sense, that is, i
f
f
U
δ, δ
=

ε
,

0 satisfies the integral identity(11)
∫
Ω
∑
j
=
1
n
∇
U
δ
x
p
-
2
∂
U
δ
x
∂
x
j
∂
φ
x
∂
x
j
d
x
=
λ
∫
Ω
U
δ
p
-
2
U
δ
φ
d
x
+
∫
Ω
F
φ
d
xfor every φ
∈
W
1
,
p
(
Ω
,
Γ
4
)
.Theorem 5.
The spectrum of problems (3) and (5) is nonempty closed set. Let λ
ε
1
,
λ
0
1 be the first eigenvalues of problems (3) and (5), respectively. Then (12)
λ
ε
1
⟶
λ
0
1
as
ε
⟶
0
.Moreover, if u
ε
,
u
0 are corresponding eigenfunctions, normalized in L
p, then up to a subsequence, (13)
u
ε
-
u
0
W
1
,
p
⟶
0
.The proofs of analogous theorems for linear boundary-value problems were given in [6, 12–15] for different types of singular perturbations. The following lemma, which is proved in Section 3.1, is necessary for our analysis.Lemma 6.
Letv
ε be a sequence of functions from W
1
,
p
(
Ω
ε
,
Γ
ε
) and assume that v
ε
⇀
v
∗ weakly in W
1
,
p
(
Ω
) when ε
→
0. Then v
∗
∈
W
1
,
p
(
Ω
,
Γ
4
)
.For the questions on existence of solutions to the discussed problems we will refer to the following general result (see [39]).Theorem 7.
LetV be reflexive separable Banach space. Assume that the operator A
:
V
→
V
′ has the following properties: (i)
Ais bounded and semicontinuous: A
<
∞, ∀
u
,
v
,
w
∈
V; the function s
→
(
A
(
u
+
s
v
)
,
w
) is continuous as a function mapping R into R.
(ii)
A is monotone: (
A
(
u
)
-
A
(
v
)
,
u
-
v
)
≥
0
∀
u
,
v
∈
V
.
(iii)
Consider(14)
A
u
,
u
u
⟶
+
∞
as
u
⟶
+
∞
.Then for any f
∈
V
′ there exists u
∈
V such that A
(
u
)
=
f
.


## 3. The Friedrichs Inequality

In our analysis we will need the Friedrichs inequality for functionsv
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
. We prove this result.Theorem 8.
The inequality(15)
∫
Ω
ε
v
ε
p
d
x
≤
K
∫
Ω
ε
∇
v
ε
p
d
xholds for any functions v
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
), where the constant K does not depend on ε
.Proof.
To demonstrate the technique of proving and avoiding the heavyn-dimensional notations we assume that n
=
2
. The case of arbitrary n
>
2 can be done by repeating all lines of the present proof.
Let the length of projection ofB on axis ξ
1 equal b. Denote (16)
B
ε
i
=
x
∈
Ω
:
ε
-
1
x
1
-
i
,
x
2
∈
B
,
i
∈
Z
.Represent the domain Ω
ε as follows: (17)
Ω
ε
=
⋃
i
=
-
N
N
Π
1
,
ε
i
+
∪
Π
1
,
ε
i
-
∪
⋃
i
=
-
N
-
1
N
Π
2
,
ε
i
∪
R
ε
,where Π
1
,
ε
i
+ is the domain (a part of the vertical strip) of the length b
ε, bounded from below by the upper part of the boundary B
ε
i and bounded from above by a part of the boundary Γ
1; the domain Π
1
,
ε
i
- is the strip of the length b
ε, bounded from above by the lower part of B
ε
i and bounded from below by the line x
2
=
0; Π
2
,
ε
i, i
=
-
N
,
…
,
N
-
1, is the strip of the length (
1
-
b
)
ε, bounded from below by the line x
2
=
0 and bounded from above by a part of Γ
1 and having the common vertical bounds with Π
1
,
ε
i
+
,
Π
1
,
ε
i
+
1
+ (see Figure 2); the domains Π
2
,
ε
-
N
-
1 and Π
2
,
ε
N are vertical strips bounded from below by x
2
=
0 and bounded from above by a part of Γ
1 and having the common vertical bounds with Π
1
,
ε
-
N
+ and Π
1
,
ε
N
+, correspondingly. The sum of widths for Π
2
,
ε
-
N
-
1 and Π
2
,
ε
N is (
1
-
b
)
ε
. Finally, R
ε is the remaining part of Ω
ε having the bound belonging to Γ
1.
Letω
>
ε. Define by Π
1
,
ε
ω
,
i
+ a domain Π
1
,
ε
i
+
⋂
{
x
2
<
ω
} (see Figure 2). Moreover, let Γ
1
,
ε
ω
,
i be the segment of the line x
n
=
ω, which is upper bound for Π
1
,
ε
ω
,
i
+
. Analogously we define the sets Γ
2
,
ε
ω
,
i and Π
2
,
ε
ω
,
i. Without loss of generality we can assume that v
ε is real-valued function. Denote by C
0
1
(
Ω
¯
ε
,
Γ
ε
) functions from C
1
(
Ω
¯
ε
), vanishing in the neighborhood of Γ
ε. Assume first that v
ε
∈
C
0
1
(
Ω
¯
ε
,
Γ
ε
). Let (
x
1
,
ω
)
∈
Π
1
,
ε
i
+, and point (
x
1
,
x
~
2
) belongs to the bottom of Π
1
,
ε
i
+. Since v
ε
(
x
1
,
x
~
2
)
=
0, then the Newton-Leibnitz formula gives (18)
v
ε
x
1
,
ω
=
∫
x
~
2
ω
∂
v
ε
∂
x
2
d
x
2
.Taking the power p on both sides of that equality and using the Hölder inequality, we have (19)
v
ε
x
1
,
ω
p
=
∫
x
~
2
ω
∂
v
ε
∂
x
2
d
x
2
p
≤
ω
p
/
q
∫
x
~
2
ω
∂
v
ε
∂
x
2
p
d
x
2
≤
ω
p
/
q
∫
x
~
2
ω
∇
v
ε
p
d
x
2
.Integrate both sides of that inequality over Γ
1
,
ε
ω
,
i with respect to x
1 and assume that the function v
ε is extended by zero into B
ε
i
∪
Π
1
,
ε
i
-. One gets (20)
∫
Γ
1
,
ε
ω
,
i
v
ε
x
1
,
ω
p
d
x
1
≤
ω
p
/
q
∫
Π
1
,
ε
ω
,
i
+
∇
v
ε
p
d
x
.
Consider now the rectangleΠ
2
,
ε
i, i
=
-
N
,
…
,
N, which touches B
ε
i from the right. Draw the tangential lines to B
ε
i from both ends of segment Γ
2
,
ε
ω
,
i (see Figure 3).
It is easy to see that the angleα between the tangential lines and O
x
1 belongs to (
α
0
,
π
/
2
), where α
0
>
0 does not depend on ε and ω
. We remember that our assumption was ω
>
ε. This follows from the fact that the diameter of the set and the distance between them are of the same order. Connect all points of Γ
2
,
ε
ω
,
i with boundary of B
ε
i such that the intersection of these lines coincides with the intersection point of tangents. Thus, we have a beam of lines with directors l
(
x
)
. The angle between each line and O
x
1 belongs to (
α
0
,
π
/
2
). Let (
x
1
,
ω
)
∈
Γ
2
,
ε
ω
,
i, and (
x
~
1
,
x
~
2
)
∈
∂
B
ε
i
∩
l
(
x
1
,
ω
). Since v
ε
(
x
~
1
,
x
~
2
)
=
0, then (21)
v
ε
x
1
,
ω
=
∫
x
~
1
,
x
~
2
x
1
,
ω
∂
v
ε
∂
l
d
l
.Analogously to (19), taking the power p on both sides of that formula and using the Hölder inequality, one obtains (22)
v
ε
x
1
,
ω
p
≤
ω
sin
⁡
α
0
p
/
q
∫
x
~
1
,
x
~
2
x
1
,
ω
∂
v
ε
∂
l
p
d
l
≤
ω
sin
⁡
α
0
p
/
q
∫
x
~
1
,
x
~
2
x
1
,
ω
∇
v
ε
p
d
l
.Integrating both sides of that inequality over Γ
2
,
ε
ω
,
i with respect to x
1 and replacing the right-hand side by the greater integral, we get (23)
∫
Γ
2
,
ε
ω
,
i
v
ε
x
1
,
ω
p
d
x
1
≤
ω
sin
⁡
α
0
p
/
q
∫
Π
2
,
ε
ω
,
i
∪
Π
1
,
ε
ω
,
i
+
∪
Π
1
,
ε
ω
,
i
-
∇
v
ε
p
d
x
,
i
=
-
N
,
…
,
N
.
It remains to estimate the integral overΓ
2
,
ε
ω
,
-
N
-
1. We use the same technique as for Γ
2
,
ε
ω
,
i, i
=
-
N
,
…
,
N
. In this case one needs to consider the domain B
ε
-
N which is bordered with the strip Π
2
,
ε
-
N
-
1 from the right. Analogously, we get (24)
∫
Γ
2
,
ε
ω
,
-
N
-
1
v
ε
x
1
,
ω
p
d
x
1
≤
C
ω
p
/
q
∫
Π
2
,
ε
ω
,
-
N
-
1
∪
Π
1
,
ε
ω
,
-
N
+
∪
Π
1
,
ε
ω
,
-
N
-
∇
v
ε
p
d
x
.
LetΓ
4
ω
=
Ω
¯
∩
{
x
2
=
ω
} be the segment connecting the points (
-
1
/
2
,
ω
) and (
1
/
2
,
ω
); it means that Γ
4
ω
=
⋃
i
Γ
1
,
ε
ω
,
i
¯
∪
Γ
2
,
ε
ω
,
i
¯. Summing up inequalities (20), (23), and (24), one gets (25)
∫
Γ
4
ω
v
ε
x
p
d
x
1
≤
C
1
ω
p
/
q
∫
Ω
ε
∇
v
ε
p
d
x
. Finally, integrating (25) with respect to ω, we obtain that (26)
∫
Ω
ε
v
ε
x
p
d
x
1
d
x
2
≤
K
∫
Ω
ε
∇
v
ε
p
d
x
.
Approximating the functions fromW
1
,
p
(
Ω
ε
,
Γ
ε
) by smooth functions, we conclude that inequality (26) is valid for v
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
. The proof is complete.Figure 2
Parts ofΩ
ε.Figure 3
Tangential lines.Remark 9.
(1) Extending functions fromW
1
,
p
(
Ω
ε
,
Γ
ε
) by zero into ⋃
i
B
ε
i we obtain the Friedrichs inequality in Ω: (27)
v
ε
L
p
Ω
p
≤
K
∇
v
ε
L
p
Ω
p
f
o
r
v
ε
∈
W
1
,
p
Ω
ε
,
Γ
ε
. (2) The validity of Friedrichs inequality means that one can introduce in W
1
,
p
(
Ω
ε
,
Γ
ε
) the norm v
W
1
,
p
≡
∇
v
L
p which is equivalent to (28)
v
W
1
,
p
≡
v
L
p
+
∇
v
L
p
.


### 3.1. Proof of Lemma6

Proof.
First we point out that with the same method of proof inequality (25) is valid also for the case of an arbitrary n
>
2, where (29)
Γ
4
ω
=
Ω
¯
∩
x
n
=
ω
.Approximating the functions from W
1
,
p
(
Ω
ε
,
Γ
ε
) by smooth functions, we conclude that inequality (25) is valid for v
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
. Using (25) and keeping in mind the uniform boundedness of the sequence v
ε, we obtain that (30)
∫
Γ
4
ω
v
ε
x
1
,
…
,
x
n
-
1
,
ω
p
d
x
≤
ω
p
/
q
C
.
Now, we can pass to the limit in (30) when ε
→
0 and find that (31)
∫
Γ
4
ω
v
∗
p
d
x
≤
ω
p
/
q
C
3
.Due to the fact that ω is an arbitrary small positive number and v
∗
∈
W
1
,
p
(
Ω
), it follows from (31) that v
∗
=
0 on Γ
4
. The proof is complete.


## 3.1. Proof of Lemma6

Proof.
First we point out that with the same method of proof inequality (25) is valid also for the case of an arbitrary n
>
2, where (29)
Γ
4
ω
=
Ω
¯
∩
x
n
=
ω
.Approximating the functions from W
1
,
p
(
Ω
ε
,
Γ
ε
) by smooth functions, we conclude that inequality (25) is valid for v
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
)
. Using (25) and keeping in mind the uniform boundedness of the sequence v
ε, we obtain that (30)
∫
Γ
4
ω
v
ε
x
1
,
…
,
x
n
-
1
,
ω
p
d
x
≤
ω
p
/
q
C
.
Now, we can pass to the limit in (30) when ε
→
0 and find that (31)
∫
Γ
4
ω
v
∗
p
d
x
≤
ω
p
/
q
C
3
.Due to the fact that ω is an arbitrary small positive number and v
∗
∈
W
1
,
p
(
Ω
), it follows from (31) that v
∗
=
0 on Γ
4
. The proof is complete.


## 4. Proof of Theorem4

For the proof we need the following lemma.Lemma 10.
LetQ be an arbitrary compact set in the complex plane and λ
∈
Q
. Suppose that the estimate (32)
U
ε
p
≤
C
F
qholds uniformly in ε and λ for any solution U
ε of the boundary-value problem (7), which is normalized in L
p
(
Ω
). Then estimate (32) holds also for any solution of problem (7) when λ
∈
Q
.Proof.
Let us remember first that we denote by·
p the norm in L
p space. If U
ε
p
≠
1, then by setting V
ε
=
U
ε
/
U
ε
p, we obtain that (33)
V
ε
p
=
1and the function V
ε satisfies the identity(34)
∫
Ω
∑
j
=
1
n
∇
V
ε
x
p
-
2
∂
V
ε
x
∂
x
j
∂
φ
x
∂
x
j
d
x
=
λ
∫
Ω
V
ε
p
-
2
V
ε
φ
d
x
+
∫
Ω
F
ε
φ
d
x
,where (35)
F
ε
=
F
U
ε
p
p
-
1
.Hence, due to the assumptions we see that the estimate (36)
V
ε
p
≤
C
F
ε
qholds for V
ε
. Multiplying the last inequality by U
ε
p and using (33) and (35), we obtain the estimate (32) (probably with a different constant) for any U
ε
. The proof is complete.


### 4.1. Proof of Part 1

Step 1.
The existence of the solution to problem (7) can be proved with help of Theorem 7. Indeed, we take V
=
W
1
,
p
(
Ω
,
Γ
ε
)
. Introduce the operator A
(
u
)
=
-
‍
∑
i
∂
/
∂
x
i
∂
u
/
∂
x
i
p
-
2
∂
u
/
∂
x
i
. For u
,
v
∈
W
1
,
p
(
Ω
,
Γ
ε
) we define (37)
A
u
,
v
=
-
∫
Ω
∑
i
∂
∂
x
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
v
d
x
=
∫
Ω
∑
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
∂
v
∂
x
i
d
x
.Moreover, for u
∈
W
1
,
p
(
Ω
,
Γ
ε
) it holds that λ
|
u
|
p
-
1
u
∈
L
q
(
Ω
,
Γ
ε
), and for F
∈
L
q
(
Ω
) we can introduce the functional on W
1
,
p
(
Ω
,
Γ
ε
): (38)
λ
u
p
-
1
u
+
F
,
v
=
∫
Ω
λ
u
p
-
1
u
v
d
x
+
∫
Ω
F
v
d
x
.It is clear that U
ε is the solution to (7) iff (
A
(
U
ε
)
,
v
)
=
(
λ
|
U
ε
|
p
-
1
U
ε
+
F
,
v
) for any v
∈
W
1
,
p
(
Ω
,
Γ
ε
)
. Let us verify the properties of operator A
. We take u
∈
W
1
,
p with u
W
1
,
p
=
1
. Then (39)
A
u
W
-
1
,
q
=
A
u
,
u
=
∇
u
L
p
p
≤
C
u
W
1
,
p
p
≤
C
.One can check the semicontinuity and monotonicity of operator A either directly or by using the following general result (see [39]).Proposition 11.
If convex functionalv
→
J
(
v
) is differentiable in Gato sense, that is, there exists such continuous linear mapping v
→
J
′
(
u
)
·
v of space V onto R such that (40)
lim
s
→
0
⁡
1
s
J
u
+
s
v
-
J
u
=
J
′
·
v
,then the mapping u
→
J
′
(
u
) is monotone and semicontinuous.Let us define the functional(41)
J
v
=
1
p
∑
i
=
1
2
∫
Ω
∂
v
∂
x
i
p
d
xon the space V
=
W
1
,
p
(
Ω
,
Γ
ε
)
. It is easy to check that J is convex and differentiable in Gato sense and J
′
(
u
)
=
A
(
u
)
. Hence, A is semicontinuous and monotone operator. Finally, taking into account Remark 9 we calculate that (42)
A
u
,
u
u
=
∇
u
L
p
p
u
W
1
,
p
=
C
u
W
1
,
p
p
-
1
⟶
+
∞
a
s
u
W
1
,
p
⟶
+
∞
.Therefore by Theorem 7 for any F
∈
L
q there exists a function U
ε satisfying A
(
U
ε
)
=
λ
|
U
ε
|
p
-
2
+
F
.Step 2.
Let us derive now estimate (8). Substitute φ with U
ε in (11). Then it follows that (43)
∫
Ω
∇
U
ε
x
p
d
x
=
λ
∫
Ω
U
ε
p
d
x
+
∫
Ω
F
U
ε
d
x
,
U
ε
W
1
,
p
p
=
λ
+
1
∫
Ω
U
ε
p
d
x
+
∫
Ω
F
U
ε
d
x
≤
λ
+
1
U
ε
L
p
p
+
F
L
q
U
ε
L
p
≤
C
U
ε
W
1
,
p
U
ε
L
p
p
-
1
+
F
L
p
.Hence, (44)
U
ε
W
1
,
p
p
-
1
≤
C
U
ε
L
p
p
-
1
+
F
L
q
.Assume now that estimate (8) does not hold, that is, that there exists a sequence {
ε
k
}
ε
k
→
0 when k
→
∞, F
k
∈
L
q
(
Ω
) and λ
k such that the inequality (45)
U
ε
k
W
1
,
p
>
k
F
k
qholds for the solutions to problem (7), where (46)
ε
=
ε
k
,
λ
=
λ
k
,
F
=
F
k
.
Due to Lemma10 we may assume without loss of generality that the sequence {
U
ε
} is normalized in L
p
(
Ω
), that is, that(47)
U
ε
k
p
=
1
.
Then, by using (44) and (45), we obtain that (48)
U
ε
k
W
1
,
p
≤
C
,
F
k
q
<
C
k
,when k
→
∞
.
Hence, there exists a subsequence{
k
′
} of indexes {
k
} and U
∗
∈
W
1
,
p
(
Ω
), λ
∗ such that (49)
λ
k
′
⟶
λ
∗
∈
K
,
U
ε
k
′
⇀
U
∗
in
W
1
,
p
Ω
when
k
′
⟶
+
∞
.By (47) and (49) we have that (50)
U
∗
≠
0
.Moreover, since the operator A is bounded, it follows that A
U
ε
k
′
W
-
1
,
q
≤
C. Therefore we conclude the existence of χ
∈
W
-
1
,
q such that A
(
U
ε
k
′
)
⇀
χ weakly in W
-
1
,
q
. Suppose that v is an arbitrary fixed function from C
0
∞
(
Ω
¯
,
Γ
ε
). Then v
∈
W
1
,
p
(
Ω
,
Γ
ε
) for all small ε. Substitute in the integral identity (11) v as a test function; U
=
U
ε
k
′, λ
=
λ
k
′, and F
=
F
k
′, when ε
=
ε
k
′. We have that (51)
∫
Ω
∑
j
=
1
n
∇
U
ε
k
′
x
p
-
2
∂
U
ε
k
′
x
∂
x
j
∂
v
x
∂
x
j
d
x
=
λ
k
′
∫
Ω
U
ε
k
′
p
-
2
U
ε
k
′
v
d
x
+
∫
Ω
F
k
′
v
d
x
.
Now we want to pass to the limit asε
k
′
→
0
. Due to the definition of A
(
·
), we see that (52)
χ
,
v
=
lim
ε
k
′
→
0
⁡
λ
k
′
U
ε
k
′
p
-
2
U
ε
k
′
+
F
k
′
,
v
.From the other hand, (53)
A
U
ε
k
′
,
U
ε
k
′
⟶
χ
,
U
∗
,
U
ε
k
′
p
-
2
U
ε
k
′
,
U
ε
k
′
⟶
U
∗
p
-
2
U
∗
,
U
∗
.Let us show now that χ
=
A
(
U
∗
)
. The monotonicity of operator A implies (54)
A
U
ε
k
′
-
A
U
,
U
ε
k
′
-
U
≥
0
∀
U
∈
W
1
,
p
Ω
,
Γ
4
.Passing to the limit, it yields that (55)
χ
-
A
U
,
U
∗
-
U
≥
0
∀
U
∈
W
1
,
p
Ω
.If we take U
=
U
∗
-
s
W, s
>
0, W
∈
W
1
,
p
(
Ω
,
Γ
4
), we get (56)
s
χ
-
A
U
∗
-
s
W
,
W
≥
0
⟹
χ
-
A
U
∗
-
s
W
,
W
≥
0
.Let s
→
0
. Then (57)
χ
-
A
U
∗
-
s
W
,
W
≥
0
∀
U
∈
W
1
,
p
Ω
,
Γ
4
.This exactly means that χ
=
A
(
U
∗
)
. Thus, passing to the limit in the integral identity (51) as ε
k
′
→
0, using (48), (49), and Lemma 6, we obtain that (58)
∫
Ω
∇
U
∗
x
p
-
2
∇
U
∗
x
∇
v
x
d
x
=
λ
∗
∫
Ω
U
∗
p
-
2
U
∗
v
d
x
for
v
∈
C
∞
Ω
,
Γ
4
.From the density of the embedding C
∞
(
Ω
¯
,
Γ
ε
) into W
1
,
p
(
Ω
,
Γ
4
) we can conclude that this inequality holds also for v
∈
W
1
,
p
(
Ω
,
Γ
4
)
. It follows from Lemma 6 that U
∗
∈
W
1
,
p
(
Ω
,
Γ
4
)
. Due to the fact that U
∗
≠
0 and v is an arbitrary function from W
1
,
p
(
Ω
,
Γ
4
), it follows that λ
∗
∈
K is the eigenvalue of the limit problem (5). But we assumed that K did not contain the eigenvalues of the limit problem (5). This contradiction proves estimate (8).Step 3.
Let us prove now the uniqueness of the solution of (7). Let U
ε
1 and U
ε
2 be two different solutions. Then (59)
U
ε
1
-
U
ε
1
=
0
on
⁡
Γ
ε
,
∇
U
ε
1
p
-
2
∂
U
ε
1
∂
ν
-
∇
U
ε
2
p
-
2
∂
U
ε
2
∂
ν
=
0
on
⁡
∂
Ω
.This together with equation (60)
A
U
ε
1
-
A
U
ε
2
,
U
ε
1
-
U
ε
2
=
0
=
∑
i
∫
Ω
∂
U
ε
1
∂
x
i
p
-
2
∂
U
ε
1
∂
x
i
-
∂
U
ε
2
∂
x
i
p
-
2
∂
U
ε
2
∂
x
i
∂
U
ε
1
∂
x
i
-
∂
U
ε
2
∂
x
i
d
ximplies that (61)
∂
U
ε
1
∂
x
i
=
∂
U
ε
2
∂
x
i
⟹
U
ε
1
=
U
ε
2due to the boundary conditions. The uniqueness of the solution to (10) can be proved identically.


### 4.2. Proof of Part 2

Letλ
∈
K be an arbitrary fixed number and assume that the sequence {
ε
k
}
→
0 when k
→
+
∞
. Thinking analogously as in Part 1 of the proof, we have existence of a subsequence {
k
′
} and a function U
∗ such that U
ε
k
′
⇀
U
∗ in W
1
,
p
(
Ω
,
Γ
4
) (and strongly in L
p
(
Ω
)) when k
′
→
+
∞
. In addition, A
(
U
ε
k
′
)
⇀
χ weakly in W
-
1
,
q
. It can be shown exactly in the same way that χ
=
A
(
U
∗
)
.Now by using Lemma6 we obtain that the limit function U
∗
∈
W
1
,
p
(
Ω
,
Γ
4
)
. Passing to the limit in the integral identity (51) by means of the same reasoning as in the proof of Part 1 of the theorem, we get that (62)
∫
Ω
∇
U
∗
x
p
-
2
∇
U
∗
x
∇
v
x
d
x
=
λ
∫
U
∗
p
-
2
U
∗
v
d
x
+
∫
Ω
F
v
d
x
,which coincides with the integral identity of problem (10). Since the solution to problem (10) is unique we conclude that U
∗
=
U
0
. In addition, from (49) we find that (63)
U
ε
k
⟶
U
0
strongly  in
L
p
Ω
and  weakly  in
W
1
,
p
Ω
,
A
U
ε
k
⇀
A
U
0
in
W
-
1
,
q
when
ε
⟶
0
.From the integral identities we have that(64)
∫
Ω
∇
U
ε
k
p
-
2
∇
U
ε
k
-
∇
U
0
p
-
2
∇
U
0
·
∇
U
ε
k
-
U
0
d
x
=
λ
∫
Ω
U
ε
k
p
-
2
U
ε
k
U
0
d
x
-
∫
Ω
∇
U
0
p
-
2
∇
U
0
∇
U
ε
k
-
∇
U
0
d
x
+
∫
Ω
F
U
ε
k
-
U
0
d
x
.Due to convergences (63) the right-hand side tends to zero in the limit; therefore(65)
lim
k
→
∞
⁡
∫
Ω
∇
U
ε
k
p
-
2
∇
U
ε
k
-
∇
U
0
p
-
2
∇
U
0
·
∇
U
ε
k
-
U
0
d
x
=
0
.By using the inequality (66)
2
1
-
p
a
-
b
p
≤
a
p
-
2
a
-
b
p
-
2
b
·
a
-
b
,
p
≥
2
,with a
=
∇
U
ε
k and b
=
∇
U
0 we can show the strong convergence of gradients: (67)
lim
k
→
∞
⁡
∫
Ω
∇
U
ε
k
-
∇
U
0
p
d
x
=
0
.Thus, we have proved that, up to a subsequence, (68)
U
ε
-
U
0
W
1
,
p
⟶
0
.The proof is complete.


## 4.1. Proof of Part 1

Step 1.
The existence of the solution to problem (7) can be proved with help of Theorem 7. Indeed, we take V
=
W
1
,
p
(
Ω
,
Γ
ε
)
. Introduce the operator A
(
u
)
=
-
‍
∑
i
∂
/
∂
x
i
∂
u
/
∂
x
i
p
-
2
∂
u
/
∂
x
i
. For u
,
v
∈
W
1
,
p
(
Ω
,
Γ
ε
) we define (37)
A
u
,
v
=
-
∫
Ω
∑
i
∂
∂
x
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
v
d
x
=
∫
Ω
∑
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
∂
v
∂
x
i
d
x
.Moreover, for u
∈
W
1
,
p
(
Ω
,
Γ
ε
) it holds that λ
|
u
|
p
-
1
u
∈
L
q
(
Ω
,
Γ
ε
), and for F
∈
L
q
(
Ω
) we can introduce the functional on W
1
,
p
(
Ω
,
Γ
ε
): (38)
λ
u
p
-
1
u
+
F
,
v
=
∫
Ω
λ
u
p
-
1
u
v
d
x
+
∫
Ω
F
v
d
x
.It is clear that U
ε is the solution to (7) iff (
A
(
U
ε
)
,
v
)
=
(
λ
|
U
ε
|
p
-
1
U
ε
+
F
,
v
) for any v
∈
W
1
,
p
(
Ω
,
Γ
ε
)
. Let us verify the properties of operator A
. We take u
∈
W
1
,
p with u
W
1
,
p
=
1
. Then (39)
A
u
W
-
1
,
q
=
A
u
,
u
=
∇
u
L
p
p
≤
C
u
W
1
,
p
p
≤
C
.One can check the semicontinuity and monotonicity of operator A either directly or by using the following general result (see [39]).Proposition 11.
If convex functionalv
→
J
(
v
) is differentiable in Gato sense, that is, there exists such continuous linear mapping v
→
J
′
(
u
)
·
v of space V onto R such that (40)
lim
s
→
0
⁡
1
s
J
u
+
s
v
-
J
u
=
J
′
·
v
,then the mapping u
→
J
′
(
u
) is monotone and semicontinuous.Let us define the functional(41)
J
v
=
1
p
∑
i
=
1
2
∫
Ω
∂
v
∂
x
i
p
d
xon the space V
=
W
1
,
p
(
Ω
,
Γ
ε
)
. It is easy to check that J is convex and differentiable in Gato sense and J
′
(
u
)
=
A
(
u
)
. Hence, A is semicontinuous and monotone operator. Finally, taking into account Remark 9 we calculate that (42)
A
u
,
u
u
=
∇
u
L
p
p
u
W
1
,
p
=
C
u
W
1
,
p
p
-
1
⟶
+
∞
a
s
u
W
1
,
p
⟶
+
∞
.Therefore by Theorem 7 for any F
∈
L
q there exists a function U
ε satisfying A
(
U
ε
)
=
λ
|
U
ε
|
p
-
2
+
F
.Step 2.
Let us derive now estimate (8). Substitute φ with U
ε in (11). Then it follows that (43)
∫
Ω
∇
U
ε
x
p
d
x
=
λ
∫
Ω
U
ε
p
d
x
+
∫
Ω
F
U
ε
d
x
,
U
ε
W
1
,
p
p
=
λ
+
1
∫
Ω
U
ε
p
d
x
+
∫
Ω
F
U
ε
d
x
≤
λ
+
1
U
ε
L
p
p
+
F
L
q
U
ε
L
p
≤
C
U
ε
W
1
,
p
U
ε
L
p
p
-
1
+
F
L
p
.Hence, (44)
U
ε
W
1
,
p
p
-
1
≤
C
U
ε
L
p
p
-
1
+
F
L
q
.Assume now that estimate (8) does not hold, that is, that there exists a sequence {
ε
k
}
ε
k
→
0 when k
→
∞, F
k
∈
L
q
(
Ω
) and λ
k such that the inequality (45)
U
ε
k
W
1
,
p
>
k
F
k
qholds for the solutions to problem (7), where (46)
ε
=
ε
k
,
λ
=
λ
k
,
F
=
F
k
.
Due to Lemma10 we may assume without loss of generality that the sequence {
U
ε
} is normalized in L
p
(
Ω
), that is, that(47)
U
ε
k
p
=
1
.
Then, by using (44) and (45), we obtain that (48)
U
ε
k
W
1
,
p
≤
C
,
F
k
q
<
C
k
,when k
→
∞
.
Hence, there exists a subsequence{
k
′
} of indexes {
k
} and U
∗
∈
W
1
,
p
(
Ω
), λ
∗ such that (49)
λ
k
′
⟶
λ
∗
∈
K
,
U
ε
k
′
⇀
U
∗
in
W
1
,
p
Ω
when
k
′
⟶
+
∞
.By (47) and (49) we have that (50)
U
∗
≠
0
.Moreover, since the operator A is bounded, it follows that A
U
ε
k
′
W
-
1
,
q
≤
C. Therefore we conclude the existence of χ
∈
W
-
1
,
q such that A
(
U
ε
k
′
)
⇀
χ weakly in W
-
1
,
q
. Suppose that v is an arbitrary fixed function from C
0
∞
(
Ω
¯
,
Γ
ε
). Then v
∈
W
1
,
p
(
Ω
,
Γ
ε
) for all small ε. Substitute in the integral identity (11) v as a test function; U
=
U
ε
k
′, λ
=
λ
k
′, and F
=
F
k
′, when ε
=
ε
k
′. We have that (51)
∫
Ω
∑
j
=
1
n
∇
U
ε
k
′
x
p
-
2
∂
U
ε
k
′
x
∂
x
j
∂
v
x
∂
x
j
d
x
=
λ
k
′
∫
Ω
U
ε
k
′
p
-
2
U
ε
k
′
v
d
x
+
∫
Ω
F
k
′
v
d
x
.
Now we want to pass to the limit asε
k
′
→
0
. Due to the definition of A
(
·
), we see that (52)
χ
,
v
=
lim
ε
k
′
→
0
⁡
λ
k
′
U
ε
k
′
p
-
2
U
ε
k
′
+
F
k
′
,
v
.From the other hand, (53)
A
U
ε
k
′
,
U
ε
k
′
⟶
χ
,
U
∗
,
U
ε
k
′
p
-
2
U
ε
k
′
,
U
ε
k
′
⟶
U
∗
p
-
2
U
∗
,
U
∗
.Let us show now that χ
=
A
(
U
∗
)
. The monotonicity of operator A implies (54)
A
U
ε
k
′
-
A
U
,
U
ε
k
′
-
U
≥
0
∀
U
∈
W
1
,
p
Ω
,
Γ
4
.Passing to the limit, it yields that (55)
χ
-
A
U
,
U
∗
-
U
≥
0
∀
U
∈
W
1
,
p
Ω
.If we take U
=
U
∗
-
s
W, s
>
0, W
∈
W
1
,
p
(
Ω
,
Γ
4
), we get (56)
s
χ
-
A
U
∗
-
s
W
,
W
≥
0
⟹
χ
-
A
U
∗
-
s
W
,
W
≥
0
.Let s
→
0
. Then (57)
χ
-
A
U
∗
-
s
W
,
W
≥
0
∀
U
∈
W
1
,
p
Ω
,
Γ
4
.This exactly means that χ
=
A
(
U
∗
)
. Thus, passing to the limit in the integral identity (51) as ε
k
′
→
0, using (48), (49), and Lemma 6, we obtain that (58)
∫
Ω
∇
U
∗
x
p
-
2
∇
U
∗
x
∇
v
x
d
x
=
λ
∗
∫
Ω
U
∗
p
-
2
U
∗
v
d
x
for
v
∈
C
∞
Ω
,
Γ
4
.From the density of the embedding C
∞
(
Ω
¯
,
Γ
ε
) into W
1
,
p
(
Ω
,
Γ
4
) we can conclude that this inequality holds also for v
∈
W
1
,
p
(
Ω
,
Γ
4
)
. It follows from Lemma 6 that U
∗
∈
W
1
,
p
(
Ω
,
Γ
4
)
. Due to the fact that U
∗
≠
0 and v is an arbitrary function from W
1
,
p
(
Ω
,
Γ
4
), it follows that λ
∗
∈
K is the eigenvalue of the limit problem (5). But we assumed that K did not contain the eigenvalues of the limit problem (5). This contradiction proves estimate (8).Step 3.
Let us prove now the uniqueness of the solution of (7). Let U
ε
1 and U
ε
2 be two different solutions. Then (59)
U
ε
1
-
U
ε
1
=
0
on
⁡
Γ
ε
,
∇
U
ε
1
p
-
2
∂
U
ε
1
∂
ν
-
∇
U
ε
2
p
-
2
∂
U
ε
2
∂
ν
=
0
on
⁡
∂
Ω
.This together with equation (60)
A
U
ε
1
-
A
U
ε
2
,
U
ε
1
-
U
ε
2
=
0
=
∑
i
∫
Ω
∂
U
ε
1
∂
x
i
p
-
2
∂
U
ε
1
∂
x
i
-
∂
U
ε
2
∂
x
i
p
-
2
∂
U
ε
2
∂
x
i
∂
U
ε
1
∂
x
i
-
∂
U
ε
2
∂
x
i
d
ximplies that (61)
∂
U
ε
1
∂
x
i
=
∂
U
ε
2
∂
x
i
⟹
U
ε
1
=
U
ε
2due to the boundary conditions. The uniqueness of the solution to (10) can be proved identically.


## 4.2. Proof of Part 2

Letλ
∈
K be an arbitrary fixed number and assume that the sequence {
ε
k
}
→
0 when k
→
+
∞
. Thinking analogously as in Part 1 of the proof, we have existence of a subsequence {
k
′
} and a function U
∗ such that U
ε
k
′
⇀
U
∗ in W
1
,
p
(
Ω
,
Γ
4
) (and strongly in L
p
(
Ω
)) when k
′
→
+
∞
. In addition, A
(
U
ε
k
′
)
⇀
χ weakly in W
-
1
,
q
. It can be shown exactly in the same way that χ
=
A
(
U
∗
)
.Now by using Lemma6 we obtain that the limit function U
∗
∈
W
1
,
p
(
Ω
,
Γ
4
)
. Passing to the limit in the integral identity (51) by means of the same reasoning as in the proof of Part 1 of the theorem, we get that (62)
∫
Ω
∇
U
∗
x
p
-
2
∇
U
∗
x
∇
v
x
d
x
=
λ
∫
U
∗
p
-
2
U
∗
v
d
x
+
∫
Ω
F
v
d
x
,which coincides with the integral identity of problem (10). Since the solution to problem (10) is unique we conclude that U
∗
=
U
0
. In addition, from (49) we find that (63)
U
ε
k
⟶
U
0
strongly  in
L
p
Ω
and  weakly  in
W
1
,
p
Ω
,
A
U
ε
k
⇀
A
U
0
in
W
-
1
,
q
when
ε
⟶
0
.From the integral identities we have that(64)
∫
Ω
∇
U
ε
k
p
-
2
∇
U
ε
k
-
∇
U
0
p
-
2
∇
U
0
·
∇
U
ε
k
-
U
0
d
x
=
λ
∫
Ω
U
ε
k
p
-
2
U
ε
k
U
0
d
x
-
∫
Ω
∇
U
0
p
-
2
∇
U
0
∇
U
ε
k
-
∇
U
0
d
x
+
∫
Ω
F
U
ε
k
-
U
0
d
x
.Due to convergences (63) the right-hand side tends to zero in the limit; therefore(65)
lim
k
→
∞
⁡
∫
Ω
∇
U
ε
k
p
-
2
∇
U
ε
k
-
∇
U
0
p
-
2
∇
U
0
·
∇
U
ε
k
-
U
0
d
x
=
0
.By using the inequality (66)
2
1
-
p
a
-
b
p
≤
a
p
-
2
a
-
b
p
-
2
b
·
a
-
b
,
p
≥
2
,with a
=
∇
U
ε
k and b
=
∇
U
0 we can show the strong convergence of gradients: (67)
lim
k
→
∞
⁡
∫
Ω
∇
U
ε
k
-
∇
U
0
p
d
x
=
0
.Thus, we have proved that, up to a subsequence, (68)
U
ε
-
U
0
W
1
,
p
⟶
0
.The proof is complete.


## 5. Proof of Theorem5

Define byΣ the spectrum of problem (69)
-
Δ
p
u
x
=
λ
u
p
-
2
u
in
Ω
,
u
=
0
on
⁡
γ
,
where
γ
⊂
∂
Ω
,
∂
u
∂
p
ν
≡
∇
u
p
-
2
∇
u
,
ν
=
0
o
n
∂
Ω
∖
γ
¯
.That is,Σ
=
{
λ
∈
R
:
t
h
e
p
r
o
b
l
e
m (69) h
a
s
t
h
e
n
o
n
t
r
i
v
i
a
l
w
e
a
k
s
o
l
u
t
i
o
n
}
.Theorem 12.
The spectrumΣ is nonempty closed set, Σ
⊂
(
0
,
∞
)
.Proof.
To show that the spectrum is nonempty, one needs to prove the existence of weak solution to problem (69). We can make use of Theorem 7. We take V
=
W
1
,
p
(
Ω
,
γ
) and A
(
u
)
=
-
∑
i
∂
/
∂
x
i
∂
u
/
∂
x
i
p
-
2
∂
u
/
∂
x
i
. For u
,
v
∈
W
1
,
p
(
Ω
,
γ
) we define (70)
A
u
,
v
=
-
∫
Ω
∑
i
∂
∂
x
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
v
d
x
=
∫
Ω
∑
i
∂
u
∂
x
i
p
-
2
∂
u
∂
x
i
∂
v
∂
x
i
d
x
.One needs to verify the properties of operator A
. First, the boundedness is fulfilled since (71)
A
u
,
u
≤
C
u
W
1
,
p
p
.To check the semicontinuity, monotonicity of the operator, and the property (
A
(
u
)
,
u
)
/
u
→
+
∞
a
s
u
W
1
,
p
→
+
∞ one can in the same way as in the proof of Theorem 4. Let us omit the details.
As soon as we conclude that the conditions of Theorem7 are fulfilled, we conclude the existence of the solution. Thus, the spectrum is nonempty. Check now the positiveness of Σ
. Let λ
∈
Σ and the corresponding eigenfunction u
∈
W
1
,
p
(
Ω
,
γ
)
. It yields that (72)
∫
Ω
∇
u
p
d
x
=
λ
∫
Ω
u
p
d
x
,
therefore
λ
=
∫
Ω
∇
u
p
d
x
∫
Ω
u
p
d
x
>
0
.Let λ
k
∈
Σ correspond to eigenfunctions u
k and λ
k
→
λ
. Without loss of generality one may assume that u
k
L
p
=
1
. Since {
λ
k
} is bounded, it follows from (73)
∫
Ω
∇
u
k
p
d
x
=
λ
k
∫
Ω
u
k
p
d
xthat u
k
W
1
,
p
(
Ω
,
γ
)
≤
C which implies the existence of u
∈
W
1
,
p
(
Ω
,
γ
) such that up to a subsequence(74)
u
k
⇀
u
weakly  in
W
1
,
p
Ω
,
γ
,
u
k
⟶
u
strongly  in
L
p
Ω
.Moreover, (75)
A
u
k
=
A
u
k
,
u
k
≤
C
u
k
L
p
p
-
1
,and therefore there exists χ such that (76)
A
u
k
⇀
χ
weakly  in
W
-
1
,
q
Ω
,
γ
.These convergences imply that u
L
p
=
1 and (77)
∫
Ω
χ
∇
v
d
x
=
λ
∫
Ω
u
p
-
2
u
v
d
x
for
v
∈
W
1
,
p
Ω
,
γ
.Thus it remains to show that (78)
∫
Ω
χ
∇
v
d
x
=
∫
Ω
∇
u
p
-
2
∇
u
∇
v
d
x
.For every v
∈
W
1
,
p
(
Ω
,
γ
) it holds that (79)
0
≤
∫
Ω
∇
u
k
p
-
2
∇
u
k
-
∇
v
p
-
2
∇
v
,
∇
u
k
-
∇
v
d
x
=
∫
Ω
λ
k
u
k
p
-
2
u
k
u
k
-
v
d
x
-
∫
Ω
∇
v
p
-
2
∇
v
∇
u
k
-
∇
v
d
x
.Passing to the limit as k
→
∞, one obtains that (80)
0
≤
∫
Ω
λ
u
p
-
2
u
u
-
v
d
x
-
∫
Ω
∇
v
p
-
2
∇
v
∇
u
-
∇
v
d
x
=
∫
Ω
χ
∇
u
-
∇
v
d
x
-
∫
Ω
∇
v
p
-
2
∇
v
∇
u
-
∇
v
d
x
.Take now v
=
u
-
t
w, w
∈
W
1
,
p
(
Ω
,
γ
), t
∈
R
+
. Then (81)
0
≤
∫
Ω
χ
-
∇
u
-
t
w
p
-
2
∇
u
-
t
w
∇
w
d
x
.Passing to the limit as t
→
+
0, we get (82)
0
≤
∫
Ω
χ
-
∇
u
p
-
2
∇
u
∇
w
d
x
.From this inequality we deduce the validity of (78).Let us mention the properties of the first eigenvalue to spectral problem (69). The following theorem directly follows from the results in [34].Theorem 13.
The smallest eigenvalue(83)
λ
1
≔
inf
u
∈
W
1
,
p
Ω
,
γ
∖
0
⁡
∫
Ω
∇
u
p
d
x
∫
Ω
u
p
d
xis simple and isolated. Moreover, if u
1
,
u
2 are two eigenfunctions corresponding to λ
1, then they are proportional: u
1
=
k
u
2
.Now we come back to our spectral problems (3) and (5). The existence of normalized solution and properties of spectrum follows by Theorems 12 and 13. The uniqueness of the normalized solution is a direct consequence of Theorem 13. Concerning the regularity, the known fact (see [40]) says that if eigenfunctions u
∈
W
0
1
,
p, then u
∈
L
∞
. The statements of Theorem 5 can be proved similarly to the proof of statement (2) of Theorem 4.Indeed, sinceλ
ε
1 is isolated, that is, ∃
δ
>
0
:
|
λ
ε
1
-
λ
ε
2
|
>
δ (λ
ε
2 is the second eigenvalue to (3)), there exists a compact containing λ
ε
1
. Hence, there exists a subsequence {
ε
k
}
k
=
1
∞
→
0 such that λ
ε
k
1
→
λ
∗
. The boundedness of u
ε
k in L
p implies the existence of limit u
∗ (strong in L
p and weak in W
1
,
p). The boundedness of operator A
(
u
ε
k
) in the dual space W
-
1
,
q gives the existence of weak limit (84)
χ
=
lim
k
→
∞
⁡
A
u
ε
k
=
lim
k
→
∞
⁡
∂
∂
x
j
∑
j
=
1
n
∇
u
ε
k
x
p
-
2
∂
u
ε
k
x
∂
x
j
.Thus, passing to the limit in the identity (85)
∫
Ω
∑
j
=
1
n
∇
u
ε
k
x
p
-
2
∂
u
ε
k
x
∂
x
j
∂
v
x
∂
x
j
d
x
=
λ
ε
k
1
∫
Ω
u
ε
k
p
-
2
u
ε
k
v
d
x
,we obtain (86)
∫
Ω
∑
j
=
1
n
∇
u
∗
x
p
-
2
∂
u
∗
x
∂
x
j
∂
v
x
∂
x
j
d
x
=
λ
∗
∫
Ω
u
∗
p
-
2
u
∗
v
d
x
.Having in mind the same arguments as in the proof of Theorem 4 one can show that λ
∗
=
λ
0
1
,
u
∗
=
u
0, and χ
=
A
(
u
0
)
=
-
∑
i
∂
/
∂
x
i
∂
u
0
/
∂
x
i
p
-
2
∂
u
0
/
∂
x
i which means that problem (5) is the limit one for (3). Analogously as in Theorem 4 one can derive that (87)
u
ε
k
-
u
0
W
1
,
p
⟶
0
.Thus, Theorem 5 is completely proved.The question on convergence of the full spectrum for similarp-Laplace boundary-value problem is studied, for example, in [33, 35]. We do not cover this studying for our problem since in the applications of the results of the present paper we will use only the fact about converging of the first eigenelements.


## 6. The Estimate of the Solution in a Neighborhood of the Eigenvalue

Let us derive an estimate for the solution to problem (7) in a neighborhood of λ
0
1
.Lemma 14.
Letλ be close to λ
0
1 and λ
ε
1 converges to λ
0
1
. Then the following estimate holds: (88)
U
ε
W
1
,
p
p
≤
C
F
L
q
2
λ
ε
1
-
λ
.Proof.
Since the functionU
ε
∈
W
1
,
p
(
Ω
ε
,
Γ
ε
), it satisfies Friedrichs inequality U
ε
L
p
p
≤
K
∇
U
ε
L
p
p
. Moreover, it is clear that the best constant in the inequality is 1
/
λ
ε
1, where λ
ε
1 is the first eigenvalue to (3). Thus, (89)
U
ε
L
p
p
≤
1
λ
ε
1
∇
U
ε
L
p
p
.By using estimates (43), (8), (89), integral identity with v
=
U
ε, and equivalent norm in W
1
,
p, it yields that (90)
∇
U
ε
L
p
p
≤
λ
U
ε
L
p
p
+
F
L
p
U
ε
L
p
≤
λ
λ
ε
1
∇
U
ε
L
p
p
+
F
L
q
2
⟹
λ
ε
1
-
λ
∇
U
ε
L
p
p
≤
C
F
L
q
2
,
w
h
e
r
e
λ
ε
1
<
C
=
const
.Replacing ∇
U
ε
L
p
p with the equivalent norm U
ε
W
1
,
p
p, we obtain (88).



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*Source: 392479-2015-09-30.xml*