# Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

Authors: Mohammed El Mokhtar Ould El Mokhtar
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.


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## Body

## 1. Introduction

In this paper, we study the existence of nontrivial solutions of the following problem:P
λ
,
μ
-
Δ
u
-
μ
y
-
2
u
=
y
-
a
γ
u
γ
-
2
u
1
+
λ
g
x
in
R
N
,
y
≠
0
,
u
>
0
,where y
∈
R
k, and let k and N be integers such that N
≥
3 and k belongs to 1
,
…
,
N. 2
∗
=
2
N
/
N
-
2 is the critical Sobolev exponent, γ
≤
2
∗, 0
≤
a
<
1, g is a continuous function on R
N, and λ and μ are parameters which we will specify later.We denote pointx in R
N by the pair y
,
z
∈
R
k
×
R
N
-
k, D
0
1,2
=
D
0
1,2
R
k
∖
0
×
R
N
-
k, and H
μ
=
H
μ
R
k
∖
0
×
R
N
-
k, the closure of C
0
∞
R
k
∖
0
×
R
N
-
k with respect to the norms (1)
u
=
∫
R
N
∇
u
2
1
/
2
,
u
μ
=
∫
R
N
∇
u
2
-
μ
y
-
2
u
2
d
x
1
/
2with μ
<
μ
¯
k
=
k
-
2
/
2
2 for k
≠
2.From the Hardy inequality, it is easy to see that the normu
μ is equivalent to u.We define the weighted Sobolev spaceD
≔
H
μ
∩
L
γ
R
N
,
y
-
b
d
x
∩
L
2
R
N
,
y
-
2
d
x with b
=
a
γ, which is a Banach space with respect to the norm defined by N
(
u
)
≔
u
μ
+
(
∫
R
N
‍
y
-
b
u
γ
d
x
)
1
/
γ.My motivation of this study is the fact that such equations arise in the search for solitary waves of nonlinear evolution equations of the Schrödinger or Klein-Gordon type (cf. [1–3]). Roughly speaking, a solitary wave is a nonsingular solution which travels as a localized packet in such a way that the physical quantities corresponding to the invariances of the equation are finite and conserved in time. Accordingly, a solitary wave preserves intrinsic properties of particles such as the energy, the angular momentum, and the charge, whose finiteness is strictly related to the finiteness of the L
2-norm. Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, and plasma physics (see, e.g., [4]).Several existence and nonexistence results are available in the casek
=
N, and we quote, for example, [5–7] and the references therein. When μ
=
0, g
x
≡
1; problem P
λ
,
μ has been studied in the famous papers by Brézis and Nirenberg [8] and Xuan [9] which consider the existence and nonexistence of nontrivial solutions to quasilinear Brézis-Nirenberg-type problems with singular weights.Concerning the existence result in the casek
<
N, we cite [10, 11] and the references therein. As noticed in [10], for μ
<
0 and a
=
0, Badiale and Rolando have considered the problem P
0
,
μ. They established the existence of nontrivial nonnegative radial solution when β
∈
0,2 and γ
∈
2
β
,
2
∗ or β
∈
2
,
+
∞ and γ
∈
2
∗
,
2
β; in addition, if the function f
u
=
u
γ
-
1
u is odd, then P
0
,
μ has infinitely many radial solutions. In [5], Badiale et al. proved the nonexistence of nonzero classical solutions when k
≤
N and the pair β
,
γ belongs to the light gray region. That is, β
,
γ
∈
A
=
A
1
∪
A
2
∪
A
3, where (2)
A
1
≔
β
,
γ
∈
R
2
:
β
∈
0,2
,
γ
∉
2
β
,
2
∗
,
γ
≥
2
∖
2
,
2
∗
,
A
2
≔
β
,
γ
∈
R
2
:
β
∈
2
,
N
,
γ
∉
2
∗
,
2
β
,
γ
≥
2
,
A
3
≔
β
,
γ
∈
R
2
:
β
∈
N
,
+
∞
,
γ
∈
2
,
2
∗
.Since our approach is variational, we define the functionalI
λ
,
μ on D by (3)
I
λ
,
μ
u
≔
1
2
u
μ
2
-
1
γ
∫
R
N
y
-
b
u
γ
1
+
λ
g
x
d
x
.We say that u
∈
D is a weak solution of the problem P
λ
,
μ if it is a nontrivial nonnegative function and satisfies (4)
I
λ
,
μ
′
u
,
v
≔
∫
R
N
∇
u
∇
v
-
μ
y
-
2
u
v
-
y
-
b
u
γ
-
2
u
v
1
+
λ
g
x
=
0
,
for
v
∈
D
.Throughout this work, we consider the following regions R
1, R
2, such that (5)
R
1
≔
2
,
γ
∈
R
2
:
γ
∈
2
2
-
2
a
,
2
∗
,
R
2
≔
2
,
γ
∈
R
2
:
γ
∈
2
,
2
2
-
2
awith 2
2
-
2
a
=
2
N
/
N
-
2
-
2
a.Concerning the perturbationg, we assume G
g
∈
L
∞
R
N
,
g
x
>
0
∀
x
∈
R
N
.In our work, we prove the existence of at least one critical point of I
λ
,
μ by Ekeland’s variational principle in [12].We will state our main result.Theorem 1.
Assume that2
<
k
≤
N, μ
<
μ
¯
k, 0
<
a
<
1, and G hold.
If2
,
γ
∈
R
1
∪
R
2, then there exists Λ
∗
>
0 such that the problem P
λ
,
μ has at least one nontrivial solution for any λ
>
Λ
∗.This paper is organized as follows. In Section2, we give some preliminaries. Section 3 is devoted to the proof of Theorem 1.


## 2. Preliminaries

We list here a few integrals inequalities. The first inequality that we need is the weighted Hardy inequality [13] (6)
μ
¯
k
∫
R
N
y
-
2
v
2
d
x
≤
∫
R
N
∇
v
2
d
x
,
∀
v
∈
H
μ
.The starting point for studying P
λ
,
μ is the Hardy-Sobolev-Maz’ya inequality that is peculiar to the cylindrical case k
<
N and that was proved by Gazzini and Musina in [14]. It states that there exists positive constant C
γ such that (7)
C
γ
∫
R
N
v
γ
d
x
2
/
γ
≤
∫
R
N
∇
v
2
-
μ
y
-
2
v
2
d
x
,for μ
=
0; equation of P
λ
,
μ is related to a family of inequalities given by Caffarelli et al. [15], for any v
∈
C
c
∞
(
R
k
∖
0
×
R
N
-
k
). The embedding H
μ
↪
L
γ
R
N
,
y
-
b
d
x is compact, where b
=
a
γ and L
γ
R
N
,
y
-
b
d
x is the weighted L
γ space with respect to the norm (8)
u
γ
,
b
2
=
∫
R
N
y
-
b
v
γ
d
x
2
/
γ
.Definition 2.
Assume2
≤
k
<
N, 0
<
μ
≤
μ
¯
k, and 2
<
γ
<
2
∗. Then, the infimum S
μ
,
γ defined by (9)
S
μ
,
γ
=
S
μ
,
γ
k
,
γ
≔
i
n
f
v
∈
D
∖
0
∫
R
N
∇
v
2
-
μ
y
-
2
v
2
d
x
∫
R
N
y
-
b
v
γ
d
x
2
/
γis achieved on H
μ.Lemma 3.
Letu
n
⊂
D be a Palais-Smale sequence (P
S
δ for short) of I
λ
,
μ such that(10)
I
λ
,
μ
u
n
⟶
δ
,
I
β
,
λ
,
μ
′
u
n
⟶
0
i
n
D
′
d
u
a
l
o
f
D
a
s
n
⟶
∞
,for some δ
∈
R. Then, u
n
⇀
u in D and I
β
,
λ
,
μ
′
u
=
0.Proof.
From (10), we have (11)
1
2
u
n
μ
2
-
1
γ
∫
R
N
y
-
b
u
n
γ
1
+
λ
g
x
d
x
=
δ
+
o
n
1
,
u
n
μ
2
-
∫
R
N
y
-
b
u
n
γ
1
+
λ
g
x
d
x
=
o
n
1
,
for
n
large
,where o
n
1 denotes o
n
1
→
0 as n
→
∞. Then, (12)
δ
+
o
n
1
=
I
λ
,
μ
u
n
-
1
γ
I
β
,
λ
,
μ
′
u
n
,
u
n
=
γ
-
2
2
γ
u
n
μ
2
,and u
n is bounded in D. Going if necessary to a subsequence, we can assume that there exists u
∈
D such that (13)
u
n
⇀
u
in
D
,
u
n
⟶
u
in
L
γ
R
N
,
y
-
b
d
x
,
u
n
⟶
u
a.e.  in
R
N
.Consequently, we get, for all v
∈
C
0
∞
R
k
∖
0
×
R
N
-
k, (14)
∫
R
N
∇
u
∇
v
-
μ
y
-
2
u
v
-
y
-
b
u
γ
-
2
u
v
1
+
λ
g
x
=
0
,which means that (15)
I
β
,
λ
,
μ
′
u
=
0
.


## 3. Existence Result

Firstly, we require the following lemmas.Lemma 4.
Letu
n
⊂
D be a P
S
δ sequence of I
λ
,
μ for some δ
∈
R. Then, (16)
u
n
⇀
u
i
n
Dand either(17)
u
n
⟶
u
o
r
δ
≥
I
λ
,
μ
u
+
γ
-
2
2
γ
S
μ
,
γ
γ
/
γ
-
2
.Proof.
We know thatu
n is bounded in D. Up to a subsequence if necessary, we have that (18)
u
n
⇀
u
in
D
u
n
⟶
u
a.e.  in
R
N
.Denote v
n
=
u
n
-
u, and then v
n
⇀
0. As in Brézis and Lieb [16], we have (19)
∫
R
N
y
-
b
u
n
γ
=
∫
R
N
y
-
b
v
n
γ
+
∫
R
N
y
-
b
u
γ
,
u
n
μ
2
=
v
n
μ
2
+
u
μ
2
.From Lebesgue theorem and by using the assumption G, we obtain (20)
lim
n
→
∞
⁡
∫
R
N
g
x
y
-
b
u
n
γ
d
x
=
lim
n
→
∞
⁡
∫
R
N
g
x
y
-
b
u
γ
d
x
.Then, we deduce that (21)
I
λ
,
μ
u
n
=
I
λ
,
μ
u
+
1
2
v
n
μ
2
-
1
γ
∫
R
N
y
-
b
v
n
γ
+
o
n
1
,
I
λ
,
μ
′
u
n
,
u
n
=
v
n
μ
2
-
∫
R
N
y
-
b
v
n
γ
+
o
n
1
.From the fact that v
n
⇀
0 in D, we can assume that (22)
lim
n
→
∞
⁡
v
n
μ
2
=
lim
n
→
∞
⁡
∫
R
N
y
-
b
v
n
γ
=
α
≥
0
.Assuming that α
>
0, we have by definition of S
μ
,
γ
(23)
α
≥
S
μ
,
γ
l
2
/
γ
,and so (24)
α
≥
S
μ
,
γ
γ
/
γ
-
2
.Then, we get (25)
δ
≥
I
λ
,
μ
u
+
γ
-
2
2
γ
S
μ
,
γ
γ
/
γ
-
2
.Therefore, if not, we obtain α
=
0. That is, u
n
→
u in D.Lemma 5.
Suppose that2
<
k
≤
N, μ
<
μ
¯
k, and G hold. If 2
,
γ
∈
R
1
∪
R
2, then there exist Λ
∗
>
0 and ϱ and ν positive constants such that, for all λ
>
Λ
∗,(i)
there existω
∈
R
N such that I
λ
,
μ
ω
<
0,
(ii)
we have(26)
I
λ
,
μ
u
≥
ν
>
0
f
o
r
u
μ
=
ϱ
0
.Proof.
(i) Lett
0
>
0 where t
0 is small, and ϕ
∈
C
0
∞
R
k
∖
0
×
R
N
-
k such that ϕ
≢
0. Choosing Λ
∗
=
t
0
ϕ
1
-
γ, then, if λ
>
Λ
∗ large enough, (27)
I
λ
,
μ
t
0
ϕ
≔
t
0
2
2
ϕ
μ
2
-
t
0
γ
γ
∫
R
N
y
-
b
ϕ
γ
1
-
t
0
γ
γ
∫
R
N
y
-
b
ϕ
γ
λ
g
x
<
t
0
2
2
ϕ
μ
2
-
t
0
γ
γ
∫
R
N
y
-
b
ϕ
γ
1
-
t
0
γ
∫
R
N
y
-
b
ϕ
g
x
<
0
.Thus, if ω
=
t
0
ϕ, we obtain that I
λ
,
μ
ω
<
0.
(ii) By the Holder inequality and the definition ofS
μ
,
γ and since γ
>
2, we get for all u
∈
D
∖
0
(28)
I
λ
,
μ
u
≔
1
2
u
μ
2
-
1
γ
∫
R
N
y
-
b
u
γ
1
+
λ
g
x
d
x
≥
1
2
u
μ
2
-
1
γ
S
μ
,
γ
u
μ
γ
1
+
λ
g
∞
.If λ
>
Λ
∗, then there exist ν
>
0 and ϱ
0
>
0 small enough such that (29)
I
λ
,
μ
u
≥
ν
>
0
for
u
μ
=
ϱ
0
.We also assume that t
0 is small enough such that t
0
ϕ
μ
<
ϱ
0. Thus, we have (30)
c
1
=
inf
⁡
I
λ
,
μ
u
:
u
∈
B
ϱ
0
<
0
,
where
B
ϱ
0
=
u
∈
D
,
N
u
≤
ϱ
0
.Using Ekeland’s variational principle, for the complete metric space B
¯
ρ
0 with respect to the norm of D, we can prove that there exists a P
C
c
1 sequence u
n
⊂
B
¯
ρ
0 such that u
n
⇀
u
1 for some u
1 with N
(
u
1
)
≤
ρ
0.
Now, we claim thatu
n
→
u
1. If not, by Lemma 4, we have (31)
c
1
≥
I
λ
,
μ
u
1
+
γ
-
2
2
γ
S
μ
,
γ
γ
/
γ
-
2
≥
c
1
+
γ
-
2
2
γ
S
μ
,
γ
γ
/
γ
-
2
>
c
1
,which is a contradiction.
Then, we obtain a critical pointu
1 of I
λ
,
μ for all λ
>
Λ
∗ large enough satisfying (32)
c
1
=
γ
-
2
2
γ
u
1
μ
2
>
0
.Proof of Theorem1.
From Lemmas4 and 5, we can deduce that there exists at least a nontrivial solution u
1 for our problem P
λ
,
μ with positive energy.



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*Source: 494907-2015-10-28.xml*