Áp dụng phương pháp giải tích nghiên cứu một số bài toán elliptic suy biến tt tiếng anh

MINISTRY OF EDUCATION AND TRAINING
HANOI PEDAGOGICAL UNIVERSITY 2

BUI KIM MY

ANALYTICAL METHODS FOR STUDYING SOME
DEGENERATE ELLIPTIC PROBLEMS

SUMMARY OF DOCTORAL THESIS IN
MATHEMATICS

Major: Mathematical Analysis
Code: 9 46 01 02

HA NOI, 2019


This dissertation has been written at Hanoi Pedagogical University 2

Supervisor: Assoc. Prof. Dr. Cung The Anh

Referee 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
......................................................
Referee 2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
......................................................
Referee 3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
......................................................
The thesis shall be defended at the University level Thesis Assessment Council at Hanoi Pedagogical University 2 on . . . . . . . . .

This thesis can be found in:
- The National Library of Vietnam;

- HPU2 Library Information Centre.


INTRODUCTION

1. Motivation and history of the problem
Many type of elliptic equations are associated with the study of the steady
of evolutionary processes in physics, chemistry, mechanics and biology and
many important classes of nonlinear elliptic equations are also started from
the problems of differential geometry (see, for instance, Ambrosetti and Malchiodi (2007), Evans (1998), Gilbarg and Trudinger (1998), Quittner and Souplet (2007), Willem (1986)). Therefore, studying these classes are meaningful
in science and technology. In addition, the study of elliptic equations motivates and provides ideas for the development of tools and the results of many
analytical fields, such as theory of functional spaces, nonlinear analysis, . . . .
Especially, the development of these issues lead to progress in the theory of
elliptic equations. Thus, the theory of elliptic equations has been attracting
the attention of many scientists in the world.
As mentioned above, the research on elliptic equations by using analytical
methods has been studing by many domestic and foreign mathematicians. In
recent decades, a lot of qualitative results for many classes of problems involving both nondegenerate elliptic operators and degenerate elliptic operators
are obtained (see, for example, Ambrosetti and Malchiodi (2007), Quitter and
Souplet (2007), Willem (1996), Figueiredo (1996) and Kogoj (2018)). Among
the degenerate operators, ∆λ -Laplace operator is an important class which is
following form
N

∂xi (λ2i (x)∂xi u),

∆λ u =
i=1

where λi are functions satisfying some suitable conditions. This operator is first

introduced by Franchi and Lanconelli in 1982, and it is recently reconsidered.
It was named as ∆λ -Laplacians by Kogoj and Lanconelli in 2012. Especially,
it contains many important elliptic operators such as Laplace operator ∆u =
N

uxi xi , Grushin operator Gs u = ∆x u + |x|2s ∆y u (see Grushin (1971)), and

i=1

strongly degenerate operator Pα,β u = ∆x u + |x|2α ∆y u + |y|2β ∆z u, . . . (see N.M.
Tri et. al. (2002, 2012)), etc.
1


We now recall some recent important results related to the existence and
qualitative properties of solutions to elliptic equations and systems which are
involving the content of my thesis.
• Semilinear elliptic equations.
In the last decades, the boundary value problem for semilinear elliptic
equation has form

−∆u = f (x, u), x ∈ Ω,
(1)

u = 0,
x ∈ ∂Ω.
has been studied by many authors. Many important problems arise in
studying progress of the above equation, for instance, existence, regularity, qualitative estimates, the effect of the domain topology on the number
of solutions to the equations . . . . Many methods are used to study problem (1) such as the method of sub-supersolutions (see Evans (1998)), the
topological degree method (see Li (1989)), . . . . However, one of the most

effective methods in order to study the existence of weak solutions is the
variational methods. The idea of the method is that we transform the
problem (1) into finding the critical of the differentiable functional J associated to problem (1).
The following Ambrosetti-Rabinowitz condition introduced by Ambrosetti
and Rabinowitz (1973)
(AR)

∃R0 > 0, θ > 2 sao cho
0 < θF (x, s) ≤ sf (x, s),

∀|s| ≥ R0 , ∀x ∈ Ω,

plays an important role in their studies. This condition not only ensures
that the Euler-Lagrange functional associated to problem (1) has a mountain pass geometry, but also guarantees that Palais-Smale sequences of
the Euler-Lagrange functional is bounded. With this (AR) condition, one
can use the classical version of the Mountain Pass Theorem of Ambrosetti
and Rabinowitz to study the existence of solutions (see e.g. AmbrosettiRabinowitz (1973), Ambrosetti (1986)). Although (AR) condition is quite
natural and important but there are many problems where the nonlinear
term f (x, u) does not satisfy the (AR) condition, and thus this condition is
2


restrictive to many nonlinearities. Because of this reason, in recent years,
some authors has studied the problem (1) by trying to drop the (AR)
condition, for instance, Schechter and Zou (2004), Liu and Wang (2004),
Miyagaki and Souto (2008), Liu (2010), Lam and Lu (2013, 2014), Binlin
et. al. (2015).
The existence of nontrivial weak solutions of problem (1) where Laplace
operator replaced by degenerate operators has been studied by many authors, e.g., V.V. Grushin (1971), N.M. Tri (1998), N.T.C. Thuy and N.M.
Tri (2002), P.T. Thuy and N.M. Tri (2012, 2013).

In 2017 many authors concerned with the Dirichlet boundary problem
for semilinear elliptic equations involved the strongly degenerate elliptic
∆λ . Specifically, it is the following problem

−∆ u + V (x)u = f (x, u), x ∈ Ω,
λ
(2)

u = 0,
x ∈ ∂Ω,
where Ω is a bounded domain in RN , N ≥ 2. Some results in existence,
multiplicity and regularity of weak solutions to problem (2) has been considered by Kogoj and Lanconelli (2012), D.T. Luyen and N.M. Tri (2015,
2018), Luyen (2017), Chen and Tang (2018), Rahal (2018), where V (x) is
potential function and the nonlinearities allow is discontinuous but (AR)
condition is still required (see also the survey paper of Kogoj (2018)).
Therefore, we can see that for the degenerate elliptic equations, most
of all results are only obtained in the cases where the nonlinearity has
standard growth (i.e., has subcritical growth and satisfies (AR) condition).
In my knowledge, there are still many open problems in this topic, for
instance, studying the existence of weak solutions to problem (4) when the
nonlinearity term f (x, u) has subcritical or critical growth, . . . .
• Semilinear elliptic systems of Hamiltonian type.
Beside studying the scalar elliptic equations, the elliptic systems are
also of interest to many mathematicians, one of typically elliptic systems
is class of Hamiltonian as follows:

3






−∆u = |v|p−1 v,


−∆v = |u|q−1 u,




u = v = 0,

x ∈ Ω,
x ∈ Ω,

(3)

x ∈ ∂Ω,

where p, q > 1 and Ω is a bounded domain in RN , N ≥ 3 with smooth
boundary ∂Ω. For system (3), we know that the critical hyperbola is
1
1
N −2
+
=
.
p+1 q+1
N
For exponents (p, q) lying on or above this curve, that is,

1
N −2
1
+
≤
,
p+1 q+1
N
then nonexistence of positive classical solutions of systems (3) in starshaped bounded domain has been proved in works of Pucci and Serrin
(1986), Mitidieri (1993). In the degenerate operator, some results in existence and nonexistence to the Hamiltonian/gradient systems are obtained
by N.T. Chung (2014) and by N.M. Chuong et. al. (2004, 2005).
In case of (p, q) below the critical hyperbola, by using variational
method and Fountain theorem of Bartsch and Figueiredo (1996), the existence of weak solutions of (3) has shown (see some works of Peletier and
van der Vorst (1992), Hulshof and van der Vorst (1993), de Figueiredo and
Felmer (1994) and a survey paper of Figueiredo (1996)).
However, the corresponding results for degenerate elliptic systems are
still very few; for example, existence, multiplicity and nonexistence to systems (3) when Laplace operator is replaced by the strongly degenerate ∆λ
is not considered.
• Some Liouville type theorems for elliptic equations and systems.
In recent years, one of the very hot topics is the study Liouville type
theorems for elliptic equations and systems. The Liouville-type theorem is
the nonexistence of solutions in entire space or in half-space. The classical
Liouville-type theorem stated that a bounded harmonic (or holomorphic)
function defined in entire space must be constant. This theorem, known
as the Liouville Theorem, was first announced by Liouville (1844). Later,
4


Cauchy (1844) published the first proof of the above stated theorem (see
also Axler, Bourdon and Ramey (2001)). This classical result has been

extended to nonnegative solutions of the semilinear elliptic equations in
whole space RN by Gidas and Spruck (1980, 1981), Chen and Li (1991).
The Liouville theorem for semilinear elliptic equations or inequalities on
a cone Σ in RN was also studied by Dolcetta, Berestycki and Nirenberg
(1995).
Recently, Liouville type theorems for degenerate elliptic equations have
been attracted the interest of many mathematicians. The classical Liouville theorem was generalized to p-harmonic functions on the whole space
RN and on exterior domains by Serrin and Zhou (2002). The Liouville
type theorem for semilinear elliptic inequality involving the Grushin operator has been established by Dolcetta and Cutr`ı in (1997), D’Ambrosio v
Lucente (2003), Monticelli (2010), Yu (2014). The Liouville type theorems
for elliptic systems and systems of inequalities has also attracted the interest of many mathematicians, for instance, Souto (1995), Serrin and Zou
(1996), Mitidieri and Pohozaev (2001), Souplet (2009). In the case elliptic system involving the Grushin operator, some Liouville type for stable
solutions established by Hung and Tuan (2017).
Therefore, we see that some Liouville type theorems are just obtained
for weak degenerate operators and there are still few results. The corresponding results for strongly degenerate operators such as ∆λ are still open
in many cases.
Summary, for analysis as above, we would see that, beside the results are
known, many problems in elliptic equations and systems involving the strongly
degenerate ∆λ still open, for instance:
• The existence and multiplicity of weak solutions to semilinear strongly
degenerate elliptic has form

−∆ u = f (x, u)
x ∈ Ω,
λ
(4)

u=0
x ∈ ∂Ω,
where Ω is a bounded domain in RN , N ≥ 2 and the nonlinear term does

not satisfy the Ambrosetti-Rabinowitz condition.
5


• Existence and nonexistence of solutions to
erate elliptic system



−∆λ u = |v|p−1 v


−∆λ v = |u|q−1 u




u=v=0

a Hamiltonian strongly degen-

x ∈ Ω,
x ∈ Ω,

(5)

x ∈ ∂Ω,

with p, q > 1 and Ω is a bounded domain in RN , N ≥ 3.
• Establishing Liouville type theorems for semilinear elliptic inequality and

systems of inequalities involving the strongly degenerate operator ∆λ :
− ∆λ u ≥ up

x ∈ RN ,


−∆ u ≥ v p
λ
−∆ v ≥ uq
λ

x ∈ RN ,

(N ≥ 2, p > 1),

(6)

(N ≥ 2, p, q > 0).

(7)

and systems

N

x∈R ,

Therefore, our thesis focus on study existence, nonexistence, multiplicity and
establish some Liouville type theorems for some degenerate elliptic problems
involving the ∆λ -Laplace operator.

2. Purpose of thesis
This thesis focus on study some class of elliptic equations and systems
involving the ∆λ -Laplace operator. Namely, that is the following important
issues:
• To study the existence of weak solutions;
• To study multiplicity of solutions;
• To study nonexistence positive classical solutions in starshaped like domain;
• To study some Liouville type theorems on the nonexistence of solutions in
entire space.

6


3. Object and scope of thesis
Research scope:
• Content 1. To study existence and multiplicity of solutions in the subcritical case of the semilinear degenerate elliptic equations involving the ∆λ Laplace operator when the nonlinear term does not satisfy the AmbrosettiRabinowitz condition.
• Content 2. To study existence and nonexistence of solutions to a Hamiltonian elliptic systems involving the ∆λ -Laplace operator.
• Content 3. To study some Liouville type theorems for semilinear inequality elliptic system involving the ∆λ -Laplace operator.
4. Research methods
• To study the existence and multiplicity of solutions: Variational methods.
• To study nonexistence of positive classical solutions: Establishing suitable
Pohozaev type identities and exploiting geometry structure of the domain.
• To study Liouville type theorems: Using the rescaled-test functions methods and establishing suitable integral estimates.

5. Results of thesis
The thesis achieved the following main results:
• Proving the existence of nontrivial weak solutions to problem (4) when
the nonlinearity has subcritical polynomial growth and does not satisfy
the Ambrosetti-Rabinowitz condition. In additions, if the nonlinear term
is odd with respect to the second variable, we proved the multiplicity of

weak solutions to problem (4). This is the main content of Chapter 2.
• Proving the nonexistence of positive classical solutions to Hamiltonian system (5) in the starshaped like domain. Proving the multiplicity of weak
solutions of systems (5) when (p, q) below critical hyperbola. This is the
main content of Chapter 3.
7


• Establishing some Liouville type theorems on the nonexistence of nonnegative classical solutions for inequality (6) and elliptic inequalities (7) in
entire space. This is the main content of Chapter 4.
6. Structures of thesis
Beside Introduction, Conclusion, Authors works related to the thesis and References, the thesis includes 4 chapters:
• Chapter 1. Preliminaries;
• Chapter 2. Existence of solutions to a semilinear degenerate elliptic equation;
• Chapter 3. Existence and nonexistence of solutions to a degenerate Hamiltonian system;
• Chapter 4. Liouville type theorems for degenerate elliptic inequalities.

8


Chapter 1
PRELIMINARIES

In this chapter, we recall some concepts and results about the Definition
strongly degenerate elliptic ∆λ -Laplace operator, some function spaces, some
results on embeddings, eigenvalues, eigenfunctions of ∆λ operator, some results
on the variational methods and the critical points theory in order to prove the
main results of the thesis in the following chapters.
1.1. ∆λ -Laplace operator
We consider the operator of the form
N

∂xi (λ2i ∂xi ),

∆λ :=
i=1

where ∂xi = ∂x∂ i , i = 1, . . . , N. Here the functions λi : Rn → R are continuous,
strictly positive and of class C 1 outside the coordinate hyperplanes, i.e., λi >
N

0, i = 1, . . . , N in R \

= {(x1 , . . . , xN ) ∈ R

, where

N

N

xi = 0}. We

:
i=1

assume that λi satisfy the following properties:
1) λ1 (x) ≡ 1, λi (x) = λi (x1 , . . . , xi−1 ), i = 2, . . . , N ;
2) For every x ∈ RN , λi (x) = λi (x∗ ), i = 1, . . . , N , where
x∗ = (|x1 |, . . . , |xN |) if x = (x1 , . . . , xN );
3) There exists a constant ρ ≥ 0 such that

0 ≤ xk ∂xk λi (x) ≤ ρλi (x) ∀k ∈ {1, . . . , i − 1}, i = 2, . . . , N,
N
and for every x ∈ RN
+ := {(x1 , . . . , xN ) ∈ R : xi ≥ 0 ∀i = 1, . . . , N };

4) There exists a group of dilations {δt }t>0
δt : RN → RN , δt (x) = δt (x1 , . . . , xN ) = (t 1 x1 , . . . , t N xN ),
where 1 ≤
i − 1, i.e.,

1

≤

2

≤ ··· ≤

N,

λi (δt (x)) = t i −1 λi (x),

such that λi is δt -homogeneous of degree
∀x ∈ RN , t > 0, i = 1, . . . , N.
9


This implies that the operator ∆λ is δt -homogeneous of degree two, i.e.,
∀u ∈ C ∞ (RN ).

∆λ (u(δt (x))) = t2 (∆λ u)(δt (x)),

We denote by Q is the homogeneous dimension of RN with respect to the group
of dilations {δt }t>0 , i.e.,
Q :=

+ ··· +

1

N.

The homogeneous dimension Q plays a crucial role, both in the geometry and
the functional associated to the operator ∆λ .
1.2. Some function spaces and embeddings
◦

∞
For 1 ≤ p < +∞, denote by W 1,p
λ (Ω) is the closure of C0 (Ω) in the norm

 p1

u

◦

W 1,p
λ

|∇λ u|p dx ,

=
Ω

where ∇λ = (λ1 ∂x1 , . . . , λ1 ∂xN ).
◦

◦

1,2
We see that W 1,p
λ (Ω) are Banach spaces and when p = 2, W λ (Ω) is a Hilbert
space with the inner products

∇λ u · ∇λ v dx,

(u, v) =
Ω

with norm
u

1,2

1
2

2

|∇λ u| dx

=

.

Ω

We next define Wλ2,p (Ω) as the space of all functions u such that
u ∈ Lp (Ω), λi (x)

∂
∂u
∈ Lp (Ω), λi (x)
∂xi
∂xi

λj (x)

∂u
∂xj

∈ Lp (Ω),

i, j = 1, 2, . . . , N,

with the norm

u

Wλ2,p

 p1

N

|u|p + |∇λ u|p +

=

λi (x)
i,j=1

Ω

∂
∂u p
(λj (x)
)| dx .
∂xi
∂xj

It is easy to see that Wλ2,p (Ω) are Banach spaces. In particular, Wλ2,2 (Ω) is a
Hilbert space with the following inner products
N

N

(u, v)W 2,2
λ

∂v
∂
∂u
∂
∂v
∂u
, λi
)L2 +
λi
(λj
), λi
(λj
)
= (u, v)L2 + (λi
∂x
∂x
∂x
∂x
∂x
∂x
i
i
i
j
i
j
i,j=1
i=1
10

,
L2


f (x)g(x)dx with f, g ∈ L2 (Ω).

where (f, g)L2 =
Ω

The following result was established by Kogoj and Lanconelli which frequently used in thesis.
Proposition 1.1. Assume that the functions λi , i = 1, 2, . . . , N satisfy conditions 1) − 4) as in Section 1.1 and Q > 2. Then the embedding
◦

∗

∗
pλ
W 1,p
λ (Ω) → L (Ω), where pλ :=

pQ
,
Q−p

is continuous. Moreover, the embedding
◦

γ
W 1,p

λ (Ω) → L (Ω)

is compact for every γ ∈ [1, p∗λ ).
We now prove the following important result.
Proposition 1.2. Assume that the functions λi , i = 1, 2, . . . , N satisfy conditions 1) − 4) as in Section 1.1 and Q > 4. Then the embedding
◦

γ
Wλ2,2 (Ω) ∩ W 1,2
λ (Ω) → L (Ω)

is continuous if 1 ≤ γ ≤

2Q
.
Q−4

We consider the following homogeneous Dirichlet problem:

−∆ u = f (x) inΩ,
λ

u = 0 on ∂Ω.
◦

(1.1)
◦

1,2
Proposition 1.3. The operator −∆λ : W 1,2

λ (Ω) → (W λ (Ω)) is surjective,
◦

◦

1,2
where (W 1,2
λ (Ω)) is the dual space of W λ (Ω).

Corollary 1.1. For each f ∈ L2 (Ω) problem (1.1) has unique weak solution
◦

u ∈ W 1,2
λ (Ω).
◦

◦

By Proposition 1.3, the inverse operator T = (−∆λ )−1 : (W 1,2
λ (Ω)) →

W 1,2
λ (Ω) of the operator −∆λ is well defined. Then, we have following proposition.
11


Proposition 1.4. The inverse operator T of −∆λ is positive, self-adjoint and
compact in L2 (Ω).
By Proposition 1.4, there exists a sequence of eigenfunctions ϕj ∈ L2 (Ω) of
T which is an orthogonal in L2 (Ω) corresponding to eigenvalues {γj }∞

j=1 with
γj → 0 as j → +∞.
Since

◦

T : L2 (Ω) → W λ1,2 (Ω) ⊂ L2 (Ω)
◦

this implies ϕj ∈ W 1,2
λ (Ω) for all j = 1, 2, . . . . Moreover, since
ϕj = T −1 (T ϕj ) = T −1 (γj ϕj ) = γj (−∆λ ϕj ),
thus
−∆λ ϕj =

1
ϕj ,
γj

∀j = 1, 2, . . . .
◦

1,2
Therefore, the operator −∆λ has a sequence of eigenfunctions {ϕj }∞
j=1 in W λ (Ω)
corresponding to eigenvalues {µj = γ1j }∞
j=1 such that

0 < µ1 ≤ µ2 ≤ · · · ≤ µj ≤ · · · ,

µj → +∞ as j → +∞.

1.3. Some results of critical points theory
We will use the following version of Mountain Pass Theorem.
Theorem 1.1. Let X be a real Banach space and let J ∈ C 1 (X, R) satisfy the
(C)c condition for any c ∈ R, J(0) = 0, and
(i) There are constants ρ, α > 0 such that J(u) ≥ α

∀ u = ρ;

(ii) There is an u1 ∈ X, u1 > ρ such that J(u1 ) ≤ 0.
Then c = inf max J(γ(t)) ≥ α is a critical value of J, where
γ∈Γ 0≤t≤1

Γ = {γ ∈ C 0 ([0, 1], X) : γ(0) = 0, γ(1) = u1 }.
Let X be a reflexive and separable Banach space. We know that there exist
sequences {ej } ⊂ X, {ϕj } ⊂ X ∗ such that
(i) ϕi , ei = δi,j , where δi,j = 1 if i = j and δi,j = 0 otherwise;
12


∗

w
∞
∗
(ii) span{ej }∞
j=1 = X and span {ϕj }j=1 = X .

Xj . We define

Let Xj = Rej then X =
j≥1

k

Yk =

Xj

and Zk =

j=1

Xj .

(1.2)

j≥k

Because the Mountain Pass Theorem also holds when functionals satisfy the
(C)c condition, thus we can establish multiplicity results for problem (2.1) by
using the following Fountain Theorem of Bartsch.
Theorem 1.2. Assume that J ∈ C 1 (X, R) satisfies the (C)c condition for all
c ∈ R and J(u) = J(−u). If for every k ∈ N, there exists ρk > rk such that
(i) ak = max ϕ(u) ≤ 0;
u∈Yk
u =ρk

(ii) bk = inf ϕ(u) → +∞, k → ∞;

u∈Zk
u =rk

then J has a sequence of critical points {uk } such that J(uk ) → +∞.
We next recall some concepts in order to study existence of weak solutions
to Hamiltonian system in Chapter 3.
Definition 1.5. Let E be a Hilbert space and a functional Φ ∈ C 1 (E, R).
Given a sequence F = (En ) of finite dimensional subspaces of E such that
En ⊂ En+1 , n = 1, 2, . . ., and ∪∞
n=1 En = E. We say that
(i) sequence (zk ) ⊂ E with zk ∈ Enk , nk → ∞, is a (P S)Fc -sequence if
Φ(zk ) → c and (1 + zk )(Φ |Enk )(zk ) → 0.
(ii) Φ satisfies (P S)Fc , at level c ∈ R, if every sequence (P S)Fc -sequence has a
subsequence converging to a critical point of Φ.
We will use the Fountain Theorem established by Bartsch and de Figueiredo
to prove the existence of infinitely many of weak solutions to our problem.
We decompose the Hilbert space E into direct sum E = E + ⊕ E − , denote
E1± ⊂ E2± ⊂ · · · be an increasing sequence of finite dimensional subspaces of
−
±
+
±
E ± , respectively, such that ∪∞
n=1 En = E and let En = En ⊕ En , n = 1, 2, . . . .

13


Theorem 1.3. Assume Φ : E → R is C 1 (E, R) and satisfies the following
conditions:

(Φ1) Φ satisfies (P S)Fc , with F = (En ), n = 1, 2, . . . and c > 0;
(Φ2) There exists a sequence rk > 0, k = 1, 2, . . . , such that for some k ≥ 2,
bk := inf{Φ(z) : z ∈ E + , z⊥Ek−1 , z = rk } → +∞ as k → ∞;
(Φ3) There exists a sequence of isomorphisms Tk : E → E, k = 1, 2, . . . , with
Tk (En ) = En for all k and n, and there exists a sequence Rk > 0, k =
1, 2, . . . , such that, for z = z + +z − ∈ Ek+ ⊕E − and Rk = max{ z + , z − },
one has
Tk z > rk and Φ(Tk z) < 0,
where rk is obtained in (Φ2);
(Φ4) dk := sup{Φ(Tk (z + + z − )) : z + ∈ Ek+ , z − ∈ E − , z + , z − ≤ Rk } < +∞;
(Φ5) Φ is even, i.e., Φ(z) = Φ(−z).
Then Φ has an unbounded sequence of critical values.
We notice that, if Φ mapping bounded sets in E into bounded sets in R then
the (Φ4) condition is satisfied.
1.4. Some standard conditions on the nonlinearity term
In this section, we introduce some standard conditions, for instance, AmbrosettiRabinowitz (AR) condition, subcritical polynomial growth (SCP) and (SCPI),
and critical growth conditions on the nonlinearity f (x, s) and give some related
results to problem (1).

14


Chapter 2
EXISTENCE OF SOLUTIONS TO A SEMILINEAR
DEGENERATE ELLIPTIC EQUATION

In this chapter, we study the Dirichlet boundary problem for semilinear degenerate elliptic equations involving the ∆λ -Laplace operator in a bounded
domain Ω ⊂ RN , N ≥ 2, where the nonlinearity has subcritical polynomial
growth and does not satisfy the Ambrosetti-Rabinowitz condition. Firstly, we
prove the existence of at least a weak solutions and next we use assumption on

the nonlinear term is odd function we get the multiplicity of weak solutions to
the problem.
This chapter is written based on the paper [1].
2.1. Setting of the problem
In this paper, we study the existence of nontrivial weak solutions to the
following problem

−∆ u = f (x, u), x ∈ Ω,
λ
(2.1)
u = 0,
x ∈ ∂Ω,
where Ω is a bounded domain in RN , N ≥ 2. Here the nonlinear terms f (x, u)
has subcritical polynomial growth and satisfies the following assumptions:
(f 1) f : Ω × R → R is continuous and f (x, 0) = 0 for all x ∈ Ω;
F (x, u)
(f 2) lim
= +∞ uniformly on x ∈ Ω, where F (x, u) =
u2
|u|→+∞

u

f (x, t)dt;
0

2F (x, u)
< µ1 uniformly on x ∈ Ω, where µ1 > 0 is the first
|u|2
|u|→0

eigenvalue of the operator −∆λ in Ω with homogeneous Dirichlet boundary
conditions;
(f 3) lim sup

(f 4) There exist C∗ ≥ 0, θ ≥ 1 such that
H(x, t) ≤ θH(x, s) + C∗

∀t, s ∈ R, 0 < |t| < |s|, ∀x ∈ Ω,

1
where H(x, u) = uf (x, u) − F (x, u).
2
15


(SCP I) f has subcritical polynomial growth on Ω, i.e.,
2Q
f (x, s)
∗
=
0,
2
=
,Q > 2
∗
λ
Q−2
|s|→+∞ |s|2λ −1
lim

where Q denotes the homogeneous dimension of RN with respect to a group
of dilations.
Remark 2.1. In problem (2.1) we do not require the (AR) condition imposed
on the nonlinear terms.
We recall the definition of weak solutions of problem (2.1).
◦

Definition 2.1. A function u ∈ W 1,2
λ (Ω) is called a weak solutions of (2.1) if
∇λ u∇λ ϕ dx =

∀ϕ ∈ C0∞ (Ω).

f (x, u)ϕ dx,
Ω

Ω

Define the Euler-Lagrange functional associated with problem (2.1) as follows
Jλ (u) =

1
2

|∇λ u|2 dx −
Ω

F (x, u)dx,
Ω

u
0 f (x, s)ds. By the hypotheses on f ,
◦
◦
1,2
1
W λ (Ω) and Jλ ∈ C (W 1,2
λ (Ω), R) with

where F (x, u) =
well-defined on

◦

∇λ u∇λ vdx −

Jλ (u)v =
Ω

we can see that Jλ is

f (x, u)vdx,

∀ v ∈ W 1,2
λ (Ω).

Ω

One can also check that the critical points of Jλ are the weak solutions of
problem (2.1), and thus we can use a suitable version of the Mountain Pass

Theorem to study the existence of weak solutions to problem (2.1).
2.2. Existence of nontrivial weak solutions
The main result of this section is the following theorem.
Theorem 2.1. Assume that f has subcritical polynomial growth on Ω, i.e.
(SCPI) condition holds, and satisfies (f 1) − (f 4). Then problem (2.1) has
a nontrivial weak solution.
We prove Theorem 2.1 by verifying that all conditions of Lemma 1.1 are
satisfied. First, we check the condition (i) in this lemma.
16


Lemma 2.1. Assume that f satisfies conditions (f 1), (f 3) and (SCP I). Then
there exist α, ρ > 0 such that
Jλ (u) ≥ α

◦

∀u ∈ W 1,2
λ , u

1,2

= ρ.

Next, we check the condition (ii) in Lemma 1.1.
Lemma 2.2. Assume that f satisfies (f 2). Then Jλ (tu) → −∞ as t → +∞
◦

for all functions u ∈ W 1,2
λ (Ω) \ {0}.

We now show the (C)c conditions is satisfied.
Lemma 2.3. If f (1) − (f 4) and (SCP I) are satisfied, then Jλ satisfies the (C)c
condition for all c ∈ R.
Proof of Theorem 2.1: Combining Lemmas 2.1-2.3, we deduce that problem (2.1) has a nontrivial weak solution.
2.3. The multiplicity of weak solutions
Next, when f (x, s) is an odd function in s we obtain the result on the
existence of infinitely many weak solutions to problem (2.1).
Theorem 2.2. Assume that (f 1) − (f 4) hold and
(1) there exist a, b > 0 and q ∈ (2, 2∗λ ) such that
(SCP)

|f (x, s)| ≤ a + b|s|q−1

(2) f (x, −s) = −f (x, s),

∀(x, s) ∈ Ω × R;

∀(x, s) ∈ Ω × R.

Then problem (2.1) has a sequence of solutions {un } such that Jλ (un ) → +∞.

17


Chapter 3
EXISTENCE AND NONEXISTENCE OF SOLUTIONS TO A
HAMILTONIAN DEGENERATE ELLIPTIC

In this chapter, we study nonexistence of classical solutions and existence
of infinitely many weak solutions to a semilinear degenerate elliptic system

involving ∆λ -Laplace operator in a bounded domain.
This chapter is written based on the paper [3].
3.1. Setting of the problem
We consider the following semilinear degenerate elliptic system of Hamiltonian type



−∆λ u
= |v|p−1 v, x ∈ Ω,


(3.1)
−∆λ v
= |u|q−1 u, x ∈ Ω,



 u = v = 0,
x ∈ ∂Ω,
where p, q > 1, and Ω is a bounded domain in RN , N ≥ 3 with smooth boundary
∂Ω.
We now define some functional spaces which are used to study problem (3.1).
◦

By definition of the spaces W λ1,2 (Ω) v Wλ2,2 (Ω) as in Chapter 1, we consider
the operator

◦

A : Wλ2,2 (Ω) ∩ W λ1,2 (Ω) → L2 (Ω),

(3.2)

where A = −∆λ with the homogeneous Dirichlet boundary condition.
We denote E s = D(As ), with s > 0, the space with the inner product
As u As v dx,

(u, v)E s =

u, v ∈ E s ,

Ω

where
∞
s

D(A ) = {ϕ =

∞

∞
2
µ2s
j aj

aj ϕj , aj ∈ R|
j=1

j=1

s

aj µsj ϕj .

< +∞} and A ϕ =
j=1

where ϕj are eigenfunctions of A corresponding to eigenvalues µj , j = 1, 2, . . . .
We notice that, as a consequence of Proposition 1.2 and interpolation theorem, we obtain following important embeddings.
18


Lemma 3.1. Suppose that Q > 4. Then, the following embeddings
E s → Lq+1 (Ω) and E t → Lp+1 (Ω)
1
1 2s 1
1 2t
≥ − ,
≥ − , respectively. Moreover, these
q+1
2 Q p+1
2 Q
embeddings are compact if the corresponding inequalities are strict.
are continuous if

For s, t ≥ 0 such that s + t = 1, we consider E = E s × E t , a Hilbert space
with the inner product
(z, η)E = (u, ϕ)E s + (v, ψ)E t , for z = (u, v), η = (ϕ, ψ) ∈ E.
We consider the bilinear form

(As uAt ψ + As ϕAt v) dx.

B((u, v), (ϕ, ψ)) =
Ω

Now we define the functional Φ : E = E s ×E t → R associated to the problem
(3.1) by
(As uAt ψ + As ϕAt v) dx −

Φ(z) =
Ω

H(u, v)dx,
Ω

where

|v|p+1 |u|q+1
+
.
H(u, v) =
p+1
q+1

One can check that Φ is well-defined on E and Φ ∈ C 1 (E, R) with
(As u At ψ + At v As φ)dx −

Φ (u, v)(φ, ψ) =
Ω

(uq φ + v p ψ)dx.
Ω

One can also see that the critical points of Φ are the weak solutions of the
problem (3.1) in the sense following definition.
Definition 3.1. We say that z = (u, v) ∈ E = E s × E t is a weak solution of
(3.1) if
As u At ψ dx −
Ω

v p ψdx = 0 ∀ψ ∈ E t ,
Ω

At v As φdx −
Ω

uq φ dx = 0 ∀φ ∈ E s .
Ω

19


3.2. Nonexistence of positive classical solutions
In this section, we prove nonexistence result of our problem when the domain
Ω is starshaped in the sense of definition below.
We first consider the following vector field
N

T :=

i xi
i=1

∂
,
∂xi

(3.3)

and this vector field is the generator of the group of dilation {δt }t>0 . Here, a
function u is δt -homogeneous of degree m if and only if T u = mu.
Definition 3.2. A domain Ω is called δt -starshaped with respect to the origin
if 0 ∈ Ω and T, ν ≥ 0 at every point of ∂Ω, where ν is the outward normal
vector and ·, · denotes the inner product in RN .
We will denote by Λ2 (Ω) the linear space of the functions u ∈ C(Ω) such
that
Xj u, Xj2 u, j = 1, . . . , N,
exist in the weak sense of distributions in Ω and can be continuously extended
∂
.
to Ω. Here Xj := λj
∂xj
We obtain useful following lemma.
Lemma 3.2. For any u, v ∈ Λ2 (Ω), we have
[T (u) ∇λ v, νλ + T (v) ∇λ u, νλ ]dS

[T (u)∆λ v + T (v)∆λ u] dx =
Ω

∂Ω

−

∇λ u, ∇λ v T, ν dS + (Q − 2)

∇λ u, ∇λ v dx,

(3.4)

Ω

∂Ω

where T is the vector field in (3.3), ν is the outward normal to Ω, νλ =
(λ1 ν1 , . . . , λN νN ) and ∇λ = (λ1 ∂x1 , . . . , λN ∂xN ).
The following theorem is the main result of this section.
Theorem 3.1. Assume N ≥ 3, and p, q > 1 satisfy
1
Q−2
1
+
≤
p+1 q+1
Q

(3.5)

If Ω is bounded and δt -starshaped with respect to the origin, then problem (3.1)
has no nontrivial nonnegative solution u ∈ Λ2 (Ω).
20

3.3. Existence of infinitely many of weak solutions
In this section we show the existence of infinitely many solutions to the
systems (3.1).
We have the following result on the existence of infinitely many solutions to
the systems (3.1).
Theorem 3.2. If p, q > 1, Ω is a smooth and bounded domain in RN and
1
Q−2
1
+
>
,
p+1 q+1
Q

(3.6)

then the problem (3.1) has infinitely many weak solutions.
Here, in order to prove Theorem 3.2 we will check conditions (Φ1) − (Φ4) in
Theorem 1.3 in Chapter 1 are satisfied.

21


Chapter 4
LIOUVILLE TYPE THEOREMS FOR DEGENERATE ELLIPTIC
SYSTEM OF INEQUALITIES

In this chapter, we study some Liouville type theorems, that is the nonexistence of positive classical solutions for the elliptic system of inequalities involving the ∆λ -Laplace operator in entire space RN , N ≥ 2.
This chapter is written based on the paper [2].
4.1. Setting of the problem
We will establish Liouville type theorems for the elliptic system of inequalities

−∆ u ≥ v p , x ∈ RN ,
λ
(4.1)
−∆ v ≥ uq , x ∈ RN ,
λ

p, q > 0. We will consider two cases:
• Case 1. The exponents p, q > 1 and such that
2(p + 1) 2(q + 1)
,
} ≥ Q − 2.
max{
pq − 1 pq − 1
To do this, we use the so-called rescaled test-functions method and exploit
the homogeneous properties of the operator ∆λ . The idea of this method
is the following: we fix a constant R > 0 and consider the equations or
systems in bounded domains, next multiply by suitable test functions and
use suitable computations, then let R tends to infinity, we obtain new
equations or systems in the whole space, hence obtain the results that our
solutions are trivial.
• Case 2. The exponents p, q > 0 such that pq > 1 and satisfy
1
1
Q−2
+

≥
.
p+1 q+1
Q−1
To do this, we exploit the interesting idea of Souto (1995) for the Laplace
operator, namely it is to reduce the problem to a question concerning a
scalar equation by introducing a new function w = uv and then use the
Liouville type theorem for elliptic inequality.
22


4.2. Liouville type theorem for p, q > 1
Our main results are the following theorems.
Theorem 4.1. Let p, q > 1. Then system (4.1) does not possess positive classical solutions u, v ∈ C 2 (RN ) provided
max{a, b} ≥ Q − 2,
where a =

2(q + 1)
2(p + 1)
,b=
.
pq − 1
pq − 1

As a direct consequence of Theorem 4.1, when u = v and p = q, we get the
following Liouville type theorem for the elliptic inequality
− ∆λ u ≥ up trong RN .
Corollary 4.1. Assume 1 < p ≤
ity (4.2), then u ≡ 0.

Q
Q−2 .

(4.2)

If u is a nonnegative solution of inequal-

4.3. Liouville type theorem for p, q > 0
The main result in this section is following.
Theorem 4.2. Assume p, q > 0 such that pq > 1, and
1
1
Q−2
+
≥
.
p+1 q+1
Q−1
Then system (4.1) does not possess any positive classical solutions.

23