DSpace at VNU: Solutions of elliptic problems of p-Laplacian type in a cylindrical symmetric domain

DSpace at VNU: Solutions of elliptic problems of p-Laplacian type in a cylindrical symmetric domain tài liệu, giáo án, b...

Acta Math Hungar., 135 (1–2) (2012), 42–55

DOI: 10.1007/s10474-011-0163-6 First published online October 6, 2011

SOLUTIONS OF ELLIPTIC PROBLEMS OF

p-LAPLACIAN TYPE IN A CYLINDRICAL

SYMMETRIC DOMAIN

N T CHUNG 1 ,∗and H Q TOAN2

1 Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet,

Dong Hoi, Vietnam e-mail: ntchung82@yahoo.com 2

Faculty of Mathematics, Mechanics and Informatics, University of Natural Sciences, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

e-mail: hq toan@yahoo.com

(Received February 27, 2011; revised May 16, 2011; accepted May 16, 2011)

Abstract. We consider the p-Laplacian type elliptic problem



− div (a(x, ∇u)) = h(x)|u| q−2 u + g(x) in Ω,

where Ω = Ω1× Ω2 ⊂ R N

is a bounded domain having cylindrical symmetry, Ω1⊂ R mis a bounded regular domain and Ω2 is a k-dimensional ball of radius R, centered in the origin and m + k = N , m  1, k  2 Under some suitable con-ditions on the functions a and h, using variational methods we prove that the problem has at least one resp at least two solutions in two cases: g = 0 and

g = 0.

1 Introduction

In this article, we are concerned with nonlinear elliptic equations in which the divergence form operator − diva(x, ∇u) is involved Such operators appear in many nonlinear diffusion problems, in particular the

mathematical modeling of non-Newtonian fluids The p-Laplacian operator

Δp u = − div|∇u| p −2 ∇u is a special case of the operator− diva(x, ∇u)

We refer the readers to some recent works [8,12,13] on elliptic problems of

p-Laplacian type in the general case.

∗Corresponding author.

Key words and phrases: elliptic problem, p-Laplacian type, cylindrical symmetric domain,

mountain pass theorem.

2010 Mathematics Subject Classification: 35J65, 35J20.

0236-5294/$ 20.00 c 2011 Akad´emiai Kiad´o, Budapest, Hungary

In [8], P De N´apoli et al studied the Dirichlet problem of the form

(1.1)



− diva(x, ∇u) = f (x, u) in Ω,

where Ω⊂ R N (N  3) is a bounded domain with smooth boundary, the

function a : Ω × R N → R N satisfies the condition

(1.2) a(x, ξ)  C

1 +|ξ| p −1

for all ξ ∈ R N , a.e x ∈ Ω.

Under condition of Ambrosetti–Rabinowitz type, i.e., there exists μ > p such

that

0 < μF (x, t) := μ

 t

0

f (x, s) ds  f(x, t)t for all t ∈ R\{0}, and a.e x ∈ Ω, the authors showed the existence of at

least one solution for (1.1) using the arguments of classical mountain pass type [1] In [12], A Krist´aly et al studied (1.1) with condition (1.2) in the

case when f (x, t) = λf (t) is (p − 1)-sublinear at infinity, i.e.,

lim

t →∞

f (t)

|t| p −1 = 0.

It is worth observing that in this case the nonlinear term f does not satisfy

the Ambrosetti–Rabinowitz type condition Using the three critical point theorem by G Bonanno [4], the authors proved that problem (1.1) has at least three weak solutions If Ω =RN, problem (1.1) with condition (1.2) was studied in [13], in which M Mih˘ailescu gave some sufficient conditions

on the nonlinearities to obtain some existence and multiplicity results

In a recent paper, D M Duc et al [10] have studied (1.1) in the case

when the function a satisfies

(1.3) a(x, ξ)  C

θ(x) + σ(x) |ξ| p −1

for all ξ ∈ R N , a.e x ∈ Ω,

in which θ and σ are two non-negative measurable functions satisfying

θ ∈ L p−1 p (Ω), σ ∈ L1

loc(Ω), and σ(x)  1 for a.e x ∈ Ω It is clear that

condi-tion (1.3) is weaker than (1.2) This interesting idea comes from the moun-tain pass theorem for weakly continuously differentiable functionals in [9] Using this result, they then showed in [10] that problem (1.1) has at least

one weak solution provided that the functions a and f satisfy some

fur-ther suitable conditions Regarding some extensions of [10], the readers may consult in [7,14–16,19], in which the authors studied the existence of solutions for problem (1.1) with condition (1.3) and subcritical

nonlinear-ities relying essentially on the compactness embedding W01,p (Ω)  → L q(Ω),

Acta Mathematica Hungarica 135, 2012

44 N T CHUNG and H Q TOAN

1 q < p  = N−p N p If q = p  (critical case) or q > p  (superciritical case) the

compactness of this embedding is no longer valid, so variational arguments cannot be applied as usual Generally speaking, some kinds of geometric and topological properties of the domain lead to the solvability of elliptic problems; for example, the symmetry of the domain could improve Sobolev embeddings

Motivated by the recent results on the effect of the topology of the do-main in [5,6,20], in the present work, we will investigate (1.1) with condi-tion (1.3) in the case when Ω = Ω1× Ω2 ⊂ R N is a bounded domain having cylindrical symmetry, Ω1 ⊂ R m is a bounded regular domain and Ω2 is a

k-dimensional ball of radius R, centered in the origin and m + k = N , and

m  1, k  2 Using variational methods, we prove some existence and

mul-tiplicity results for (1.1) in the critical and supercritical cases q  p  So the results introduced here are better than those of [10] and its previous extensions

Let 1 < p < N , and consider a : Ω × R N → R N , a = a(x, ξ) the continu-ous derivative with respect to ξ of the continucontinu-ous function A : Ω × R N → R,

A = A(x, ξ), that is, a(x, ξ) = ∂A(x,ξ) ∂ξ Suppose that a and A satisfy the

following hypotheses:

(A1) a(x, 0) = A(x, 0) = 0 for a.e x ∈ Ω.

(A2)a(x, ξ)  C

θ(x) + σ(x) |ξ| p −1

for all ξ ∈ R N , a.e x ∈ Ω, where

θ ∈ L p−1 p (Ω), σ ∈ L1

loc(Ω), θ(x)  0, and σ(x)  1 for a.e x ∈ Ω.

(A3) There exists k0 > 0 such that

A



x, ξ + ψ

2

 1

2A(x, ξ) +

1

2A(x, ψ) − k0σ(x)|ξ − ψ| p

for all ξ, ψ ∈ R N , a.e x ∈ Ω, that is, A is p-uniformly convex.

(A4) 0 a(x, ξ) · ξ  pA(x, ξ) holds for all ξ ∈ R N , a.e x ∈ Ω.

(A5) There exists k1> 0 such that A(x, ξ)  k1σ(x) |ξ| p for all ξ ∈ R N,

a.e x ∈ Ω;

There are many functions a : Ω × R N → R N satisfying all conditions (A1)–(A5), see for example [7,10]

Let W01,p(Ω) be the usual Sobolev space under the norm u 1,p=

( Ω|∇u| p

dx)1p and define the subspace

W 0,s 1,p(Ω) ={u ∈ W 1,p

0 (Ω) : u(x1, x2) = u

x1, |x2|, ∀ x = (x1, x2)∈ Ω},

which is a closed subspace of W01,p(Ω)

Now, we introduce the space

X =



u ∈ W 1,p 0,s(Ω) :



Ω

σ(x) |∇u| p dx < ∞

Acta Mathematica Hungarica 135, 2012

With the method as those used in [10], we can show that X is a Banach

space with respect to the norm

 

Ω

σ(x) |∇u| p dx

1

p

.

Moreover, the continuous embedding X  → W 1,p

0,s (Ω) holds true since σ(x) 1

for a.e x ∈ Ω.

Firstly, we consider problem (1.1) in the case when f (x, u) = h(x) |u| q −2 u,

then the problem becomes

(1.4)



− diva(x, ∇u) = h(x) |u| q −2 u in Ω,

Throughout this paper, we always assume that h : Ω → [0, ∞) is a H¨older

continuous function satisfying:

(H1) h(x1, x2) = h

x1, |x2| for all x = (x1, x2)∈ Ω = Ω1× Ω2, h(x1, 0)

= 0 for all x1 ∈ Ω1;

(H2) l h > 0, where

lh:= sup



λ > 0 : sup

x ∈Ω

h(x)

|x2| λ < ∞

.

We point out the fact that there are many functions satisfying the

condi-tions (H1) and (H2) For example, if h(x) = |x2| l for all x = (x1, x2)∈ Ω

= Ω1× Ω2, l is a positive real number, then problem (1.4) is a H´enon

equa-tion of p-Laplacian type in a cylindrical symmetric domain Regarding the

H´enon equations, we refer the readers to some interesting works [3,17,18] Definition 1.1 We say that a function u ∈ X is a weak solution of

problem (1.4) if and only if



Ω

a(x, ∇u) · ∇ϕ dx −

Ωh(x)|u| q −2 uϕ dx = 0

for all ϕ ∈ X.

The first result of ours concerning problem (1.4) is given by the following theorem

Theorem 1.2 Assume that the conditions (A1)–(A5) and (H1)–(H2)

are satisfied Then there exists a positive number τ such that for any q ∈

(p, p  + τ ), p  =N N p −p , problem (1.4) has at least one nontrivial weak solution.

Next, a natural question is to see what happens if the above problem is affected by a certain perturbation For this purpose, we shall consider the

Acta Mathematica Hungarica 135, 2012

46 N T CHUNG and H Q TOAN

perturbed problem for (1.4), in which q ∈ (p, p  + τ ), the number τ defined

by Theorem1.2, and f (x, u) = h(x) |u| q −2 u + g(x), i.e.,

(1.5)



− diva(x, ∇u) = h(x) |u| q −2 u + g(x) in Ω,

where g is a function which belongs to the dual space of W 0,s 1,p(Ω), denoted

by W −1,p

0,s (Ω)⊂ X  , X  is the dual space of X.

Definition 1.3 We say that a function u ∈ X is a weak solution of

problem (1.5) if and only if



Ω

a(x, ∇u) · ∇ϕ dx −

Ωh(x)|u| q −2 uϕ dx −

Ω

gϕ dx = 0

for all ϕ ∈ X.

We obtain a multiplicity result for problem (1.5) as follows

Theorem 1.4 Assume that the conditions (A1)–(A5) and (H1)–(H2)

are satisfied Then there exists a constant ε > 0 such that for any g ∈

W −1,p

0,s (Ω) with 0 < g −1 < ε, problem (1.5) has at least two nontrivial weak

solutions.

Since q ∈ (p, p  + τ ) with the number τ defined by Theorem1.2, problems (1.4) and (1.5) contain the subcritical, critical and supercritical cases The paper is organized as follows In Section 2, we give some prelimi-naries and in Section 3, we prove the main theorems

2 Preliminaries

In this section, we give some useful results which are used in the proofs

of our main theorems Firstly, the following result concerning the function

A may be found in [8,10]

Lemma 2.1 Assume that all conditions (A1)–(A5) are satisfied Then: (i) The function A satisfies the growth condition

A(x, ξ)  c0

θ(x) |ξ| + σ(x)|ξ| p

for all ξ ∈ R N and a.e x ∈ Ω;

(ii) A(x, tξ)  A(x, ξ)t p for all t  1, and for all ξ ∈ R N , a.e x ∈ Ω.

Define the functionals Λ, I and J : X → R by

Λ(u) =



Ω

q



Ω

h(x) |u| q dx +



Ω

g(x)u dx, Acta Mathematica Hungarica 135, 2012

and J (u) = Λ(u) − I(u) for all u ∈ X.

Due to the presence of σ ∈ L1

loc(Ω), the functional Λ (and thus J ) may not belong to C1(X,R) So, we cannot apply the mountain pass theorem

by A Ambrosetti and P H Rabinowitz [1] To overcome this difficulty, we need to recall a useful concept in [9] (see also in [10,19]) as follows

Definition 2.2 Let Y be a Banach space We say that a map F :

Y → R is weakly continuously differentiable on Y if and only if

(i) for any u ∈ Y , there exists a linear map DF(u) from Y into R such

that

lim

t →0

F(u + tϕ) − F(u)

(ii) for any ϕ ∈ Y , the map u → DF(u)(ϕ) is continuous on Y

Remark 2.3 If we suppose further that for every u ∈ Y , the map

ϕ → DF(u)(ϕ) is continuous and linear on Y , then F is Gˆateaux

differ-entiable on Y For information and connection on functionals that are not

Gˆateaux differentiable but only differentiable along directions of a certain

subspace of Y , we refer to [2]

Denote by C w1(Y,R) the set of weakly continuously differentiable

func-tionals on Y Then, it is clear that C1(Y, R) ⊂ C1

w (Y, R), where C1(Y,R) is the set of all continuously Fr´echet differentiable functionals on Y Now let

F ∈ C1

w (Y, R) We put for any u ∈ Y ,

D F(u)

Y  = sup{D F(u)(h), h ∈ Y with h

Y = 1},

where D F(u)

Y  may be ∞.

We say thatF satisfies the Palais–Smale condition at level c ∈ R (denote

by (PS)c) if any sequence{um} ⊂ Y for which

possesses a convergent subsequence If this is true at every level c then

we simply say that F satisfies the Palais–Smale condition (denote by (PS))

on Y

Hence, our idea is to obtain the existence of solutions for problem (1.1) using the weak version of the mountain pass theorem for weakly continuously differentiable functionals by D M Duc [9] In our arguments, we need the following results whose proofs are standard and simple

Lemma 2.4 Assume that all conditions (A1)–(A5) are satisfied Then: (i) The functional Λ is weakly lower semicontinuous in W 0,s 1,p (Ω), i.e., if

{um} converges weakly to u in W 1,p

0,s (Ω) then we have Λ(u) lim infm →∞ Λ(u m)

Acta Mathematica Hungarica 135, 2012

48 N T CHUNG and H Q TOAN

(ii) It holds that

1

2Λ(u) +

1

2Λ(v) − Λ



u + v

2

 k0u − v p

X for all u, v ∈ X.

The following result is a natural generalization of W Wang [20, Theorem 2.7], whose proof is standard so we omit it

Theorem 2.5 Assume that Ω = Ω1× Ω2,dim (Ω1) = m 1, dim (Ω2)

= k  2, and h : Ω → [0, ∞) is a H¨older continuous function satisfying the

conditions (H1), (H2) Then, there is a positive number τ = τ (h, p, m, k)

depending on h, p, m, k (well-defined) such that the embedding

W 0,s 1,p (Ω)  → L r (h, Ω), r ∈ (1, p  + τ ),

is compact, where L r (h, Ω) is the usual Lebesgue space with weight h.

3 Proof of the main results

In our arguments, the proof of Theorem1.4contains the existence result which is stated as in Theorem1.2 So, for the sake of brevity, we will deal only with problem (1.5) in details As we mentioned in Section 2, in order

to prove our main results, we will use critical point theory Let us define the

functional J : X → R by

J (u) =



Ω

A(x, ∇u) dx −1

q



Ωh(x)|u| q

dx −

Ω

gu dx, = Λ(u) − I(u),

(3.1)

where

Λ(u) =



Ω

q



Ω

h(x) |u| q

dx +



Ω

gu dx

(3.2)

for all u ∈ X.

By Theorem 2.5, for any q ∈ (p, p  + τ ), similar arguments as in [10]

ensure that the functional J is well-defined and weakly continuously differ-entiable on X Moreover, we have

DJ (u)(ϕ) =



Ω

a(x, ∇u) · ∇ϕ dx −

Ω

h(x) |u| q −2 uϕ dx −

Ω

gϕ dx

for all u, ϕ ∈ X Thus, weak solutions of problem (1.5) are exactly the

crit-ical points of the functional J

Lemma 3.1 The functional J is weakly lower semiconituous on X Acta Mathematica Hungarica 135, 2012

Proof Let a sequence {um} ⊂ X be weakly convergent to u in X.

Since the embedding X  → W 1,p

0,s(Ω) is continuous, {um} converges weakly

to u in W 0,s 1,p(Ω) By Lemma2.4, it follows that

(3.3) Λ(u) lim infm →∞ Λ(u m ).

On the other hand, Theorem 2.5 and the fact that {um} converges weakly

to u in W 0,s 1,p(Ω) imply that {um} converges strongly to u in L q (h, Ω) for all

q ∈ (p, p  + τ ) Therefore,

m →∞



Ω

h(x) |um| q dx =



Ω

h(x) |u| q dx.

We also deduce since {um} converges weakly to u in W 1,p

0,s(Ω) that

m →∞



Ω

gu m dx =



Ω

gu dx.

Combining (3.4) and (3.5), we have

m→∞ I(um ) = I(u).

Finally, we conclude from (3.3) and (3.6) that

lim inf

m →∞ J (u m) = lim infm →∞ Λ(u m)− I(um)

 Λ(u) − I(u) = J(u) and thus, the functional J is weakly lower semicontinuous on X. 

The following lemma shows that the functional J has the geometry of

the mountain pass theorem

Lemma 3.2 (i) There exist ε > 0, η > 0, and α > 0 such that J (u)  α

for all u ∈ X with u X = η, provided that 0 < g −1 < ε.

(ii) For η > 0 as in (i), there exists e ∈ X such that e X > η and

J (e) < 0.

Proof (i) For any ε > 0, using Young’s inequality we deduce that





Ω

gu dx

 g −1 u 1,p (ε p1u X).

 1

ε1p

g −1

 ε

p u p

p  ε p p

g p 

−1 ,

1

1

p  = 1

for all u ∈ X.

Acta Mathematica Hungarica 135, 2012

50 N T CHUNG and H Q TOAN

Hence, by Theorem2.5, it follows that for any ε > 0 and for all u ∈ X,

J (u) =



Ω

A(x, ∇u) dx −1

q



Ω

h(x) |u| q dx −



Ω

gu dx

(3.7)



Ω

k1σ(x)|∇u| p

qS q q u q

X − ε

p u p

p  ε p

p

g p 

−1

=



k1− ε

qS q q u q −p

X

u p

p  ε p p

g p 

−1

where S q is the best constant in the embedding W 0,s 1,p (Ω)  → L q (h, Ω).

By fixing ε ∈ (0, k1p), we can find η > 0, ε > 0 and α > 0 such that the

conclusion of the lemma holds true For example, we can take

η =

M qS q q 1

q−p , ε = p

1

p ε p1

q p (q−p)

qS q q q p (q−p) , α = 1

2M

q q−p

qS q q q q−p ,

where M = 12(k1− ε

p)> 0.

(ii) Let ϕ0 ∈ C ∞

0 (Ω)∩ W 1,p

0,s(Ω) such that Ωh(x)ϕ0(x)q

dx > 0 Then

for any t > 1 we obtain by Lemma 2.1that

J (tϕ0) =



ΩA(x, t∇ϕ0) dx − t q

q



Ωh(x)|ϕ0| q

dx − t

Ω

gϕ0dx

 t p



Ω

A(x, ∇ϕ0) dx − t q

q



Ωh(x)|ϕ0| q

dx − t

Ω

gϕ0dx,

which approaches−∞ as t → +∞ since q > p > 1 

Lemma 3.3 The functional J satisfies the Palais–Smale condition in X.

Proof Let{um} be a Palais–Smale sequence for the functional J in X,

i.e

(3.8) J (u m)→ c, DJ(um)→ 0 in X  as m → ∞,

where X  is the dual space of X.

In order to prove that J satisfies the Palais–Smale condition, we first

prove that {um} has a bounded subsequence Indeed, assume by

contradic-tion thatum X → ∞ as m → ∞ Then, using relation (3.8) and (A4)–(A5),

we deduce that for m large enough the following inequalities hold:

c + 1 + um X  J(u m)−1

q DJ (u m )(u m)

(3.9)

Acta Mathematica Hungarica 135, 2012

Ω



A(x, ∇um)−1

q a(x, ∇um)· ∇um



dx −



1−1 q



Ω

gu m dx

 k1



1− p q

um p



1− 1 q

g −1 um X

Dividing the above inequality by um p

X and letting m → ∞ we obtain a

contradiction since 1 < p < q This implies that {um} has a bounded

subse-quence, still denoted by {um} Since X → W 1,p

0,s(Ω) is continuous, {um} is

bounded in W 0,s 1,p (Ω) and thus there exists u ∈ W 1,p

0,s(Ω) such that, passing to

a subsequence, still denoted by{um}, it converges weakly to u in W 1,p

0,s(Ω)

By (3.8), it follows from Lemma2.4(i), Theorem 2.5that

= k1



Ωσ(x)|∇u| p

dx

Ω

(3.10)

 lim infm →∞ Λ(u m) = lim

m →∞



I(um ) + J (u m)

= I(u) + c < ∞,

which yields that u ∈ X and then the sequence um − u X is bounded Combining this with (3.8), we conclude that DJ (u m )(u m − u) → 0 as

On the other hand, using H¨older’s inequality we have



Ω

h(x)

|un| q −2 u − |u| q −2 u

(u n − u) dx



Ω

h(x)

|un| q −1+|u| q −1

|un − u| dx

=



Ω(h1(x) |un|)q −1 h1(x) |un − u| dx +

Ω(h1(x) |u|)q −1 h1(x) |un − u| dx

(un q −1

L q (h,Ω)+u q −1

L q (h,Ω))un − u L q (h,Ω)

This implies using Theorem 2.5that

(3.11) lim

n →∞



Ω

h(x)

|un| q −2 u − |u| q −2 u

(u n − u) dx = 0.

Moreover we deduce, since{um} converges weakly to u in W 1,p

0,s(Ω), that

m→∞



Ω

g(um − u) dx = 0.

Acta Mathematica Hungarica 135, 2012

Acta Mathematica Hungarica 135, 2012

Ω

... functional J is weakly lower semiconituous on X Acta Mathematica Hungarica 135, 2012

Proof...  = 1

for all u ∈ X.

Acta Mathematica Hungarica 135, 2012