DSpace at VNU: Multiple solutions for a class of N-Kirchhoff type equations via variational methods

DSpace at VNU: Multiple solutions for a class of N-Kirchhoff type equations via variational methods tài liệu, giáo án, b...

DOI 10.1007/s13398-014-0177-3

O R I G I NA L PA P E R

Multiple solutions for a class of N-Kirchhoff type

equations via variational methods

Nguyen Thanh Chung · Hoang Quoc Toan

Received: 1 March 2014 / Accepted: 2 June 2014

© Springer-Verlag Italia 2014

Abstract In this article, we consider the following N -Kirchhoff type problem

⎧

⎪

⎪−M





 |∇u| N d x



 N u = λf (x, u) + μg(x, u) in ,

u = 0 on ∂,

where is a bounded smooth domain ofRN , N ≥ 2, M :R+0 →Ris a continuous function,

 N u = div(|∇u| N−2∇u), f, g :  ×R→Rare two Carathéodory functions andλ, μ are

positive parameters Using variational method, we show the existence of at least three weak solutions for the problem

Keywords N -Kirchhoff type equations· Multiple solutions · Variational method

Mathematics Subject Classifications (2010) 35J55· 35J65

1 Introduction

In this article, we consider the following N -Kirchhoff type problem

⎧

⎪

⎪

−M

⎛

⎝



|∇u| N d x

⎞

⎠ N u = λf (x, u) + μg(x, u) in ,

u = 0 on ∂,

(1.1)

N T Chung (B)

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi,

Quang Binh, Vietnam

e-mail: ntchung82@yahoo.com

H Q Toan

Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam e-mail: hq_toan@yahoo.com

where is a bounded smooth domain ofRN , N ≥ 2, N u = div(|∇u| N−2∇u), f, g :

 ×R→Rare two Carathéodory functions andλ, μ are positive parameters.

Throughout this paper, we assume that M:R+0 →Ris a continuous and nondecreasing

function, and there exist m0 > 0 and α > 1 such that

(M0) M(t) ≥ m0t α−1 for all t∈R+0 := [0, ∞).

According to the definition given by Adimurthi in [1,2], we say that a functionϕ has a

subcritical growth if, for everyδ > 0,

lim

|t|→∞supx∈

|ϕ(x, t)|

Such a notion of criticality is motivated by the Trudinger-Moser inequality after the celebrated papers [11,20] In the present paper we will assume that the nonlinearities f , g belong to

the classAof the Carathéodory functionsϕ :  ×R→Rhaving a subcritical growth and satisfying the following condition:

for every L > 0, sup

Problems of the above type arise for instance in conformal geometry When N = 2, the prescribed Gaussian curvature equation is a subcritical equation on a two dimensional

manifold M with metric tensor g of the type

− u = k(x) exp (2u) + h(x), (1.4) where is the Laplace-Beltrami operator associated to the metric g, k is the Gaussian

curvature and h is a given Hölder function The solvability of nonlinear boundary value

problems in the presence of an exponential nonlinearity is an interesting topic, we refer

to recent papers [9,12,16,19] In this paper, motivated by the ideas introduced in [16] and the multiplicity result for Kirchhoff type problems in [3,18], we want to handle problem (1.1) with exponential nonlinearities To our best knowledge, this is the first contribution

to nonlocal problems in this direction Further information on nonlocal problem involving

p-Laplace operators and p (x)-Laplace operators, we refer to [4 8,10,13–15]

Definition 1.1 Let X be a real Banach space We denote by W X the class of functionals

A : X →Rpossessing the following property: if{um} is a sequence in X weakly converging

to u ∈ X and lim infm→∞A (u m ) ≤ A(u), then {u m} has a subsequence strongly converging

to u.

The key in our argument is the following result which was presented in the paper by Ricceri [17]

Proposition 1.2 (see [ 17 ]) Let X be a separable and reflexive real Banach space; : X →

Ra coercive, sequentially weakly lower semicontinuous C1functional, belonging to W X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on

X∗; J : X →Ra C1functional with compact derivative Assume that

minimum u0, with (u0) = J(u0) = 0 Finally, assume that

max

 lim sup

u →+∞

J (u) (u) , lim sup u→u

J (u) (u)



≤ 0

and that

sup

u∈Xmin{ (u), J(u)} > 0.

Set

θ∗:= inf



(u)

J (u) : u ∈ X, min{ (u), J(u)} > 0



.

Then, for each compact interval ⊂ (θ∗, +∞), there exists a number σ > 0 with the

following property: for every λ ∈ and every C1 functional  : X → Rwith compact derivative, there exists μ∗> 0 such that for each μ ∈ [0, μ∗], the equation

(u) = λJ(u) + μ(u)

has at least three solutions whose norms are less than σ

2 Main result

In this section, we will discuss the existence of weak solutions for problem (1.1) When there

is no misunderstanding, we always use c to denote a positive constant For simplicity, we denote by X the Sobolev space W1,N

0 () endowed the norm

u =

⎛

⎝



|∇u| N d x

⎞

⎠

1

N

From the Rellich Kondrachov theorem, X is continuously embedded in L r () for every

r ≥ 1, that is there exists a constant Sr > 0 such that

We also denote by|u|r the norm of u in the space L r (), i.e., |u| r = |u| r d x1

r

It is also well-known that if L φ () is the Orlicz-Lebesgue space generated by the function φ(t) = exp(|t| N−1) − 1, i.e.,

L φ () :=

⎧

⎨

⎩u :  →R, measurable :



φ(u(x)) dx < ∞

⎫

⎬

⎭ , and L φ∗() is the linear hull of L φ () equipped with the norm

u L φ∗ ()= inf

⎧

⎨

⎩λ >0:



φ



u (x) λ



d x≤ 1

⎫

⎬

⎭ ,

X is continuously embedded in L φ∗() Throughout the sequel we will use the following

facts:

(T M ) For every u ∈ X, for every δ > 0, exp(δ|u(.)| N−1) ∈ L1();

(T M2) There exists a constant C N depending on N and on the measure of  such that, for

every 0< δ ≤ α N, whereα N = Nw N1−1

N−1, beingw N−1the measure of the(N − 1)

dimensional surface of the unit sphere inRN,

sup

u ≤1



exp(δ|u(x)| N−1) dx ≤ C N

Due to the assumption(M0) and the above results, problem (1.1) can be treated variationally

in X , i.e., solutions of problem (1.1) will be obtained as critical points of a suitable functional

defined on X

If f ∈A, we put

F (x, t) =

t

0

f (x, s) ds, (x, t) ∈  ×R

and



M (t) =

t

0

M (s) ds, t ≥ 0.

Furthermore, let us define the functionalsρ, , J : X = W1,N

0 () →Rby

ρ(u) =



|∇u| N d x , (u) = 1

N Mρ(u),

J (u) =



Some simple computations show that the functionals and J are of class C1in X , and

their derivatives are given by

(u)(v) = M

⎛

⎝



|∇u| N d x

⎞

⎠



|∇u| N−2∇u∇v dx,

J(u)(v) =



f (x, u)v dx for all u, v ∈ X.

Moreover, since f ∈A , the mapping J: X → Xis compact, see [16, Lemma 3].

Definition 2.1 We say that u ∈ X is a weak solution of problem (1.1) if

M

⎛

⎝



|∇u| N d x

⎞

⎠



|∇u| N−2 ∇u∇v dx − λ



f (x, u)v dx − μ



g (x, u)v dx = 0

for allv ∈ X.

Theorem 2.2 Let f ∈A be such that

(F1) sup u∈X



F (x, u(x)) dx > 0;

(F2) lim sup t→0supx ∈ F (x,t)

|t| N α ≤ 0;

(F3) lim sup |t|→+∞supx ∈ F (x,t)

|t| N α ≤ 0, where F(x, t) =t

0

f (x, s) ds.

Under such hypotheses, if we set

θ∗:= 1

N inf

⎧

⎪

⎪

⎪

⎪



M





 |∇u| N d x







F (x, u(x)) dx : u ∈ X,



F (x, u(x)) dx > 0

⎫

⎪

⎪

⎪

⎪

,

then for each compact interval ⊂ (θ∗, +∞), there exists a number σ > 0 with the

following property: for every λ ∈ and every g ∈ A there exists μ∗> 0 such that, for each

μ ∈ [0, μ∗], problem ( 1.1 ) has at least three weak solutions whose norms are less than σ

The following result is useful in proving the main result of this paper

Lemma 2.3 (i) The functional is sequentially weakly lower semicontinuous;

(ii) belongs to the class W X.

Proof (i) Let {um} ⊂ X be a sequence that converges weakly to u in X Then we have



|∇u| N d x≤ lim inf

m→∞



Combining (2.3) with the continuity and monotonicity of the function t → M (t),

we get

lim inf

m→∞ (u m ) = 1

N lim infm→∞ M

⎛

⎝



|∇um| N d x

⎞

⎠

≥ 1

N M

⎛

⎝lim inf

m→∞



|∇um| N d x

⎞

⎠

≥ 1

N M

⎛

⎝



|∇u| N d x

⎞

⎠

Thus, the functional is sequentially weakly lower semicontinuous.

(ii) It is well known thatρ(u) belongs to W X Since M is continuous and strictly

increas-ing we deduce that belongs to W X

Proof of Theorem 2.2 We wish to apply Proposition1.2by taking X = W1,N

0

J are as before In view of [16, Lemma 3], J is well defined and continuously Gâteaux differentiable with compact derivative Jgiven by

J(u)(v) =



f (x, u)v dx

for all u , v ∈ X.

Moreover, by Lemma2.3, is a sequentially weakly lower semicontinuous and C1

functional belonging toW X, and a simple computation shows that it is also coercive In fact,

by(M0), we have

(u) = 1

N M

⎛

⎝



|∇u| N d x

⎞

⎠

≥ m0

N α

⎛

⎝



|∇u| N d x

⎞

⎠

α

= m0

from which we have the coercivity of It is evident that u0= 0 is the only global minimum

of and that (u0) = J(u0) = 0.

Moreover, it is easy to see that, if u ≤ r then (u) ≤ 1

N M(r N ) and so is bounded

on each bounded subset of X

Now, let us show that the operator  : X → X∗is invertible on X On account of the well-known Minty-Browder theorem (see [21, Theorem 26.A(d)]), it suffices to prove that

is strictly convex, hemicontinuous and coercive in the sense of monotone operators.

So, let u , v ∈ X with u = v and λ, μ ∈ [0, 1] with λ + μ = 1 Since the operator

ρ: X → X∗given by

ρ(u)(v) =



|∇u| N−2∇u∇v dx

is strictly montone, by [21, Proposition 25.10],ρ is strictly convex Moreover, since M is

nondecreasing the function M is convex in [0, +∞) Thus, we have



M (ρ(λu + μv)) <  M (λρ(u) + μρ(v)) ≤ λ  M (ρ(u)) + μ  M (ρ(u)).

This shows that is strictly monotone.

For any u ∈ X, by (M0), one has

(u)(u)

M (ρ(u)) |∇u| N d x

u

≥ m0(ρ(u)) α

u

= m0 u N α−1 ,

from which we have the coercivity of  Standard arguments ensure that is hemicontinu-ous Thus, in view of [21, Theorem 26.A(d)] there exists −1: X∗→ X and it is bounded.

Let us prove that −1is continuous by showing that it is sequentially continuous.

Let{wm} ⊂ X∗be a sequence strongly converging tow ∈ X∗and let u

m = −1(w m ),

m = 1, 2, , and u = −1(w) Then, {u m} is bounded in X and without loss of generality,

we can assume that it converges weakly to a certain u0 ∈ X Since {wm} converges strongly

tow, it is easy to see that

lim

m→∞ (u m )(u m − u0) = lim m→∞ w m (u m − u0) = 0

lim

m→∞ M

⎛

⎝



|∇um| N d x

⎞

⎠



|∇um| N−2∇um (∇u m − ∇u0) dx = 0. (2.6)

Since{um} is bounded in X, passing to a subsequence, if necessary, we may assume that



|∇um| N d x → t0≥ 0 as m → ∞.

If t0 = 0 then {um} converges strongly to u0 = 0 in X and the proof is finished because of

the continuity and injectivity of  If t

0> 0, it follows from the continuity of the function

M that

M

⎛

⎝



|∇um| N d x

⎞

⎠ → M(t0) as m → ∞.

Thus, by(M0), for sufficiently large m, we have

M

⎛

⎝



|∇um| N d x

⎞

From (2.6) and (2.7), it follows that

lim

m→∞



|∇um| N−2∇um (∇u m − ∇u0) dx = 0. (2.8)

From (2.8) and the fact that{um } converges weakly to u0in X we deduce that {um} converges strongly to u0in X The continuity and injectivity of imply that{um} converges strongly

to u, so −1is continuous.

Let us prove that

lim sup

u→0

J (u)

By the assumption(F2), for every  > 0 there exists some constant η1 > 0 such that, for

every x ∈  and |t| ≤ η1,

As f belongs to A, for fixedδ > 0 and q > Nα, there exists c > 0 such that, for every

x ∈  and |t| ≥ η1,

F (x, t) ≤ c|t| qexp



δ|t| N−1



Then, for every x ∈  and t ∈R, one has

F (x, t) ≤ |t| N α + c expδ|t| N−1



After choosing p > 1, we apply Hölder’s inequality to get



exp(δ|u(x)| N−1)|u(x)| q d x

≤

⎡

⎣



exp



p δ u N−1



|u(x)|

u

 N

−1

d x

⎤

⎦

1 ⎛

⎝



|u(x)| pq

d x

⎞

⎠

1

p

where pis the conjugate of p.

By combining the previous inequality with(T M2) and bearing in mind that X is

contin-uously embedded in L r () for every r ≥ 1, for u ≤α N

p δ

N−1

N

, one has

J (u) ≤ (S N α ) N α u N α + c(Spq ) q C

1

N u q

≤ (SN α ) N α N α

m0 (u) + c(S pq ) q C

1

N



N α

m0 (u)

q

N α

Therefore,

J (u)

(u) ≤ (SN α ) N α

N α

m0 + c(S pq ) q C

1

N



N α

m0

q

N α ( (u)) q −Nα N α

Since q > Nα and the fact that (u) → 0 as u → 0, claim (2.9) immediately follows Let us prove now that

lim sup

u →∞

J (u)

By the assumption(F3), for every  > 0, there exists some constant η2 > 0 such that for

every x ∈  and |t| > η2,

From condition (1.3), there exists some constant K > 0 such that for every x ∈ ,

sup

|t|≤η2

Then, for every x ∈  and t ∈R,

F (x, t) ≤ K η2+ |t| N α

and so

J (u) ≤ K η2meas () + 



Since X is continuously embedded into L N α (), we get

J (u) (u) ≤

N α

m0 ·K η2meas ()

u N α +  N α

m0 ·



 |u| N α d x

u N α

≤ N α

m0 ·K η2meas ()

u N α +  N α

m0 · S N α

and claim (2.15) follows at once

In view of (2.9) and (2.15), we get

max

 lim sup

u →∞

J (u) (u) , lim sup u→0

J (u) (u)



≤ 0

and all the assumptions of Proposition1.2 are satisfied Notice that, if G :  ×R →

Ris defined by G (x, t) =  g (x, s) ds, then the functional (u) =  G (x, u) dx is

continuously Gâteaux differentiable in X , it has compact derivative, and for every u , v ∈ X,

(u)(v) =



g (x, u)v dx.

Then, Proposition1.2applies and there existsσ > 0 such that for every λ ∈ , it is possible

to findμ∗ > 0 verifying the following condition: for each μ ∈ [0, μ∗], the functional

− λJ − μ has at least three critical points whose norms are less than σ It is clear that

critical points of − λJ − μ are precisely weak solutions of problem (1.1) The proof is completed

Acknowledgments The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript This work is supported by Vietnam National Foundation for Science and Technology Development (Grant N.101.02.2014.03).

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