Multiplicity Of Solutions For ! p (x)-Laplacian Elliptic Kirchho¤

Multiplicity Of Solutions For ! p (x)-Laplacian Elliptic Kirchho¤

Type Equations

Anass Ourraoui

Received 11 March 2019

Abstract

In this work, we deal with a class of anisotropic nonlocal type equations, by means of the variational approach, we prove the existence of in…nitely many solutions under suitable assumptions.

1 Introduction

The investigation of the problems concerning anisotropic variable exponent has drawn the attention of many authors, since there are some physical phenomena which can be modelled by such kind of equations, the reader can …nd several models in mathematical physics where this class of problems appears, like those in electrorheological ‡uids [8], thermorheological ‡uids [2] and image restoration [5].

The purpose of the present paper is to study the nonlocal anisotropic p(x)-Laplacian Dirichlet problems of the form

XN i=1

Mi

Z j@xiuj^pi(x)

pi(x) dx @xi j@xiuj^{pi(x) 2}@xiu =f(x; u); forx2 u= 0; forx2@ ;

(1)

where R^N (N 3) is a bounded open set with smooth boundary,p_i; i= 1; :::N;are continuous functions on ; and for eachi= 1; :::; N; M_i : [0;1)![0;1); f : R!Rare continuous functions with the potential

F(x; t) = Z t

0

f(x; s)ds:

This kind of equations with variable exponent growth conditions enable the study of equations with more complicated nonlinearities since the di¤erential operator _~p(x)(u) :=PN

i=1@_xi j@_xiuj^{pi(x) 2}@_xiu allows a distinct behavior for partial derivatives in various directions.

Some articles have been interested in the existence of solutions for this kind of problems in the case when the nonlinearityf veri…es the Ambrosetti-Rabinowitz type conditions(AR);(see below), for instance in [3]

and [7], the authors obtained the existence and multiplicity of solutions under the well-known Ambrosetti- Rabinowitz type condition:

(AR) 9 > p⁺_M; K >0 such that x2 ;jtj K)0 F(x; t) f(x; t)t:

The role of (AR) condition is to ensure the boundness of the Palais-Smale sequences of the Euler-Lagrange functional. This is very crucial in the applications of critical point theory, especially for the nonlocal equa- tions. A distinguishing feature is that we use(Cc)Cermai condition which is weaker than the (P.S) condition.

Furthermore, problem (1) is related also to the stationary version of a model, the so-called Kirchho¤

equation, which is introduced by Kirchho¤ [9]. To be more precise, Kirchho¤ established a model given by the equation

@²u

@t²

0

h + E 2L

Z L 0

@u

@x

2dx @²u

@x² = 0; (2)

Mathematics Sub ject Classi…cations: 35J30 , 35J60, 35J92.

yDepartment of Mathematics (FSO), University Mohamed I, Oujda, Morocco

124

which extends the classical D’Alembert’s wave equation by considering the e¤ects of the changes in the length of the strings during the vibrations. A distinct feature is that (2) contains a nonlocal coe¢ cient

0

h +_2L^E RL

0 j^@u@xj²dx which depends on the average _2L¹ RL 0

@u

@x

2dx, and hence the equation is no longer a pointwise equation.

This work is organized as follows: In section 2, we introduce some preliminary results on the anisotropic variable exponent Sobolev spaces. In section 3, we consider problem (1) and obtain some results on existence and multiplicity for (1) by using the variational method. Finally, we give the proof of the main results.

2 Preliminaries

In order to deal with the problem (1), we recall some auxiliary results. For convenience, we only recall some basic facts which will be used later, we refer to [4,6,10].

Forq2C+( ), we introduce the Lebesgue space with variable exponent de…ned by L^{q( )}( ) = u:uis a measurable real-valued function,

Z

ju(x)j^q(x)dx <1 ; whereC+( ) =fq2C( ;R) : infx2 q(x)>1g:This space, endowed with the Luxemburg norm,

jujq(:)=kukL^{q( )}( )= inf >0 :

Z u(x) ^q(x)

dx 1 ; is a separable and re‡exive Banach space. We also have an embedding result.

Proposition 1 Assume that is bounded and q₁,q₂2C₊( ) such thatq₁ q₂ in . Then the embedding L^{q2( )}( ),!L^{q1( )}( )is continuous.

Furthermore, the Hölder-type inequality Z

u(x)v(x)dx 2kukL^{q( )}( )kvkL^{q0( )}( ) (3) holds for all u2L^{q( )}( )and v2L^{q0( )}( ); whereL^{q0( )}( ) the conjugate space ofL^{q( )}( ), with1=q(x) + 1=q⁰(x) = 1. Moreover, we denote

q⁺= sup

x2

q(x); q = inf

x2 q(x) and foru2L^{q( )}( ), we have the following properties:

kukL^{q( )}( )<1 (= 1;>1) , Z

ju(x)j^q(x)dx <1 (= 1;>1);

kukL^{q( )}( )>1 ) kuk^q_L^{q( )}_{( )} Z

ju(x)j^q(x)dx kuk^q_L^{+q( )}_{( )}; (4)

kukL^{q( )}( )<1 ) kuk^q_L^{+q( )}_{( )} Z

ju(x)j^q(x)dx kuk^q_L^{q( )}_{( )}; (5) kukL^{q( )}( )!0 (! 1) ,

Z

ju(x)j^q(x)dx!0 (! 1):

To recall the de…nition of the isotropic Sobolev space with variable exponent,W^{1;q( )}( ), we set W^{1;q( )}( ) =fu2L^{q( )}( ) :@xiu2L^{q( )}( )for alli2 f1; : : : ; Ngg;

endowed with the norm

kukW^{1;q( )}( )=kukL^{q( )}( )+ XN i=1

k@xiukL^{q( )}( ): The space W^{1;r( )}( );k kW^{1;r( )}( ) is a separable and re‡exive Banach space.

Now, we consider ~p: !R^N to be the vectorial function

~

p(x) = (p1(x); : : : ; pN(x)) withp_i2C₊( ) for alli2 f1; : : : ; Ngand we put

p_M(x) = maxfp₁(x); : : : ; p_N(x)g; p_m(x) = minfp₁(x); : : : ; p_N(x)g: The anisotropic space with variable exponent is

W^{1;~p( )}( ) =fu2L^{pM( )}( ) :@xiu2L^{pi( )}( ) for alli2 f1; : : : ; Ngg and it is endowed with the norm

kukW^{1;~p( )}( )=kukL^{pM( )}( )+ XN

i=1

k@_xiukL^{pi( )}( ):

The space W^{1;~p( )}( );k kW^{1;~p( )}( ) is a re‡exive Banach space. Furthermore, an embedding theorem takes place for all the exponents that are strictly less than a variable critical exponent, which is introduced with the help of the notations

p(x) = N

PN

i=11=p_i(x); q^?(x) =

(N q(x)=[N q(x)] ifq(x)< N;

1 ifq(x) N:

Proposition 2 Let R^N be a bounded open set and p_i 2C₊( ) for alli2 f1; : : : ; Ng. Ifq2C( ;R), 1 q(x) < maxfp (x); p_M(x)g for all x 2 , then we have the compact and continuous embedding W^{1;~p( )}( ),!L^{q( )}( ).

We denote byX =W₀^1;!p(:)( )the closure ofC₀¹( )inW^{1;~p( )}( ). The space W₀^{1;~p( )}( );kuk_W0^{1;~p( )}_{( )} is a re‡exive Banach space, where

kuk=kuk_W^{1;~p( )}

0 ( )=

XN i=1

k@_xiukL^{pi( )}( ):

3 Main Results

Proposition 3 ([7]) Putting

I(u) = XN i=1

I_i(u) = XN i=1

Z 1

p_i(x)j@_xiu)j^pi(x)dx;

thenI2C¹(X;R)and the derivative operatorI⁰ of I is I⁰(u):v =

XN i=1

Z

j@_xiuj^{pi(x) 2}@_xiu @_xiv dx:

(i) The functionalI⁰ is of (S+)type, whereI⁰ is the Gâteaux derivative of the functional I:

(ii) I⁰ is a bounded homeomorphism and strictly monotone operator.

For the functionM_i; i= 1; :::; N, we make the following assumptions.

(M0) 9 m0>0 such that

Mi(t) m0 for allt 0: (6)

(M₁) 9 0< <1such that

Mci(t) (1 )Mi(t)t for allt 0; (7)

where Mc_i(t) =Rt

0M_i(s)ds.

(M₂) M_i is a di¤erentiable and decreasing function onR⁺.

For the functionf we assume the following conditions are satis…ed.

(f₀) jf(x; t)j C(1+jtj^{q(x) 1})for all(x; t)2 RwithC 0and1< q(x)< p (x), wherep (x) = _N^{N p(x)}_p(x) ifp(x)< N,p (x) = +1ifp(x) N.

(f1) lim_jtj!1 ^F(x;t)

jtj

p+ 1M

= +1, uniformly for a.e. x2 , whereF(x; t) =Rt

0f(x; s)ds.

(f2) There exists 1 such that G(x; t) G(x; st) for (x; t) 2 R and s 2 [0;1], where G(x; t) = f(x; t)t ^p

+ M

1 F(x; t).

(f3) limt!0F(x;t)

jtj^p+M = 0, uniformly for a.e. x2 .

Under (AR), any Palais Smale (PS) sequence of the corresponding energy functional is bounded, which plays an important role in the application of variational methods. Indeed, there are many superlinear functions which do not satisfy (AR) condition. For instance the function below does not satisfy (AR)

f(x; t) = p⁺_M 1 jtj

p+ M

1 2tln(jtj); (8)

where 2(0;1). But it is easy to see the above function (8) satis…es (f₀)–(f₃).

Now we are ready to state our result.

Theorem 4 Assume that (M0)–(M2) and (f0)–(f3) are satis…ed. Moreover, we assume that (f₄) f(x; t) = f(x; t)for allx2 andt2R.

If q > p⁺_M, then problem (1) has a sequence of weak solutions f u_kg¹k=1 such that J( u_k) ! +1 as k!+1.

Example 1 Fori= 1; :::; N; Mi(t) = 2 +_1+t¹2; i= 1; :::; N. It is easy to see thatMi satis…es the conditions (M0)–(M2). It is clear that the above function (8) satis…es (f0)–(f3).

De…ne

J(u) = XN i=1

Mci(Ii(u)) Z

F(x; u)dx; u2X;

whereMc_i(t) =Rt

0M_i(s)ds:

De…nition 1 We call the weak solution for problem (1) a functionu2X satisfying:

XN i=1

Mi

Z 1

pi(x)j@xiuj^pi(x)dx Z

j@xiuj^{pi(x) 2}@xiu @xiv dx Z

f(x; u)v dx= 0 for allv2X:

4 Proofs

First of all, we start with the following compactness result which plays a crucial role.

Lemma 5 Under assumptions (M₀)–(M₂) and (f₀)–(f₂),J veri…es the Cerami condition.

Proof. For allc2R, let’s prove that J satis…es the …rst assertion (i) of Cerami condition. Let fung X be bounded, J(un) ! c and J⁰(un) ! 0. Without loss of generality, we assume that un * u, then J⁰(un)(un u)!0:Thus we have

J⁰(un)(un u) = XN i=1

Mi

Z j@_xiu_nj^pi(x) pi(x) dx

! Z

j@xiunj^{pi(x) 2}@xiun(@xiun @xiu)dx Z

f(x; un)(un u)dx!0:

In view of (f₀) and Propositions2, it follows thatR

f(x; u_n)(u_n u)dx!0:Therefore, we have XN

i=1

Mi

Z j@xiunj^pi(x) pi(x) dx

Z

j@xiunj^{pi(x) 2}@xiun(@xiun @xiu)dx!0:

From (M0), we infer that Z

j@xiunj^{pi(x) 2}@xiun(@xiun @xiu)dx!0:

This shows thatu_n !uinX. Afterwards, we claim thatJ satis…es the assertion (ii) of Cerami condition.

Otherwise, there existc2R andfu_ng X such that:

J(un)!c; kunk ! 1; kJ⁰(un)kkunk !0: (9) By (9), we have thatJ(un) ¹

p⁺_MJ⁰(un)(un)!c asn!+1. Denote wn = _k^u_uⁿ

nk, thenfwngis bounded.

Up to a subsequence, for somew2X, we obtain

w_n* w inX; w_n !w inL^q(x)( ); w_n(x)!w(x) a.e. in : Ifw 0, we can de…ne a sequenceft_ng Rsuch that

J(tnun) = max

t2[0;1]J(tun):

For any B > 0, putting b_n = 2BN^{pm 1}p⁺_M

1

pmw_n = K!_n, since b_n ! 0 in L^q(x)( ) and jF(x; t)j C(1 +jtj^q(x)), by the continuity of the Nemytskii operator, we see that F(:; bn)!0 in L¹( )as n!+1, therefore

nlim!1

Z

F(x; bn)dx= 0: (10)

Putting

i;n= p⁺_M ifk@_xiu_nkL^pi(x) <1;

p_m ifk@xiunkL^pi(x) >1:

Similarly, from Remark 2.1 in [11], by using (4) and (5) we have for anyn;

XN i=1

Z

j@_xiu_nj^pi(x)dx

XN i=1

k@_xiu_nk_L^i;npi(x)( )

XN i=1

k@xiunk^p_L^mpi(x)( )

X

fi: i;n=p⁺_Mg

k@xiunk^p_L^mpi(x)( ) k@xiunk^p

+ M

L^pi(x)( )

N PN

i=1k@xiunkL^pi(x)

N

!p_m

N

= kunk^pm N^{pm 1} N:

Fornlarge enough, 2BN^{pm 1}p⁺_M

1

pm =kunk 2(0;1): From (M0) and by virtue of the last inequalities we obtain

J(t_nu_n) J(b_n) XN i=1

Z Ii(bn) 0

m₀ds Z

F(x; b_n)dx m₀

p⁺_M kb_nk^pm N^{pm 1}

N p⁺_M

Z

F(x; b_n)dx m₀ kK!_nk^pm

p⁺_MN^{pm 1} N p⁺_M

! Z

F(x; b_n)dx

m0

K^pm 2p⁺_MN^{pm 1}

N p⁺_M

! :

That is,J(t_nu_n)!+1. FromJ(0) = 0andJ(u_n)!c, we know thatt_n2(0;1)and hJ⁰(tnun); tnuni=tn

d dt t=tn

J(tun) = 0:

According to (M2) and (f2), we get c= lim

n!+1 J(un) 1

p⁺_M hJ⁰(un); uni

= lim

n!+1

"_N X

i=1

Mci Ii(un) 1 p⁺_M

XN i=1

Mi Ii(un) Z

j@xiunj^pi(x)dx+1 p⁺_M

Z

G(x; un)dx

#

n!lim+1

"_N X

i=1

Mci Ii(tnun) 1 p⁺_M

XN i=1

Mi Ii(tnun) Z

t^p_n^i(x)j@xiunj^pi(x)dx+1 p⁺_M

Z G(x; tnun) dx

#

= lim

n!+1

1 J(tnun) 1

p⁺_M hJ⁰(tnun); tnuni = +1;

which is impossible. Ifw6= 0, then the set =fx2 :w(x)6= 0ghas positive Lebesgue measure, forx2 we havejun(x)j !+1as n!+1. Hence by (f1) we deduce

F(x; u_n) jun(x)j

p+ M 1

jw_n(x)j

p+ M

1 !+1 as n!+1: (11)

Noting that1< ^p

+ M

1 :Then

kunk ku_nk

p+ M 1

!0:

Now, becauseJ(u_n)!c, by (11), we deduce via the Fatou Lemma that C1

(pm)¹¹

c+o(1) kunk

p+ M 1

Z F(x; un) kunk

p+ M 1

dx

= Z

wn6=0

F(x; u_n) jun(x)j

p+ M 1

jw_n(x)j

p+ M

1 dx+

Z

wn=0

F(x; u_n) jun(x)j

p+ M 1

jw_n(x)j

p+ M

1 dx

= Z

wn6=0

F(x; un) ju_n(x)j

p+ M 1

jw_n(x)j

p+ M

1 dx!+1; which is absurd.

BecauseX is a separable and re‡exive Banach space [7], there existfejg¹j=1 X andffjg¹j=1 X such that

f_i(e_j) = _i;j=

(1; ifi=j;

0; ifi6=j:

X =spanfe_j :j= 1;2; : : :g; X =span^w ff_j:j = 1;2; : : :g: Fork= 1;2; : : : denote by

Xj=spanfejg; Yk= ^k_j=1Xj; Zk= ¹_j=kXj:

Lemma 6 ([1]) For q2C₊( ),q(x)< p (x)for anyx2 , de…ne _k = supfjujq(x): kuk= 1; u2Z_kg: Thenlim_{k!1 k}= 0.

Proof of Theorem 4. We use the Fountain Theorem. According to Lemma 5 and (f4), J is an even functional and satis…es condition (C). We show that forklarge enough, there exist _k> rk>0such that:

(A1) bk:= inffJ(u) : u2Zk;kuk=rkg !+1ask!+1; (A2) ak := maxfJ(u) : u2Yk;kuk= _kg 0ask!+1.

(A1): Letu2Z_k withkuk=r_k>1 (r_k will be speci…ed below), by using condition (f₀), we have J(u) =

XN i=1

M Ic i(u) Z

F(x; u)dx m0

p⁺_Mkuk^pm Z

C(juj+juj^q(x))dx m0

p⁺_Mkuk^pm Ckuk^{q( )}L^q(x)( ) Ckuk; where 2 ; 8<

:

m0

p⁺_Mkuk^pm C Ckuk ifkukL^q(x)( ) 1

m0

p⁺_Mkuk^pm C( _kkuk)^q+ ifkukL^q(x)( )>1 m0

p⁺_Mkuk^pm C( _kkuk)^q+ Ckuk

=r_k^pm m₀

p⁺_M C ^q_k⁺r_k^{q+ pm} Cr_k:

De…nerk = ^Cq

+ q+ k

m0

1 pm q+

;therefore

J(u) m0r^p_k^m 1 p⁺_M

1

q+ Crk C:

The fact1< p_m p⁺_M < q⁺implies thatr_k !+1, whenk!+1. Thus,J(u)!+1askuk !+1with u2Zk.

(A2): SincedimY_k <1and all norms are equivalent in the …nite-dimensional space, there existsd_k >0, for allu2Y_k withkuk 1, we have

(u) = XN i=1

Mci Ii(u) C1 kuk

p+ M 1

(pm)¹¹ dkkuk

p+ M 1

L

p+ 1M ( )

: (12)

By conditions (f₀), (f₁) and (f₃) we have F(x; u) 2dkjuj

p+ M

1 juj^p+M; 8(x; u)2 R: (13)

Combining (12) and (13), foru2Y_k such thatkuk= _k > r_k, we infer that J(u) d_kkuk

p+ M 1

L

p+ M 1 ( )

+ kuk^p

+ M

L^p+M( )

C₂ (pm)¹¹ kuk

p+ M

1 + C3kuk^p+M: Therefore, for _k large enough ( _k > r_k), we get from the above that

ak := maxfJ(u) : u2Yk;kuk= _kg 0:

The assertion (A2) holds, and this completes the proof of Theorem4.

Acknowledgements. The author would like to thank the anonymous referee for the suggestions and helpful comments which improved the presentation of the original manuscript.

References

[1] M. Allaoui and A. Ourraoui, Exisitence results for a class of p(x)-Kirchho¤ problem with a singular weight, Mediterr. J. Math. (2016) 13: 677. Volume 13, Issue 2, pp 677- 686.(https://doi.org/10.1007/s00009-015-0518-2)

[2] S. N. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous ‡ows, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52(2006), 19–36.

[3] M. Avci, R. A. Mashiyev and B. Cekic, Solutions of an anisotropic nonlocal problem involving variable exponent, Adv. Nonlinear Anal., 2(2013), 325–338.

[4] M. M. Boureanu, In…nitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese Journal of Math., 15(2011), 2291–2310.

[5] Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383–1406.

[6] D. E. Edmunds and J. Rákosník, Sobolev embedding with variable exponent, Studia Math., 143(2000), 267–293.

[7] X. Fan, On nonlocal !p(x)-Laplacian equations, Nonlinear Anal., 73(2010), 3364–3375.

[8] T. C. Halsey, Electrorheological ‡uids, Science, 258(1992), 761–766.

[9] G. Kirchho¤, Mechanik, Teubner, Leipzig, 1883.

[10] O. Kováµcik, J. Rákosník, On spacesL^p(x)andW^k;p(x), Czechoslovak Math. J., 41(1991), 592–618.

[11] A. Ourraoui, On a nonlocalp(:)-Laplacian equations via genus theory, Riv Mat. Univ. Parma, 6(2015), 305–316.