Existence of Solutions for

Existence of Solutions for

-Laplacian equations without Ambrosetti-Rabinowitz type condition

Zehra Yucedag

Department of Mathematics, Faculty of Science, Dicle University, 21280-Diyarbakir, Turkey

Abstract.This paper investigates the existence and multiplicity of solutions for superlinear p(x)-Laplacian equations with Dirichlet boundary condi- tions. Under no Ambrosetti-Rabinowitz’s superquadraticity conditions, we obtain the existence and multiplicity of solutions by using a variant Fountain theorem without Palais-Smale type assumptions.

Keywords: p(x)-Laplace operator; variable exponent Lebesgue-Sobolev spaces; variational approach; variant Fountain theorem

MSC: 35D05, 35J60, 35J70

1 Introduction

We consider the following superlinear elliptic problem

(

p(x)u=f(x; u) +g(x; u); in ;

u= 0; on@ ; (P)

and obtain in…nitely many solutions, where is a bounded smooth domain of R^N (N 3) and p2C with 1< p(x)< N for all x2 .

Generally, in order to search the existence of solutions for Dirichlet problems which is superlinear, it is essential to assume the following superquadraticity condition, which is known as Ambrosetti-Rabinowitz type condition [2]:

(AR) 9M >0; > p⁺ such that0< F(x; s) f(x; s)s; jsj M; x2 ; where f is nonlinear term such thatF(x; t) = ^R₀^tf(x; s)ds.

1 e-mail: [email protected] phone: (+90) 505 253 51 30 fax: (+90) 412-248 80 39

There are many paper dealing with superlinear Dirichlet problems involving p(x)-Laplace operator p(x)u:= div(jruj^{p(x) 2}ru), in which(AR)is the main assumption to get the existence and multiplicity of solutions [9,10]. However, as far as we are concerned, there are many functions which are superlinear but not satisfy(AR)[3,17].

It is well known that the main aim of using(AR)is to ensure the boundedness of the Palais-Smale type sequences of the corresponding functional. In the present paper we do not use(AR). Instead, we use a variant Fountain theorem not including Palais-Smale type assumptions (see Theorem 5).

The study of di¤erential equations and variational problems involving p(x)- growth conditions has attracted a special interest in recent years and a lot of researchers have devoted their work to this area [5,12,14,16] since there are some physical phenomena which can be modelled by such kind of equations.

In particular, we may mention some applications related to the study of elas- tic mechanics and electrorheological fuids [1,4,11,15,19]. The appearance of such physical models was facilitated by the development of variable exponent Lebesgue L^p(x) and Sobolev spacesW^1;p(x).

2 Preliminaries

At …rst, we shall mention some de…nitions and basic properties of generalized Lebesgue-Sobolev spaces L^p(x)( ), W^1;p(x)( ) and W₀^1;p(x)( ). We refer the reader to [6–8,13] for the fundamental properties of these spaces.

Set

C₊ =ⁿp;p2C ,infp(x)>1;8x2 ^o: Letp2C₊ and denote

p := inf

x2

p(x) p(x) p⁺ := sup

x2

p(x)<1:

For anyp2C+ , we de…ne the variable exponent Lebesgue space by

L^p(x)( ) =

8<

:uju: !R is measurable;

Z

ju(x)j^p(x)dx <1

9=

;; then L^p(x)( ) endowed with the norm

jujp(x)= inf

8<

: >0 :

Z u(x) ^p(x)

dx 1

9=

;; becomes a Banach space.

The modular of the L^p(x)( ) space, which is the mapping : L^p(x)( ) ! R de…ned by

(u) =

Z

ju(x)j ^p(x)dx; 8u2L^p(x)( ): (2:1)

Proposition 1 ([7,13]) If u; u_n2L^p(x)( ) (n= 1;2; :::), we have

(i) juj^p(x) <1 (= 1;>1), (u)<1 (= 1;>1) ; (ii) juj^p(x) >1 =) juj^pp(x) (u) juj^pp(x)⁺ ; (iii) juj^p(x) <1 =) juj^pp(x)⁺ (u) juj^pp(x);

Proposition 2 [7,13] If u; u_n 2 L^p(x)( ) (n = 1;2; :::), then the following statements are equivalent:

(i) lim

n!1jun uj^p(x) = 0;

(ii) lim

n!1 (u_n u) = 0;

(iii)u_n!u in measure in and lim

n!1 (u_n) = (u): The variable exponent Sobolev space W^1;p(x)( ) is de…ned by

W^1;p(x)( ) =fu2L^p(x)( ) :jruj 2L^p(x)( )g; with the norm

kuk1;p(x) =juj^p(x)+jruj^p(x); 8u2W^1;p(x)( ):

Then (W^1;p(x)( );k k1;p(x)) becomes a Banach space. The space W₀^1;p(x)( ) is de…ned as the closure of C₀¹( ) in W^1;p(x)( ) with respect to the norm k k1;p(x). Foru2 W₀^1;p(x)( ), we can de…ne an equivalent norm

kuk=jruj^p(x);

since Poincaré inequality

juj^p(x) Cjruj^p(x); 8u2W₀^1;p(x)( ) holds, whereC is a positive constant [9].

Proposition 3 [7,13] If 1 < p and p⁺ < 1, then the spaces L^p(x)( ), W^1;p(x)( ) and W₀^1;p(x)( ) are separable and re‡exive Banach spaces.

Proposition 4 [7,13] Assume that is bounded, the boundary of possesses the cone property and p 2 C₊( ). If q 2 C₊( ) and q(x) < p (x) := _N^{N p(x)}_p(x) for all x 2 , then the embedding W^1;p(x)( ) ,! L^q(x)( ) is compact and continuous.

From[18], let X be a re‡exive and separable Banach space, then there are e_j X and e_j X such that

X =spanfe_jj j = 1;2; :::g; X =spanⁿe_j j j = 1;2; :::^o; and

he_i; e_ji=

8>

<

>:

1 i=j;

0 i6=j;

whereh:; :idenotes the duality product betweenX andX :For convenience, we write

X_j =spanfe_jg; Y_k = ^k_j=1X_j; Z_k = ¹_j=kX_j: And let

B_k =fu2Y_k :kuk kg; N_k =fu2Z_k :kuk=r_kg; for _k> r_k >0:

Let consider the C¹-functionalI :X !R de…ned by I (u) :=A(u) B(u); 2[1;2]:

Now we give the following variant Fountain theorem (see [20], Theorem 2.2), which we use in the proof of the main results of the present paper:

Theorem 5 (Variant Fountain Theorem) Assume the functional I sat- is…es the followings:

(T₁) I maps bounded sets to bounded sets uniformly for 2[1;2].

Moreover, I ( u) =I (u) for all ( ; u)2[1;2] X.

(T₂) B(u) 0; B(u) ! 1 as kuk ! 1 on any …nite dimensional subspace of X.

(T₃) There exists _k > r_k >0 such that a_k( ) := inf

u2Z_k;kuk= _kI (u) 0> b_k( ) := max

u2Y_k;kuk=r_kI (u);

for all 2[1;2]and d_k( ) := inf

u2Zk;kuk k

I (u)!0as k! 1 uniformly for 2[1;2]: Then there exists n!1,u( _n)2Y_n such that

I⁰_nj^Yn(u( _n)) = 0; I _n(u( _n))!c_k 2[d_k(2); b_k(1)] as n! 1:

Particularly, if fu( n)ghas a convergent subsequence for every k, then I1 has in…nitely many nontrivial critical points fu_kg 2 Xn f0g satisfying I₁(u_k) ! 0 as k ! 1.

3 Main results

For problem (P), we make the following assumptions:

(P₁)f(x; t) = f(x; t)and g(x; t) = g(x; t) for any x2 , t2R. (P₂)Assume that f : R!R is a Carathéodory function and there exist 1< < p and c₁ >0; c₂ >0; c₃ >0 such that

c1jtj f(x; t)t c2jtj +c3jtj , for a.e. x2 and t 2R: (P₃)Assume thatg : R!Ris a Carathéodory function andp; q 2C₊ with p(x) p⁺ < q q(x)< p (x)such that

jg(x; t)j c 1 +jtj^{q(x) 1} , for a.e.x2 and t2R;

and g(x; t)t 0;for a.e. x2 and t 2R:Moreover, lim

t!0 g(x;t)

t^{p 1} = 0 uniformly for x2 .

(P₄)Assume one of the following conditions holds:

(1) lim

jtj!1

g(x; t)

t^{p 1} = 0 uniformly for x2 :

(2) lim

jtj!1

g(x; t)

t^{p 1} = 1uniformly for x2 :

Moreover, _t^f(x;t)_{p 1} and _t^g(x;t)_{p 1} are decreasing in t2R for t large enough.

(3) lim inf

jtj!1

g(x; t)t G(x; t)

jtj c >0 uniformly for x2 ; where > and >0. Moreover, lim

jtj!1 g(x;t)

t^{p 1} =1uniformly forx2 ; ^g(x;t)

t^{p 1}

is increasing in t2R for t large enough.

Theorem 6 Assume that (P₁)-(P4) hold, then problem (P) has in…nitely many solutions fukg satisfying

(u_k) :=

Z 1

p(x)jru_kj^p(x)dx

Z

G(x; u_k)dx

Z

F(x; u_k)dx!0 as k ! 1;

where :W₀^1;p(x)( )!R is the functional corresponding to problem(P) and G(x; t) =^R₀^tg(x; s)ds,F(x; t) = ^R₀^tf(x; s)ds.

Remark 7 The conditions (P₂) and (P₃) imply the functional is well de-

…ned and of class C¹. It is well known that the critical points of are weak solutions of (P). Moreover, the derivative of is given by

h ⁰(u); i=

Z

jruj^{p(x) 2}rur dx

Z

g(x; u) dx

Z

f(x; u) dx;

for any u; 2W₀^1;p(x)( ).

Let us considerC¹-functional :W₀^1;p(x)( )!R de…ned by (u) =

Z 1

p(x)jruj^p(x)dx

Z

G(x; u)dx

Z

F(x; u)dx:=A(u) K(u) B(u), where 2 [1;2]. Then B(u) 0 and B(u) ! 1 as kuk ! 1 on any …nite dimensional subspace, wheren > k > 2:

To get the proof of Theorem 6, we will apply Theorem 5. Therefore, it is enough to obtain the results of Lemma 8 and Lemma 9.

Lemma 8 Under the assumptions of Theorem 6, there exist n!1; u_n( )2 Y_n such that

0

n j^Yn (u_n( )) = 0; _n(u_n( ))!c_k2[d_k(2); b_k(1)] as n! 1:

PROOF. First, we prove that for somer_k 2(0; _k) such that b_k( ) := max

u2Yk;kuk=rk

(u)<0;

for 2 [1;2], u 2 Y_k. The norms j j and k k is equivalent on the …nite dimensional subspace Y_k. Therefore, there is a constant c >0 such that

juj ckuk; 8u2Y_k:

Moreover, by (P₃), for any " > 0 there exists C_" > 0 such that jG(x; u)j

"juj^p +C_"juj^q(x). Then, by (P₂)and Proposition 1, we have

(u) 1

p kuk^p K(u) B(u) 1

p kuk^p "

Z

juj^p dx C_"

Z

juj^q(x)dx c₁

Z

juj dx 1

p kuk^p "c^p kuk^p C"kuk^q+ c₄kuk :

Since < p < q⁺, forkuksmall enough we getb_k( ) := max

u2Yk;kuk=rk

(u)<0 for all u2Y_k.

Second, we shall show that for some0< r_k< _k such that a_k( ):= inf

u2Z_k;kuk= _k (u) 0 for 2[1;2], and u2Z_k.

Let

k(q(x)) : = sup

u2Zk;kuk=1jujq(x); _k p := sup

u2Zk;kuk=1jujp ;

k( ) : = sup

u2Zk;kuk=1juj ; _k( ) := sup

u2Zk;kuk=1juj :

Then _k(q(x))!0, _k(p ) !0; _k( ) !0 and _k( )!0 as k ! 1 (see [10]). Therefore, by(P₂) and Proposition 1, we have

(u) =A(u) K(u) B(u) 1

p⁺kuk^p+ K(u) B(u) 1

p⁺kuk^p+ "

Z

juj^p dx C_"

Z

juj^q(x)dx c₂

Z

juj dx c₃

Z

juj dx 1

p⁺kuk^p+ cjuj^pp cjuj^qq(x) cjuj cjuj 1

p⁺kuk^p+ c ^p_k p kuk^p c ^q_k (q(x))kuk^q c _k( )kuk c _k( )kuk where c = maxf"; C_";2c₂;2c₃g. Let ' 2 Z_k, k'k = 1 and 0 < t <1; then it

follows

(t') 1

p⁺t^p+ c ^p_k p t^p c ^q_k (q(x))t^q c _k( )t c _k( )t 1

p⁺t^q c ^q_k (q(x))t^q c ^p_k p +c _k( ) +c _k( ) t ; since < < p < p⁺ < q for su¢ ciently largek, by choosingc ^q_k (q(x))<

1

2p⁺, we get

(t') 1

2p⁺t^q c ^p_k p +c _k( ) +c _k( ) t : (3:1) Put _k := 2cp^{+ p}_k (p ) + 2cp⁺ _k( ) + 2cp⁺ _k( )

1

q , then, for su¢ ciently large k, _k <1. When t = _k, '2Z_k with k'k= 1, we have (t') 0. So, for su¢ ciently largek, we obtaina_k( ) := inf

u2Zk;kuk= _k (u) 0.

Finally, we prove

d_k( ) := inf

u2Zk;kuk k

(u)!0

ask ! 1 uniformly. Indeed, sinceY_k\Z_k 6=? and r_k < _k, we have

d_k( ) := inf

u2Zk;kuk k

(u) b_k( ) := max

u2Yk;kuk=rk

(u)<0:

By(3:1), for' 2Z_k, k'k= 1,0 t _k and u=t'it follows that

(u) = (t') 1

2p⁺t^q - c ^p_k p +c _k( ) +c _k( ) t c ^p_k p +c _k( ) +c _k( ) t

c ^p_k p +c _k( ) +c _k( ) _k c ^p_k p +c _k( ) +c _k( ) ;

therefored_k( )!0ask ! 1. Hence, by Theorem 5 we can …nd n!1and u_n( )2Y_n desired as the claim. The proof is completed. 2

Lemma 9 fu_n( )g¹n=1 is bounded in W₀^1;p(x)( ).

PROOF. Since ⁰_n j^Yn (u( _n)) = 0, then we have

0 (u( _n)) =A⁰(u( _n)) K⁰(u( _n)) _nB⁰(u( _n)) =o(1)ku( _n)k; or, by Proposition 1,

1 o(1) = _n

Z f(x; u( _n))u( _n) (u( n)) dx+

Z g(x; u( _n))u( _n) (u( n)) dx

n

Z f(x; u( _n))u( _n) ku( n)k^p dx+

Z g(x; u( _n))u( _n) ku( n)k^p dx

where (u( _n)) is de…ned as in(2:1). Passing to a subsequence, if necessary, ku( _n)k ! 1 as n! 1, and using (P₂) it follows

1 o(1)

Z g(x; u( _n))u( _n) ku( _n)k^p dx;

where o(1) ! 0 as n ! 1. This is a contradiction providing that (P₄) (1) holds.

Let f!_ng W₀^1;p(x)( ) and put !_n := ^{u( n)}

ku( n)k. Since k!_nk = 1, up to subse- quences, from Proposition 4 we get

!_n* ! inW₀^1;p(x)( );

!_n!! inL ^(x)( ); p(x) (x)< p (x);

!_n(x)!!(x) a.e. x2 :

Then the main concern is that either f!_ng W₀^1;p(x)( ) vanish or it does not vanish. We shall prove that none of these alternatives can occur and this contradiction will prove thatf!_ng W₀^1;p(x)( ) is bounded.

If ! 6= 0, from Proposition 1, Fatou’s Lemma, (P₂);(P₃) and for n large enough, we have

0 (u( _n)) =A⁰(u( _n)) K⁰(u( _n)) _nB⁰(u( _n)) =o(1)ku( _n)k; or

1 +o(1) = _n

Z f(x; u( _n))u( _n) (u( _n)) dx+

Z g(x; u( _n))u( _n) (u( _n)) dx

n

Z f(x; u( _n))u( _n) ku( _n)k^p dx+

Z g(x; u( _n))u( _n) ku( _n)k^p dx:

Using lim

juj!1 g(x;u)

juj^{p 1} = 1 in (P₄) (2), we get

1 +o(1)

Z g(x; u( _n))u( _n) ku( _n)k^p dx=

Z g(x; u( _n))u( _n)

ju( _n)j^p j!_nj^p dx c+

Z

f!6=0g\fju( n)j cg

g(x; u( _n))u( _n)

ju( _n)j^p j!_nj^p dx! 1; which is a contradiction. Moreover, we can get the similar result if lim

juj!1 g(x;u) juj^{p 1} = 1 in(P₄) (3).

If! 0, we can de…ne a sequence ft_ng R as in (see [17] ) such that

n(t_nu( _n)) := max

t2[0;1] ⁿ(tu( _n)):

Let!_n:= (2p⁺c)^p1 !_n with c >0. Then forn large enough, we have

n(t_nu_n) _n(!_n) A (2p⁺c)

1

p !n K (2p⁺c)

1

p !n nB (2p⁺c)

1 p !n

1

p⁺(2p⁺c)A(!_n) K(!_n) _nB(!_n) 2c K(!_n) _nB(!_n) c;

which implies that lim

n!1 ⁿ(t_nu_n)! 1 by the fact c >0 can be large arbi- trarily. Noting that _n(0) = 0and _n(u_n)!c, so0< t_n<1when n large enough. Hence we have h ⁰n(t_nu( _n)); t_nu( _n)i= 0. Thus, it follows

nlim!1[ _n(tnu( n)) 1 p_tn

D 0

n(tnu( n)); tnu( n)^E]! 1; where p_tn = ^A_A(t^{0(tnu( n))}

nu( n)). Therefore,

nlim!1[(A(t_nu( _n)) K(t_nu( _n)) _nB(t_nu( _n)) 1

p_tn(A⁰(t_nu( _n)) + 1

p_tnK⁰(t_nu( _n)) + _n 1

p_tnB⁰(t_nu( _n))]! 1;

that is,

nlim!1[ n

1

p_tnB⁰(tnu( n)) nB(tnu( n))+ 1

p_tnK⁰(tnu( n)) K(tnu( n))]! 1: Moreover, if (P₄) (2) holds, we have

1

p_tnf(x; su)su F (x; su) + 1

p_tng(x; su)su G(x; su) c;

for all s >0 and u2R, so we get a contradiction.

If(P₄) (3)holds, by (P₂), we get 1 c₂

p_n

Z

ju( _n)j dx+ 1

p_nK⁰(u( _n)) K(u( _n)): Thus,

1

p_nK⁰(u( _n)) K(u( _n))! 1: (3:2) Furthermore, using the property ofu( _n)(see Lemma 8), it follows that

b_k(1) _n 1

p_nB⁰(u( _n)) B(u( _n))

!

+ 1

p_nK⁰(u( _n)) K(u( _n)) 1

p_n 1

p_nK⁰(u( _n)) K(u( _n))

! c₂ p_n

Z

ju( _n)j dx c1

p_nK⁰(u( _n)) K(u( _n)) c;

which contradicts (3:2). Therefore fu( _n)g is bounded. The proof is com- pleted. 2

Acknowledgments

The author would like to thank Prof. Dr. R.A. Mashiyev for his generous advice and support. 2

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