p1,...,pn

Bull. Korean Math. Soc. 50 (2013), No. 1, pp. 57–71 http://dx.doi.org/10.4134/BKMS.2013.50.1.057

EXISTENCE OF THREE SOLUTIONS FOR A CLASS OF NAVIER QUASILINEAR ELLIPTIC SYSTEMS INVOLVING THE (p1 , . . . , pn )-BIHARMONIC Lin Li Abstract. In this paper, we establish the existence of at least three solutions to a Navier boundary problem involving the (p1 , . . . , pn )-biharmonic systems. We use a variational approach based on a three critical points theorem due to Ricceri [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084–3089].

1. Introduction and main results In this work, we study the existence of at least three weak solutions for the nonlinear elliptic equation of (p1 , . . . , pn )-biharmonic type under Navier boundary conditions: (P 1 )  in Ω, −∆ |∆u1 |p1 −2 ∆u1 = λFu1 (x, u1 , . . . , un ) + µGu1 (x, u1 , . . . , un )    p2 −2   (x, u , . . . , u ) in Ω, (x, u , . . . , u ) + µG = λF −∆ |∆u | ∆u 1 n 1 n u2 u2 2 2  ···    −∆ |∆un |pn −2 ∆un = λFun (x, u1 , . . . , un ) + µGun (x, u1 , . . . , un ) in Ω,    u = ∆u = 0 for 1 ≤ i ≤ n, on ∂Ω, i i

where λ, µ ∈ [0, +∞), Ω ⊂ RN (N ≥ 1) is a non-empty bounded open set with a sufficient smooth boundary ∂Ω, pi > max 1, N2 for 1 ≤ i ≤ n. F , G : Ω × Rn 7→ R are functions such that F (·, t1 , . . . , tn ), G(·, t1 , . . . , tn ) are measurable in Ω for all (t1 , . . . , tn ) ∈ Rn and F (x, ·), G(x, ·) are continuously

Received March 12, 2011. 2010 Mathematics Subject Classification. 35J48, 35J60, 47J30, 58E05. Key words and phrases. (p1 , . . . , pn )-biharmonic, Navier condition, multiple solutions, three critical points theorem. Supported by the Fundamental Research Funds for the Central Universities (No. XDJK2013D007), Scientific Research Fund of SUSE (No. 2011KY03) and Scientific Research Fund of SiChuan Provincial Education Department (No. 12ZB081). c

2013 The Korean Mathematical Society

57

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LIN LI

differentiable in Rn for a.e. x ∈ Ω. Moreover, G satisfies the condition (G)

sup

n X

|Gti (x, t1 , . . . , tn )| ≤ hs (x)

|(t1 ,...,tn )|≤s i=1

for all s > 0 and some hs ∈ L1 with G(·, 0, . . . , 0) ∈ L1 . Fi denotes the partial derivative of F with respect to i, 1 ≤ i ≤ n, so does Gi . In recent years, the three critical points theorem of B. Ricceri has been widely used to solve differential equations (see [2, 4, 6, 8, 10, 13, 14, 15, 21, 23] and references therein). Using the three critical points theorem, some authors have considered the elliptic systems. In [12], Li and Tang get three solutions for a class of quasilinear systems involving the (p, q)-Laplacian with Dirichlet boundary condition. Afrouzi and Heidarkhani [1] unify and generalize Li and Tang’s problem. In [8], El Manouni and Kbiri Alaoui consider (p, q)-Laplacian systems with Neumann conditions via Ricceri’s three critical points theorem. Li and Tang [14] consider a (p, q)-biharmonic system under Navier boundary condition. In [10], Heidarkhani and Tian study a class of gradient systems depending on two parameters, they get three solutions using Ricceri’s three critical point theorem. Later, Heidarkhani and Tian [11] using the same method study a class of gradient Kirchhoff-type systems depending on two parameters. Graef, Heidarkhani and Kong [9] get multiplicity results for multi-point boundary value problems. There seems to be increasing interest in studying fourth-order boundary value problems, because the static form change of beam or the sport of rigid body can be described by a fourth-order equation, and specially a model to study traveling waves in suspension bridges can be furnished by the fourth-order equation of nonlinearity, so it is important to Physics. More general nonlinear fourth-order elliptic boundary value problems have been studied [5, 7, 16, 17]. Particularity, Li and Tang [13] consider the p-harmonic equation with Navier boundary condition. Using the three critical points theorem of B. Ricceri, they get at least three solutions. Recently, Li and Tang [14] also use three critical points theorem to study a class of (p, q)-biharmonic systems. Here, as in [14], our main tool is Ricceri’s three critical points theorem; see Theorem 2.1 in the next section. We also recall that, again applying Ricceri’s three critical points theorem, elliptic systems have been studied in [3, 4, 9, 10, 11]. The aim of the present paper is to extend the main result of [14] to the general case. In this paper, precisely we deal with the existence of an open interval Λ ⊆ [0, +∞) and a positive real number ρ with the following property: for every λ ∈ Λ and an arbitrary function G : Ω × Rn → R measurable in Ω for all (t1 , . . . , tn ) ∈ Rn and C 1 in Rn for every x ∈ Ω satisfying (G), there is a δ > 0, such that, for each µ ∈ [0, δ] the system (P1 ) admits at least three weak solutions in W 2,p1 (Ω) ∩ W01,p1 (Ω) × · · · × W 2,pn (Ω) ∩ W01,pn (Ω) whose norms are less than ρ. Our main result is Theorem 1.1, which provides intervals for

THREE SOLUTIONS FOR (p1 , . . . , pn )-BIHARMONIC SYSTEMS

59

the parameters such that if the parameters belong to those intervals, the corresponding system has at least three solutions satisfying some boundedness properties. Our results generalize and unifies the results in [14]. Here in the sequel, X will be denoted the Sobolev space W 2,p1 (Ω)∩W01,p1 (Ω) 2,p1 (Ω) ∩ and E will be denoted the Cartesian productof Sobolev spaces W

W01,p1 (Ω), . . . , W 2,pn (Ω) ∩ W01,pn (Ω), i.e., E = W 2,p1 (Ω) ∩ W01,p1 (Ω) × · · · × W 2,pn (Ω) ∩ W01,pn (Ω) . The space E will be endowed with the norm k(u1 , . . . , un )k = ku1 kp1 +· · ·+kun kpn , kui kpi =

Z

Ω

p1 i , 1 ≤ i ≤ n. |∆ui | dx pi

Let (1)

(

supx∈Ω |u(x)|pi K = max sup 1,p kukppii u∈W 2,pi (Ω)∩W i (Ω)\{0} 0

)

, 1 ≤ i ≤ n.

Since pi > max 1, N2 , W 2,pi (Ω) ∩ W01,pi (Ω) ֒→ C 0 (Ω), 1 ≤ i ≤ n, are compact, and one has K < +∞. As usual, a weak solution of problem (P1 ) is any (u1 , . . . , un ) ∈ E such that n Z X |∆ui |pi −2 ∆ui ∆ξi dx −

(2)

=

i=1 Ω n X Z

Fui (x, u1 , . . . , un )ξi dx +

λ

i=1

Ω

Z n X µ Gui (x, u1 , . . . , un )ξi dx i=1

Ω

for every (ξ1 , . . . , ξn ) ∈ E. Now, for every x0 ∈ Ω and choice r1 , r2 with r2 > r1 > 0, such that B(x0 , r1 ) ⊂ B(x0 , r2 ) ⊆ Ω, where B(x0 , r1 ) denotes the ball with center at x0 and radius of r1 , put (3)  N p1  i  Kπ 2 ((r2 +r1 )N −(2r1 )N ) 1  , N < r24r  (r2 −r13N )(r2 +r1 ) −r1 , 2N Γ(1+ N 2 ) θi = 1 ≤ i ≤ n, 1 p N  i  Kπ 2 ((r2 +r1 )N −(2r1 )N ) 4r1 12r1  , N ≥ r2 −r1 ,  (r2 −r1 )2 (r2 +r1 ) 2N Γ(1+ N 2 ) where Γ(·) is the Gamma function. Our main result gives the following theorems.

Theorem 1.1. Suppose that r2 > r1 > 0, such that B(x0 , r2 ) ⊂ Ω and assume that there exist n + 2 positive constants c, d, si for 1 ≤ i ≤ n with si < pi , Pn (dθi )pi > Qnc pi , and a negative function α ∈ L1 (Ω) such that i=1 pi i=1

(j1 ) F (x, t1 , . . . , tn ) ≤ 0 for a.e. x ∈ Ω\B(x0 , r1 ) and all (t1 , . . . , tn ) ∈ [0, d] × · · · × [0, d];

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Pn Ri=1

(dθi )pi pi

inf (x,t1 ,...,tn )∈Ω×A F (x, t1 , . . . , tn ) > . . . , d)dx, B(x0 ,r1 ) F (x, d, i=1 pi Pn |ti |pi where A = (t1 , . . . , tn ) i=1 pi ≤ Qnc pi ; i=1 Pn si (j3 ) F (x, t1 , . . . , tn ) ≥ α(x)(1 + i=1 |ti | ) for a.e. x ∈ Ω and all ti ∈ R, 1 ≤ i ≤ n. Then there exist an open interval Λ ⊆ [0, +∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for every Carathéodory functions Gti : Ω×Rn 7→ R, satisfying (G), there exists δ > 0 such that, for each µ ∈ [0, δ], problem (P1 ) has at least three solutions whose norms in E are less than ρ. (j2 )

m(Ω) Qn c

Let f be a continuous function in Ω and gi be a C 1 function for 1 ≤ i ≤ n and ! n Y gi (ui ) F (x, u1 , . . . , un ) = f (x) i=1

n

for each (x, u1 , . . . , un ) ∈ Ω × R . More precisely, we consider the following problem  Q n  −∆ |∆u1 |p1 −2 ∆u1 = λf (x)g1′ (u1 ) gi (ui )  i=1,  i6 = 1     in Ω, +µGu1 (x, u1 , . . . , un ),  Q   n   −∆ |∆u2 |p2 −2 ∆u2 = λf (x)g2′ (u2 ) i=1,i6=2 gi (ui )    in Ω, +µGu2 (x, u1 , . . . , un ), (P2 )  · · ·   Q   n  ′ pn −2  g (u ) = λf (x)g (u ) −∆ |∆u | ∆u i i n n n  n i=1,i6=n     (x, u , . . . , u ), in Ω, +µG  1 n un   u = ∆u = 0 for 1 ≤ i ≤ n, on ∂Ω. i i

Then, by using Theorem 1.1, we have the following result:

Corollary 1.1. Assume that there exist n + 2 positive constants c, d, and si pi Pn for 1 ≤ i ≤ n with i=1 (dθpii) > Qnc pi , si < pi for 1 ≤ i ≤ n and a negative i=1 function α(x) ∈ L1 (Ω) such that (k1 ) f (x) ≤ 0 for each x ∈ Ω\B(x0 , r1 ) and gi ≤ 0 for ti ∈ [0, d], 1 ≤ i ≤ n; pi Qn Pn (k2 ) m(Ω) i=1 (dσpii) inf (t1 ,...,tn )∈A f (x) i−1 gi (ti ) R Qn f (x)dx, > c i=1 gip(d) B(x0 ,r1 ) i Pn |ti |pi c Q where A = (t1 , . . . , tn ) i=1 pi ≤ n pi ; i=1 Pn Qn (k3 ) f (x) i=1 gi (ti ) ≥ α(x)(1 + i=1 |ti |si ) for all ti ∈ R, 1 ≤ i ≤ n and a.e. x ∈ Ω. Then there exist an open interval Λ ⊆ [0, +∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for every Carathéodory functions


61

Gti : Ω × Rn 7→ R for 1 ≤ i ≤ n, satisfying (G), there exists δ > 0 such that, for each µ ∈ [0, δ], problem (P2 ) has at least three solutions whose norms in E are less than ρ. Here is a remarkable consequence of Theorem 1.1. Consider the problem (P3 )  −∆ |∆u1 |p1 −2 ∆u1 = λFu1 (u1 , . . . , un ) + µGu1 (x, u1 , . . . , un ), in Ω,     −∆ |∆u2 |p2 −2 ∆u2 = λFu2 (u1 , . . . , un ) + µGu2 (x, u1 , . . . , un ), in Ω,  ···    −∆ |∆un |pn −2 ∆un = λFun (u1 , . . . , un ) + µGun (x, u1 , . . . , un ), in Ω,    u = ∆u = 0 for 1 ≤ i ≤ n, on ∂Ω. i i

Now we state another theorem.

Theorem 1.2. Let F : Rn 7→ R be a C 1 function and there exist n + 2 positive pi Pn constants c, d, si for 1 ≤ i ≤ n and a negative constant a with i=1 (dθpii) > Qn c pi , si < pi for 1 ≤ i ≤ n such that i=1

(l1 ) F (t1 , . . . , tn ) ≤ 0 for all (t1 , . . . , tn ) ∈ [0, d] × · · · × [0, d]; pi Pn (l2 ) m(Ω) i=1 (dσpii) inf (t1 ,...,tn )∈A F (t1 , . . . , tn ) N

>

cr1N π 2 Qn F (d, . . . , d), Γ(1+ N i=1 pi 2 )

Pn |ti |pi c Q where A = (t1 , . . . , tn ) i=1 pi ≤ n pi ; i=1 P (l3 ) F (t1 , . . . , tn ) ≥ a(1 + ni=1 |ti |si ) for all ti ∈ R, 1 ≤ i ≤ n. Then there exist an open interval Λ ⊆ [0, +∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for some Carathéodory functions Gti : Ω×Rn 7→ R, satisfying (G), there exists δ > 0 such that, for each µ ∈ [0, δ], problem (P3 ) has at least three solutions whose norms in E are less than ρ. If N = 1, we can get a better result than Theorem 1.2. For simplicity, fixing Ω =]0, 1[, pi > 1, put 1 −1 (4) k = max p , i = 1, . . . , n 2pi i

then we have the following result.

Theorem 1.3. Let F : Rn 7→ R be a C 1 function and assume that there exist pi P n + 2 positive constants c, d, si and a negative constant a with ni=1 (32d) 2Kpi > Qn c pi for 1 ≤ i ≤ n, where k is given by (4), such that i=1

(m1 ) F (t1 , . . . , tn ) ≤ 0 for all (t1 , . . . , tn ) ∈ [0, d] × · · · × [0, d]; Pn (32d)pi Qc F (d, . . . , d), (m2 ) i=1 2kpi inf (t1 ,...,tn )∈A F (t1 , . . . , tn ) > 2 n i=1 pi Pn |ti |pi where A = (t1 , . . . , tn ) i=1 pi ≤ Qnc pi ; i=1 Pn (m3 ) F (t1 , . . . , tn ) ≥ a(1 + i=1 |ti |si ) for all ti ∈ R, 1 ≤ i ≤ n.

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Then there exist an open interval Λ ⊆ [0, +∞) and a positive real number ρ with the following property: for each λ ∈ Λ and for every Carathéodory functions Gti : Ω×Rn 7→ R, satisfying (G), there exists δ > 0 such that, for each µ ∈ [0, δ], problem (P4 ) ′′  − |u′′1 |p1 −2 u′′1 = λFu1 (u1 , . . . , un ) + µGu1 (x, u1 , . . . , un ), in ]0, 1[,    ′′ p2 −2 ′′ ′′   u2 = λFu2 (u1 , . . . , un ) + µGu2 (x, u1 , . . . , un ), in ]0, 1[, − |u2 | ···  ′′   − |u′′n |pn −2 u′′n = λFun (u1 , . . . , un ) + µGun (x, u1 , . . . , un ), in ]0, 1[,    u (0) − u (1) = u′′ (0) − u′′ (1) = 0 for 1 ≤ i ≤ n, i i i i has at least three solutions whose norms in W 2,p1 (0, 1) ∩ W01,p1 (0, 1) × · · · × W 2,pn (0, 1) ∩ W01,pn (0, 1) are less than ρ. 2. Proof of theorems Our analysis is based on the following three critical points theorem to transfer the existence of three solutions of the system (P1 ) into the existence of critical points of the Euler functional. Theorem 2.1 ([19], Theorem 1). Let X be a reflexive real Banach space. Φ : X 7→ R is a continuously Gˆ ateaux differentiable and sequentially weakly lower semicontinuous functional whose Gˆ ateaux derivative admits a continuous inverse on X ∗ and Φ is bounded on each bounded subset of X; Ψ : X 7→ R is a continuously Gˆ ateaux differentiable functional whose Gˆ ateaux derivative is compact; I ⊆ R an interval. Assume that lim

(Φ(x) + λΨ(x)) = +∞

kxk→+∞

for all λ ∈ I, and that there exists h ∈ R such that (5)

sup inf (Φ(x) + λ(Ψ(x) + h)) < inf sup(Φ(x) + λ(Ψ(x) + h)). λ∈I x∈X

x∈X λ∈I

Then, there exists an open interval Λ ⊆ I and a positive real number ρ with the following property: for every λ ∈ Λ and every C 1 functional J : X 7→ R with compact derivative, there exists δ > 0 such that, for each µ ∈ [0, δ] the equation Φ′ (x) + λΨ′ (x) + µJ ′ (x) = 0 has at least three solutions in X whose norms are less than ρ. For using later, we also recall the following result, Proposition 3.1 of [18]. Proposition 2.1 ([18], Proposition 3.1). Let X be a non-empty set and Φ, Ψ two real functions on X. Assume that there are r > 0 and x0 , x1 ∈ X such that Φ(x0 ) = Ψ(x0 ) = 0,

Φ(x1 ) > r,

inf

x∈Φ−1 (]−∞,r])

Ψ(x) > r

Ψ(x1 ) . Φ(x1 )


63

Then, for each h satisfying inf

x∈Φ−1 (]−∞,r])

Ψ(x) > h > r

Ψ(x1 ) , Φ(x1 )

one has sup inf (Φ(x) + λ(h + Ψ(x))) < inf sup(Φ(x) + λ(h + Ψ(x))). λ≥0 x∈X

x∈X λ≥0

Before giving the proof of Theorem 1.1, let us see the following two lemmas. Lemma 2.1. Assume that there exist two positive constants c, d with Pn (dθi )pi > Qnc pi , such that i=1 pi i=1

(j1 ) F (x, t1 , . . . , tn ) ≤ 0 for a.e. x ∈ Ω\B(x0 , r1 ) and all (t1 , . . . , tn ) ∈ [0, d] × · · · × [0, d]; pi Pn (j2 ) m(Ω) i=1 (dθpii) inf (x,t1 ,...,tn )∈Ω×A F (x, t1 , . . . , tn ) R > Qnc pi B(x0 ,r1 ) F (x, d, . . . , d)dx, i=1 Pn pi where A = (t1 , . . . , tn ) i=1 |tpi |i ≤ Qnc pi ; i=1

Then there exist u∗i ∈ W 2,pi (Ω) ∩ W01,pi (Ω) for 1 ≤ i ≤ n, such that n X ku∗i kppi i

i=1

and

pi

1 > Qn

c i=1 pi K

R c Ω F (x, u∗1 (x), . . . , u∗n (x))dx Pn Qn , m(Ω) inf F (x, t1 , . . . , tn ) > ∗ pi K (x,t1 ,...,tn )∈Ω×A i=1 j=1,j6=i pj kui kpi Pn |ti |pi c Q where A = (t1 , . . . , tn ) i=1 pi ≤ n pi . i=1

Proof. Let (6)   x ∈ Ω\B(x0 , r2 ),  0, 4 4 3 3 2 2 1 +r2 )(l −r2 )+6r1 r2 (l −r2 )) w(x) = d(3(l −r2 )−4(r , x ∈ B(x0 , r2 )\B(x0 , r1 ), (r2 −r1 )3 (r1 +r2 )   d, x ∈ B(x0 , r1 ), qP N 0 2 where u∗i (x) = w(x) for 1 ≤ i ≤ n and l = dist(x, x0 ) = i=1 (xi − xi ) . We have ( 0, x ∈ Ω\B(x0 , r2 )∪B(x0 , r1 ), ∂u∗i (x) = 12d(l2 (xi −x0i )−(r1 +r2 )l(xi −x0i )+r1 r2 (xi −x0i )) ∂xi , x ∈ B(x0 , r2 )\B(x0 , r1 ), 3 (r2 −r1 ) (r1 +r2 )

0, x ∈ Ω\B(x0 , r2 )∪B(x0 , r1 ), ∂ 2 u∗i (x) 0 2 = 12d(r1 r2 +(2l−r1 −r2 )(xi −xi ) /l−(r2 +r1 −l)l) ∂x2i , x ∈ B(x0 , r2 )\B(x0 , r1 ), 3 (

(r2 −r1 ) (r1 +r2 )

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N X ∂ 2 u∗ (x) i

i=1

∂x2i

(

=

x ∈ Ω\B(x0 , r2 )∪B(x0 , r1 ),

0, 12d((N +2)l −(N +1)(r1 +r2 )l+N r1 r2 ) , (r2 −r1 )3 (r1 +r2 ) 2

x ∈ B(x0 , r2 )\B(x0 , r1 ).

It is easy to verify that u∗i ∈ W 2,pi (Ω) ∩ W01,pi (Ω), and, in particular, one has N

ku∗i kppii = (7)

(12d)pi 2π 2 (r2 − r1 )3pi (r1 + r2 )pi Γ( N2 ) Z r2 |(N + 2)r2 − (N + 1)(r1 + r2 )r + N r1 r2 |pi rN −1 dr. × r1

Here, we obtain from (3) and (7) that θipi dpi < ku∗i kppii for 1 ≤ i ≤ n. K

(8) By the assumption

n X (dθi )pi

pi

i=1

it follows from (8) that n X ku∗i kppi i

pi

i=1

Since 0 ≤

u∗i

c > Qn

i=1

n X dpi θpi

1 > K

i

i=1

pi

pi

!

,

1 > Qn

c . p K i=1 i

≤ d for each x ∈ Ω, 1 ≤ i ≤ n, the condition (j1 ) ensures that Z F (x, u∗1 (x), . . . , u∗n (x))dx Ω\B(x0 ,r2 ) Z F (x, u∗1 (x), . . . , u∗n (x))dx ≤ 0. + B(x0 ,r2 )\B(x0 ,r1 )

Hence, by condition (j2 ) and (8), we have m(Ω) > Q n

i=1

>

K

c ≥ K

inf

F (x, t1 , . . . , tn ) Z F (x, d, . . . , d)dx pi

(x,t1 ,...,tn )∈Ω×A

pi

c Qn

i=1

c Pn

d pi θi pi

i=1

1

pi P n

i=1

B(x0 ,r1 )

pi ku∗ i kp i

pi

Z

F (x, d, . . . , d)dx

B(x0 ,r1 )

F (x, u∗1 (x), . . . , u∗n (x))dx ΩP . n Qn ∗ pi i=1 j=1,j6=i pj kui kpi

R

Lemma 2.2. Let T : E 7→ E ∗ be the operator defined by n Z X hT (u1 , . . . , un ), (ξ1 , . . . , ξn )i = |∆ui (x)|pi −2 ∆ui (x)∆ξi (x)dx i=1

Ω


65

for all (u1 , . . . , un ), (ξ1 , . . . , ξn ) ∈ E, where E ∗ denotes the dual of E. Then T admits a continuous inverse on E ∗ . Proof. Taking into account (2.2) of [20] for p ≥ 2, there exists a positive constant cp such that h|x|p−2 x − |y|p−2 y, x − yi ≥ cp |x − y|p , where h·, ·i denotes the usual inner product in RN for every x, y ∈ RN . Thus, it is easy to see that (T (u1 , . . . , un ) − T (v1 , . . . , vn ))(u1 − v1 , . . . , un − vn ) ≥ min{cp1 , . . . , cpn }

n X

kui − vi kppii

i=1

for every (u1 , . . . , un ), (v1 , . . . , vn ) ∈ E, which means that T is uniformly monotone. Therefore, since T is coercive and hemicontinuous in X (for more details, see [14, Lemma 2]), by applying Theorem 26.A of [22], we have that T admits a continuous inverse on E ∗ . Now we can give the proof of our main results. Proof of Theorem 1.1. For each (u1 , . . . , un ) ∈ E, let Φ(u1 , . . . , un ) =

n X kui kppi i

pi

i=1

Ψ(u1 , . . . , un ) =

,

Z

F (x, u1 , . . . , un )dx

Z

G(x, u1 , . . . , un )dx.

Ω

and J(u1 , . . . , un ) =

Ω

Under the condition of Theorem 1.1, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Moreover, from Lemma 2.2 the Gˆ ateaux derivative of Φ admits a continuous inverse on E ∗ . Ψ and J are continuously Gˆ ateaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of E. In particular, for each (u1 , . . . , un ), (ξ1 , . . . , ξn ) ∈ E, n Z X |∆ui |pi −2 ∆ui ∆ξi dx, hΦ′ (u1 , . . . , un ), (ξ1 , . . . , ξn )i = hΨ′ (u1 , . . . , un ), (ξ1 , . . . , ξn )i = ′

hJ (u1 , . . . , un ), (ξ1 , . . . , ξn )i =

i=1 n X

i=1 n X i=1

Ω

λ

Z

Fui (x, u1 , . . . , un )ξi dx,

Ω

µ

Z

Ω

Gui (x, u1 , . . . , un )ξi dx.

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Hence, it follows from (2) that the weak solutions of systems (P1 ) are exactly the solutions of the equation Φ′ (u1 , . . . , un ) + λΨ′ (u1 , . . . , un ) + µJ ′ (u1 , . . . , un ) = 0. Thanks to (j3 ), for each λ > 0, one has that (9)

lim

(Φ(u1 , . . . , un ) + λΨ(u1 , . . . , un )) = +∞,

k(u1 ,...,un )k→+∞

and so the first assumption of Theorem 2.1 holds. Thanks to Lemma 2.1, there exists (u∗1 , . . . , u∗n ) ∈ E such that (10)

Φ(u∗1 , . . . , u∗n ) =

n X ku∗i kppi i

i=1

and (11) m(Ω)

inf

(x,t1 ,...,tn )∈Ω×A

pi

1 > Qn

c > 0 = Φ(0, . . . , 0) p K i=1 i

F (x, t1 , . . . , tn ) >

Pn where A = (t1 , . . . , tn ) i=1

|ti |pi pi

Qn c

≤

i=1

Now, we obtain from (1) that

R c Ω F (x, u∗1 (x), . . . , u∗n (x))dx Pn Qn , ∗ pi K i=1 j=1,j6=i pj kui kpi . pi

sup |ui (x)|pi ≤ Kkui kppii for 1 ≤ i ≤ n,

x∈Ω

for each (u1 , . . . , un ) ∈ E, and then we have ) ( n n X X |ui (x)|pi kui kppii ≤K (12) sup pi pi x∈Ω i=1 i=1 for each (u1 , . . . , un ) ∈ E. Let r = that

Qn1

i=1

c pi K ,

n X kui kppi i

Φ(u1 , . . . , un ) =

i=1

by (12) one has (13)

sup x∈Ω

(

n X |ui (x)|pi i=1

pi

So, it follows from (13) and (11) that inf

for each (u1 , . . . , un ) ∈ E such

)

pi

c ≤ Qn

inf P kui kppi  Ω i ≤r (u1 ,...,un ) n i=1 pi   Z ≥ inf F (x, t1 , . . . , tn )dx  

Ω (t1 ,...,tn )∈A

i=1

pi

.

(Ψ(u1 , . . . , un )) Z F (x, u1 , . . . , un )dx 

{(u1 ,...,un )|Φ(u1 ,...,un )≤r}

=

≤ r,


≥ m(Ω)

67

inf F (x, t1 , . . . , tn ) R F (x, u∗1 (x), . . . , u∗n (x))dx c Ω Q > n Pn ku∗i kppii K i=1 pi i=1 pi R ∗ ∗ F (x, u (x), . . . , u (x))dx 1 n =r Ω Pn ku∗i kppii (x,t1 ,...,tn )∈Ω×A

i=1

pi

Ψ(u∗1 , . . . , u∗n ) =r . Φ(u∗1 , . . . , u∗n ) So, one has (14)

inf

{(u1 ,...,un )|Φ(u1 ,...,un )≤r}

(Ψ(u1 , . . . , un )) > r

Ψ(u∗1 , . . . , u∗n ) . Φ(u∗1 , . . . , u∗n )

Fixing h such that inf

(Ψ(u1 , . . . , un )) > h > r

{(u1 ,...,un )|Φ(u1 ,...,un )≤r}

Ψ(u∗1 , . . . , u∗n ) , Φ(u∗1 , . . . , u∗n )

from (10), (14) and Proposition 1, with (u01 , . . . , u0n ) = (0, . . . , 0) and (u11 , . . . , u1n ) = (u∗1 , . . . , u∗n ), we obtain (15)

sup inf (Φ(x) + λ(h + Ψ(x))) < inf sup(Φ(x) + λ(h + Ψ(x))), λ≥0 x∈X

x∈X λ≥0

and so the assumption (5) of Theorem 2.1 holds. Now, all assumptions of Theorem 2.1 are satisfied. Hence, applying Theorem 2.1, and taking into account that the critical points of the functional Φ+λΨ+µJ are exactly the weak solutions of the system (P1 ), we have the conclusion. Proof of Theorem 1.2. From (l2 ) and since Z N π2 F (d, . . . , d)dx = r1N Γ(1 + B(x0 ,r1 ) we have n X (dθi )pi m(Ω) pi i=1

inf

(t1 ,...,tn )∈A

N 2)

c F (t1 , . . . , tn ) > Qn

i=1

F (d, . . . , d),

pi

Z

F (d, . . . , d)dx.

B(x0 ,r1 )

So, we have the conclusion by Theorem 1.1.

Proof of Theorem 1.3. For each (u1 , . . . , un ) ∈ W 2,p1 (0, 1) ∩ W01,p1 (0, 1) × · · · × W 2,pn (0, 1) ∩ W01,pn (0, 1), let Φ(u1 , . . . , un ) = Ψ(u1 , . . . , un ) =

n X kui kppi i

i=1 Z 1 0

pi

,

F (u1 , . . . , un )dx,

68

LIN LI

J(u1 , . . . , un ) =

Z

G(x, u1 , . . . , un )dx. Ω

Under the condition, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Moreover, from Lemma 2.2 the Gâteaux derivative of Φ admits a continuous inverse on X ∗ . Ψ and J are continuously Gˆ ateaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of W 2,p1 (0, 1) ∩ W01,p1 (0, 1) × · · · × W 2,pn (0, 1) ∩ W01,pn (0, 1). Hence, it is well known that the weak solutions of systems are exactly the solutions of the equation Φ′ (u1 , . . . , un ) + λΨ′ (u1 , . . . , un ) + µJ ′ (u1 , . . . , un ) = 0. Thanks to (m3 ), for each λ > 0, one has that (16)

lim

(Φ(u1 , . . . , un ) + λΨ(u1 , . . . , un )) = +∞,

k(u1 ,...,un )k→+∞

and so the first assumption of Theorem 2.1 holds. Now, let us consider 1 c . r = Qn K p i=1 i

We obtain from (1) that

sup |ui (x)|pi ≤ Kkui kppii for 1 ≤ i ≤ n

x∈(0,1)

for each (u1 , . . . , un ) ∈ W 2,p1 (0, 1)∩W01,p1 (0, 1)×· · ·×W 2,pn (0, 1)∩W01,pn (0, 1), and then we have ( n ) n X |ui (x)|pi X kui kppii (17) sup ≤K pi pi x∈(0,1) i=1 i=1

for each (u1 , . . . , un ) ∈ W 2,p1 (0, 1)∩W01,p1 (0, 1)×· · ·×W 2,pn (0, 1)∩W01,pn (0, 1). Hence, for each (u1 , . . . , un ) ∈ W 2,p1 (0, 1) ∩ W01,p1 (0, 1) × · · · × W 2,pn (0, 1) ∩ W01,pn (0, 1) such that Φ(u1 , . . . , un ) =

n X kui kppi i

i=1

pi

≤ r,

by (17) one has (18)

sup x∈(0,1)

(

n X |ui (x)|pi i=1

pi

)

c ≤ Qn

i=1

pi

.

Now if we put u∗i (x) = w(x), where ( d − 16d( 41 − |x − 12 |)2 , x ∈ [0, 41 ]∪] 43 , 1], (19) w(x) = d, x ∈] 14 , 34 ],


69

it is easy to verify that (u∗1 , . . . , u∗n ) ∈ W 2,p1 (0, 1)∩W01,p1 (0, 1)×· · ·×W 2,pn (0, 1) ∩W01,pn (0, 1) and get ku∗i kppii =

(20)

Now, under the assumption of (21)

Φ(u∗1 , . . . , u∗n ) =

(32d)pi . 2

(32d)pi i=1 2Kpi

Pn

n X ku∗i kppi i

pi

i=1

Qn c

>

i=1

pi ,

we have

1 > Qn

c > 0 = Φ(0, . . . , 0). K p i=1 i

Moreover, 0 ≤ u∗i ≤ d, it follows from (m1 ), (m2 ) and (20) that (22) c P F (d, . . . , d) inf F (t1 , . . . , tn ) > Q n n (32d)pi (t1 ,...,tn )∈A 2 i=1 pi i=1 2Kpi Z 1 1 c F (u∗1 (x), . . . , u∗n (x))dx > Qn p K i=1 pi Pn ku∗i kpii 0 i=1 pi R1 ∗ c 0 F (u1 (x), . . . , u∗n (x))dx Pn Qn ≤ ∗ pi , K i=1 j=1,j6=i pj kui kpi

Pn where A = (t1 , . . . , tn ) i=1

|ti |pi pi

≤

Qn c

i=1

pi

So, it follows from (18) and (22) that inf

{(u1 ,...,un )|Φ(u1 ,...,un )≤r}

=

.

(Ψ(u1 , . . . , un )) Z

1

F (u1 , . . . , un )dx inf  P kui kppi  0 n i (u1 ,...,un ) i=1 ≤r pi   Z 1 ≥ inf F (t1 , . . . , tn )dx  

0 (t1 ,...,tn )∈A

(23)

≥ >

= =

inf

(t1 ,...,tn )∈A

c Qn

K R1

i=1

F (t1 , . . . , tn )

R1 0

pi

F (u∗1 (x), . . . , u∗n (x))dx Pn ku∗i kppii i=1

pi

F (u∗1 (x), . . . , u∗n (x))dx r 0 Pn ku∗i kppii i=1 pi Ψ(u∗1 , . . . , u∗n ) r . Φ(u∗1 , . . . , u∗n )

70

LIN LI

Fixing h such that inf

(Ψ(u1 , . . . , un )) > h > r

{(u1 ,...,un )|Φ(u1 ,...,un )≤r}

Ψ(u∗1 , . . . , u∗n ) , Φ(u∗1 , . . . , u∗n )

from (21), (23) and Proposition 1, with (u01 , . . . , u0n ) = (0, . . . , 0) and (u11 , . . . , u1n ) = (u∗1 , . . . , u∗n ), we obtain (24)

sup inf (Φ(x) + λ(h + Ψ(x))) < inf sup(Φ(x) + λ(h + Ψ(x))), λ≥0 x∈X

x∈X λ≥0

and so the assumption (5) of Theorem 2.1 holds. Now, all assumptions of Theorem 2.1 are satisfied. Hence, applying Theorem 2.1, we have the conclusion. Acknowledgments. The author would like to thank reviewers for clear valuable comments and suggestions. References [1] G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the (p1 , . . . , pn )-Laplacian, Nonlinear Anal. 70 (2009), no. 1, 135–143. [2] , Multiplicity results for a two-point boundary value double eigenvalue problem, Ric. Mat. 59 (2010), no. 1, 39–47. [3] , Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the (p1 , . . . , pn )-Laplacian, Nonlinear Anal. 73 (2010), no. 8, 2594–2602. [4] G. A. Afrouzi, S. Heidarkhani, and D. O’Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a (p1 , . . . , pn )-Laplacian, Bull. Korean Math. Soc. 47 (2010), no. 6, 1235–1250. [5] G. Bonanno and B. Di Bella. A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl. 343 (2008), no. 2, 1166–1176. [6] F. Cammaroto, A. Chinn`ı, and B. Di Bella, Multiple solutions for a Neumann problem involving the p(x)-Laplacian, Nonlinear Anal. 71 (2009), no. 10, 4486–4492. ´ On some fourth-order semilinear elliptic problems [7] J. Chabrowski and J. Marcos do O, in RN , Nonlinear Anal. 49 (2002), no. 6, 861–884. [8] S. El Manouni and M. Kbiri Alaoui, A result on elliptic systems with Neumann conditions via Ricceri’s three critical points theorem, Nonlinear Anal. 71 (2009), no. 5-6, 2343–2348. [9] J. R. Graef, S. Heidarkhani, and L. Kong, A critical points approach to multiplicity results for multi-point boundary value problems, Appl. Anal. 90 (2011), no. 12, 1909– 1925. [10] S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal. 73 (2010), no. 2, 547–554. , Three solutions for a class of gradient Kirchhoff-type systems depending on two [11] parameters, Dynam. Systems Appl. 20 (2011), no. 4, 551–562. [12] C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the (p, q)-Laplacian, Nonlinear Anal. 69 (2008), no. 10, 3322–3329. [13] , Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal. 72 (2010), no. 3-4, 1339–1347. [14] L. Li and C.-L. Tang, Existence of three solutions for (p, q)-biharmonic systems, Nonlinear Anal. 73 (2010), no. 3, 796–805.


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