(p(x),q(x))-Laplacian - Project Euclid

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 971243, 19 pages doi:10.1155/2012/971243

Research Article Multiple Solutions for a Class of Differential Inclusion System Involving the px, qx-Laplacian Bin Ge and Ji-Hong Shen Department of Mathematics, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Bin Ge, [email protected] Received 29 December 2011; Revised 28 February 2012; Accepted 2 April 2012 Academic Editor: Kanishka Perera Copyright q 2012 B. Ge and J.-H. Shen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a differential inclusion system involving the px, qx-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

1. Introduction In recent years, the study of differential equations and variational problems with pxgrowth conditions has been a new and interesting topic, which arises from nonlinear electrorheological fluids see 1 and elastic mechanics see 2. The study on variable exponent problems attracts more and more interest in recent years, and many results have been obtained on this kind of problems, for example 3–11. Elliptic systems with standard growth conditions have been the subject of a sizeable literature. We refer to the excellent survey article of de Figueiredo 12. In 11, the author obtained the existence and multiplicity of solutions for the following problem: − div |∇u|px−2 ∇u Fu x, u, v, − div |∇v|qx−2 ∇v Fv x, u, v, u v 0,

on ∂Ω,

in Ω, in Ω,

P1

2

Abstract and Applied Analysis

where Ω ⊂ RN is a bounded domain with a smooth boundary ∂Ω, N ≥ 2, p, q ∈ CΩ2 , px > 1, qx > 1, for every x ∈ Ω. The function F is assumed to be continuous in x ∈ Ω and of class C1 in u, v ∈ R. More precisely, the author was able to prove that, under suitable conditions, the system might have at least one solution or have infinite number of solutions. Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, now a question arises: whether there exist solutions for system P1 in the case where there is no continuously differentiable hypothesis required on the potential function F with respect to ts. See, for example, Fx, t, s is locally Lipschitz with respect to ts. That is the main problem which we want to solve in the present paper. To this end, we mainly discuss the existence and multiplicity of solutions for the following nonlinear differential inclusion system involving the px, qx-Laplacian: − div |∇u|px−2 ∇u ∈ λ∂u Fx, u, v, − div |∇v|qx−2 ∇v ∈ λ∂v Fx, u, v, u v 0,

in Ω, in Ω,

P

on ∂Ω,

where Ω ⊂ RN is a bounded domain with C1 -boundary ∂Ω, λ > 0 is the parameter, p, q ∈ CΩ, 1 < p− ≤ p < ∞, 1 < q− ≤ q < ∞, F : Ω × R × R → R is a function such that F·, t, s is measurable in Ω for all t, s ∈ R × R, and Fx, t, s is locally Lipschitz with respect to ts in general it can be nonsmooth, ∂t Fx, t, s∂s Fx, t, s is the subdifferential with respect to the ts-variable in the sense of Clarke 13. We emphasize that the operator − div|∇u|px−2 ∇u is said to be px-Laplacian, which becomes p-Laplacian when px ≡ p a constant. The px-Laplacian possesses more complicated nonlinearities than the p-Laplacian, for example, it is inhomogeneous and, in general, it does not have the first eigenvalue. In other words, the infimum of the eigenvalues of px-Laplacian equals 0 see 14. Specially, if Fx, ·, v, Fx, u, · ∈ C1 R for a.a. x ∈ Ω, and px p, qx q, then the problem P becomes the following problem: − div |∇u|p−2 ∇u λFu x, u, v, − div |∇v|q−2 ∇v λFv x, u, v, u v 0,

in Ω, in Ω,

P2

on ∂Ω.

There have been a large number of papers that study the existence of the solutions to P2 . For instance, when p > N, q > N, Li and Tang 15 ensured the existence of three solutions to this problem. In 16, Kristály studied the multiplicity of solutions of the quasilinear elliptic systems P1 , where Ω is a strip-like domain and λ > 0 is the parameter. Under some growth conditions on F, the author guaranteed the existence of an open interval Λ ⊂ 0, ∞, such that for each λ ∈ Λ problem P2 has at least two distinct nontrivial solutions; when p and q are real numbers larger than 1, λ 1, Boccardo and Guedes de Figueiredo 17 obtained the existence of solutions of the system P2 .


3

But up to now, to the best of our knowledge, no paper discussing the solutions of problem P with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, we study problem P from a more extensive viewpoint. More precisely, we would study the existence of at least two nontrivial solutions for the problem P as the parameter λ > λ0 for some constant λ0 . This paper is divided into three sections: in the second section we introduce some necessary knowledge on the nonsmooth analysis, basic properties of the generalized Lebesgue-space Lpx Ω and the generalized Lebesgue-Sobolev space W 1,px Ω. In the third section, we give the assumptions on the nonsmooth potential Fx, t, s and prove the multiplicity results for problem P .

2. Preliminary 2.1. Variable Exponent Sobolev Space k,px

In order to discuss problem P , we need some theories on W0 Ω which we call variable exponent Sobolev space. Firstly we review some facts on variable exponent spaces Lpx Ω and W k,px Ω. For the details see 4, 18–20. Firstly, we need to give some notations, which we shall use through this paper: C Ω p ∈ C Ω : px > 1 for any x ∈ Ω ,

p− lim px, p max px for any p ∈ C Ω . x∈Ω

2.1

x∈Ω

Obviously, 1 < p− ≤ p < ∞. Denote by UΩ the set of all measurable real functions defined on Ω. Two functions in UΩ are considered to be one element of UΩ, when they are equal almost everywhere. For p ∈ C Ω, define

Lpx Ω

u ∈ UΩ : |ux|px dx < ∞ , Ω

2.2

and with the norm |u|Lpx Ω |u|px inf{λ > 0 : Ω |ux/λ|px dx ≤ 1}, W 1,px Ω {u ∈ Lpx Ω : |∇u| ∈ Lpx Ω}, with the norm u u W 1,px Ω |u|px |∇u|px . 1,px

Denote W0 Ω as the closure of C0∞ Ω in W 1,px Ω. Hereafter, let ⎧ ⎪ ⎨ Npx , ∗ p x N − px ⎪ ⎩ ∞,

px < N, px ≥ N.

2.3

4

Abstract and Applied Analysis 1,px

Ω are separable and Lemma 2.1 see 19. (1) The spaces Lpx Ω, W 1,px Ω, and W0 reflexive Banach spaces. Moreover, Lpx Ω is uniform convex. 1,px Ω holds; that is, there exists a positive constant C such that (2) Poincare inequality in W0 |u|Lpx Ω ≤ C|∇u|Lpx Ω ,

L

qx

1,px

∀u ∈ W0

Ω.

2.4

(3) If q ∈ C Ω and qx < p∗ x for any x ∈ Ω, then the embedding from W 1,px Ω to Ω is compact and continuous. By 2 of Lemma 2.1, we know that |∇u|px and u are equivalent norms on

1,px

W0

Ω. We will use |∇u|px to replace u in the following discussions.

Lemma 2.2 see 4. The conjugate space of Lpx Ω is Lqx Ω, where 1/px 1/qx 1. For any u ∈ Lpx Ω and v ∈ Lqx Ω, one has

Ω

|uv|dx ≤

Lemma 2.3 see 4. Set ρu

Ω

1 1 |u|Lpx Ω |v|Lqx Ω . p− q−

2.5

|ux|px dx. For u, uk ∈ Lpx Ω, one has

1 for u / 0, |u|px λ ⇔ ρu/λ 1; 2 |u|px < 1 1; > 1 ⇔ ρu < 1 1; > 1; p−

p

p

p−

3 if |u|px > 1, then |u|px ≤ ρu ≤ |u|px ; 4 if |u|px < 1, then |u|px ≤ ρu ≤ |u|px ; 5 limk → ∞ |uk |px 0 ⇔ Limk → ∞ ρuk 0; 6 |uk |px → ∞ ⇔ ρuk → ∞. 1,px

In this paper, the space W0 equivalent norm:

1,qx

Ω × W0

Ω will be endowed with the following

u, v u v ,

2.6

where

∇u px

u inf λ > 0 : λ dx ≤ 1 , Ω

∇v qx

v inf λ > 0 : λ dx ≤ 1 . Ω

Similar to Lemma 2.3, we have the following. Lemma 2.4. Set ρu Ω |∇ux|px dx. For u, uk ∈ W 1,px Ω, one has 1 for u / 0, u λ ⇔ ρu/λ 1; 2 u < 1 1; > 1 ⇔ ρu < 1 1; > 1;

2.7


5

−

−

3 if u > 1, then u p ≤ ρu ≤ u p ; 4 if u < 1, then u p ≤ ρu ≤ u p ; 5 Limk → ∞ uk 0 ⇔ Limk → ∞ ρuk 0; 6 uk → ∞ ⇔ ρuk → ∞. Consider the following function: Ju

Ω

1 |∇u|px dx, px

1,px

u ∈ W0

Ω.

2.8

1,px

We know that (see [21]) J ∈ C1 W0 Ω, R and px-Laplacian operator −Δpx u px−2 − div|∇u| ∇u is the derivative operator of J in the weak sense. We denote L J : 1,px 1,px Ω → W0 Ω∗ , then Lu, v Ω |∇ux|px−2 ∇u · ∇vdx, for all u, v ∈ W0 1,px

W0

Ω.

1,px

Lemma 2.5 see 19. Set X W0

Ω, L is as above, then

1 L : X → X ∗ is a continuous, bounded and strictly monotone operator; 2 L is a mapping of type S , if un u(weak) in X and lim supn → ∞ Lun , un − u ≤ 0, then un → u in X; 3 L : X → X ∗ is a homeomorphism.

2.2. Generalized Gradient Let X be a Banach space, X ∗ its topological dual space and we denote ·, · as the duality bracket for pair X ∗ , X. A function ϕ : X → R is said to be locally Lipschitz, if for every x ∈ X we can find a neighbourhood U of x and a constant k > 0 depending on U, such that |ϕy − ϕz| ≤ k y − z , for all y, z ∈ U. The generalized directional derivative of ϕ at the point u ∈ X in the direction h ∈ X is ϕu λh − ϕu . λ u → u;λ↓0

ϕ0 u; h lim sup

2.9

The generalized subdifferential of ϕ at the point u ∈ X is defined by the ∂ϕu u∗ ∈ X ∗ ; u∗ , h ≤ ϕ0 u; h, ∀h ∈ X ,

2.10

which is a nonempty, convex and w∗ -compact set of X. We say that u ∈ X is a critical point of ϕ, if 0 ∈ ∂ϕu. For further details, we refer the reader to 12.

6

Abstract and Applied Analysis Finally we have the following Weierstrass Theorem and Mountain Pass Theorem.

Theorem 2.6. If X is a reflexive Banach space and ϕ : X → R satisfies 1 ϕ is weak lower semicontinuous, that is, xn x0 weakly in X −→ ϕx0 ≤ lim infϕxn ;

2.11

n → ∞

2 ϕ is coercive, that is, lim x → ∞ ϕx ∞, then there exists x∗ ∈ X such that ϕx∗ minx∈X ϕx. Theorem 2.7 see 22. Let ϕ : X → R be locally Lipschitz function and x0 , x1 ∈ X. If there exists a bounded open neighbourhood U of x0 , such that x1 ∈ X \ U, max{ϕx0 , ϕx1 } < infϕ ∂U

and ϕ satisfies the nonsmooth C-condition at level c, where c infγ∈T maxt∈0,1 ϕγt, T {γ ∈ C0, 1; X : γ0 x0 , γ1 x1 }, then c is a critical value of ϕ and c ≥ inf∂U ϕ.

3. Existence Results 1,px

For each u, v ∈ W0

1,qx

Ω × W0

Φu, v

Ω, define

1 1 |∇u|px dx |∇v|qx dx, Ω px Ω px Ψu, v Fx, u, vdx.

3.1

Ω

1,px

1,qx

Ω × W0 Ω to which there By a solution of 2, we mean function u, v ∈ W0 corresponds mapping Ω x → w1 , w2 with w1 x ∈ ∂u Fx, u, v, w2 x ∈ ∂v Fx, u, v for 1,px 1,qx Ω × W0 Ω, the almost every x ∈ Ω having the property that for every ξ, η ∈ W0 1 1 function x → w1 xξx, w2 xηx ∈ L Ω × L Ω and

Ω

|∇u|

px−2

∇u∇ξdx

Ω

|∇v|

qx−2

∇v∇ηdx λ

Ω

w1 ξdx λ

Ω

w2 ηdx.

3.2

Our hypotheses on nonsmooth potential Fx, t, s is as follows. HF: F : Ω × R × R → R is a function such that Fx, 0, 0 0 a.e. on Ω and satisfies the following facts: 1 for all t ∈ R, s ∈ R, x → Fx, t, s is measurable; 2 for almost all x ∈ Ω, t → Fx, t, s and s → Fx, t, s are locally Lipschitz.


7

Lemma 3.1. Suppose HF and the following conditions hold: f1 there exists α ∈ C Ω and αx < p∗ x, such that |w1 | ≤ c1 1 |t|αx−1 |s|βxαx−1/αx ,

3.3

for almost all x ∈ Ω, all t, s ∈ R and w1 ∈ ∂t Fx, t, s; f2 there exists β ∈ C Ω and βx < q∗ x, such that |w2 | ≤ c2 1 |t|αxβx−1/βx |s|βx−1 ,

3.4

for almost all x ∈ Ω, all t, s ∈ R and w2 ∈ ∂s Fx, t, s; 1,px

then ϕ·, vϕu, · is locally Lipschitz on W0

1,qx

ΩW0

Ω. 1,px

Proof. We need only to prove that ϕ·, v is locally Lipschitz on W0 By Φ·, v ∈

1,px C1 W0 Ω, R,

Ω.

we have

ϕu1 , v − ϕu2 , v Ju1 − Ju2 J u · u1 − u2 ,

3.5

where u tu1 1 − tu2 , t ∈ 0, 1. 1,px Let Br {u ∈ W0 Ω : u − u0 W 1,px Ω ≤ r}. 0 Note that Br is w-compact. Then we obtain that there exists a positive constant M, such that J u W −1,p x Ω ≤ M with 1/px 1/p x 1, for sufficiently small r. Therefore, for any u1 , u2 ∈ Br , we have |Φu1 , v − Φu2 , v| J u · u1 − u2 ≤ J uW −1,p xx Ω u1 − u2 W 1,px Ω 3.6 0 ≤ M u1 − u2 W 1,px Ω , 0

1,qx

∀v ∈ W0

Ω.

1,qx

Fixing v ∈ W0

Ω, by f1 and Lebourg mean value theorem, we have |Fx, u1 , v − Fx, u2 , v| ≤ c1 1 |u|αx−1 |v|βxαx−1/αx |u1 − u2 |.

3.7

Hence, Fx, u1 , vdx − Fx, u , vdx 2 Ω Ω ≤ c1 |u1 − u2 |dx c1 |u|αx−1 |u1 − u2 |dx c1 |v|βxαx−1/αx |u1 − u2 |dx Ω Ω Ω ≤ c1 |u1 − u2 |αx c1 |u|αx−1 |u1 − u2 |αx c1 |v|βxαx−1/αx |u1 − u2 |αx , α x

where 1/α x 1/αx 1.

α x

3.8

8

Abstract and Applied Analysis It is immediate that α |u|αx ≤ c u α , |u|αx > 1, |u| dx ≤ |u| − |u|ααx ≤ c u α , |u|αx < 1, Ω Ω ⎧ ⎨|v|β ≤ c v β , |v| α x βx > 1, βxαx−1/αx βx βx |v| dx ≤ |v| β β− ⎩ |v|βx ≤ c v , |v|βx < 1 Ω Ω

αx−1

α x

αx

3.9

are bounded. So, Fx, u1 , vdx − ≤ c u1 − u2 , Fx, u , vdx 2 Ω

Ω

3.10

1,px

Ω → Lαx Ω is a compact embedding. since W0 Therefore, ϕ·, v is locally Lipschitz. Similarly, we can prove that ϕu, · is locally Lipschitz. Theorem 3.2. Suppose that HF, f1 with α < p− , f2 with β < q− and the following conditions f3 -f4 hold: f3 there exists γi ∈ CΩ i 1, 2 with p < γ1 x < p∗ x, q < γ2 x < q∗ x and μ1 , μ2 ∈ L∞ Ω, such that lim sup t → 0,s → 0

w1 , t |t|

γ1 x

< μ1 x,

lim sup t → 0,s → 0

w2 , s |s|γ2 x

< μ2 x

3.11

uniformly for almost all x ∈ Ω and all w1 ∈ ∂t Fx, t, s, w2 ∈ ∂s Fx, t, s; f4 there exist ξ0 ∈ R, η0 ∈ R, x0 ∈ Ω and r0 > 0, such that F x, ξ0 , η0 > δ0 > 0,

a.e. x ∈ Br0 x0 ,

3.12

where Br0 x0 : {x ∈ Ω : |x − x0 | ≤ r0 } ⊂ Ω. Then there exists λ∗ > 0 such that, for each λ > λ∗ , the problem 2 has at least two nontrivial solutions. Proof. The proof is divided into four steps as follows. Step 1. We will show that ϕ is coercive in the step. Firstly, for almost all x ∈ Ω, by HF2, t → Fx, t, s is differentiable almost everywhere on R and we have d Fx, t, s ∈ ∂t Fx, t, s. dt

3.13


9

Moreover, from f1 , f2 and Young inequality, we can get that Fx, t, s Fx, 0, s Fx, 0, 0 ≤ c1 |t|

t 0

d F x, y, s dy dy

0

d Fx, 0, zdz dz

s

t

d F x, y, s dy dy 0

|t|αx |s|βxαx−1/αx |t| αx

c2 |s| |t|αxβx−1/βx |s|

1 |s|βx βx αx − 1|s|βx |t|αx αx ≤ c1 |t| |t| αx βx

3.14

βx − 1 |t|αx |s|βx c2 |s| |s| βx βx ≤ c1 |t| 2|t|αx |s|βx c2 |s| 2|s|βx |t|αx ,

βx

for almost all x ∈ Ω and t, s ∈ R. Note that 1 < αx ≤ α < p− < p∗ x and 1 < βx ≤ β < q− < q∗ x, 1,px 1,qx Ω → Lαx Ω and W0 Ω → Lβx Ω compact then, by Lemma 2.1, we have W0 embedding. Furthermore, there exists c3 , c4 > 0 such that |u|αx ≤ c3 u and |v|βx ≤ c4 v . So, for any |u|αx > 1 and u > 1, Ω

β

β

|u|αx dx ≤ |u|ααx ≤ c3α u α

3.15

and, for any |v|βx > 1 and v > 1, Ω

|v|βx dx ≤ |v|βx ≤ c4 v β .

3.16

By 3.14, 3.15, 3.31, the Holder inequality and the Sobolev embedding theorem, ¨ we have ϕu, v ≥

Ω

1 Fx, u, vdx |∇v|qx dx − λ Ω qx Ω − 1 β v q − λc1 |u|dx − 2c1 c3α u α − c1 c4 v β q Ω

1 |∇u|px dx px

− 1

u p p β − λc2 |v|dx − 2c2 c4 v β − c2 c3α u α

Ω

10

Abstract and Applied Analysis ≥

− − 1 1

u p v q − 2λc1 |1|α x |u|αx − 2λc1 |1|β x |v|βx p q

β

β

− 2c1 c3α u α − c1 c4 v β − 2c2 c4 v β − c2 c3α u α ≥

− − 1 1

u p v q − 2λc1 c|1|α x u − 2λc1 c|1|β x v

p q

β

β

− 2c1 c3α u α − c1 c4 v β − 2c2 c4 v β − c2 c3α u α −→ ∞, as u, v −→ ∞.

3.17 Step 2. We will show that the ϕ is weakly lower semicontinuous. 1,px 1,qx Ω, vn v weakly in W0 Ω by Lemma 2.13, Leting un u weakly in W0 we obtain the following results: 1,px

W0

1,qx

Ω → Lpx Ω; W0

Ω → Lqx Ω;

un → u in Lpx Ω; vn → v in Lqx Ω; un → u for a.a. x ∈ Ω; vn → v for a.a. x ∈ Ω; Fx, un x, vn x → Fx, ux, vx for a.a. x ∈ Ω. By Fatou’s Lemma, lim sup n→∞

Ω

Fx, un x, vn xdx ≤

Ω

Fx, ux, vxdx.

3.18

Thus, lim infϕun , vv lim inf n→∞

n→∞

Ω

1 |∇un |px dx px

Ω

1 |∇vn |qx dx qx

− lim sup Fx, un , vn dx n→∞ Ω 1 1 Fx, u, vdx. ≥ |∇u|px dx |∇v|qx dx − λ Ω px Ω qx Ω

3.19

ϕu

1,px

W0

Hence, by Theorem 2.6, we deduce that there exists a global minimizer u0 , v0 ∈ 1,qx Ω × W0 Ω such that ϕu0 , v0

min

1,px

u,v∈W0

1,qx

Ω×W0

ϕu, v.

Ω

Step 3. We will show that there exists λ∗ > 0 such that for each λ > λ∗ , ϕu0 , v0 < 0.

3.20


11

By the condition f4 , there exists ξ0 , η0 ∈ R such that Fx, ξ0 , η0 > δ0 > 0, a.e. x ∈ Br0 x0 . It is clear that

0 < M1 :

max

|t|≤|ξ0 |,|s|≤|η0 |

c1 |t| 2|t|α |s|β c2 |s| 2|s|β |t|α ,

− − − − c1 |t| 2|t|α |s|β c2 |s| 2|s|β |t|α < ∞.

3.21

Now we denote 1/N M1 t0 , δ0 M1 p− p q − q ξ0 ξ0 ξ0 ξ0 , , , , Kt : max r0 1 − t r0 1 − t r0 1 − t r0 1 − t Kt 1 − tN λ∗ max N , t∈t1 ,t2 δ0 t − M1 1 − tN

3.22

where t0 < t1 < t2 < 1 and δ0 is given in the condition f4 . A simple calculation shows that N the function t → δ0 tN − M1 1 − tN is positive whenever t > t0 and δ0 tN 0 − M1 1 − t0 0. Thus λ∗ is well defined and λ∗ > 0. We will show that, for each λ > λ∗ , the problem P has two nontrivial solutions. In order to do this, for t ∈ t1 , t2 , let us define ⎧ ⎪ 0, ⎪ ⎪ ⎨ ηt x ξ0 , ⎪ ⎪ ξ0 ⎪ ⎩ r0 − |x − x0 |, r0 1 − t

if x ∈ Ω \ Br0 x0 , if x ∈ Btr0 x0 ,

3.23

if x ∈ Br0 x0 \ Btr0 x0 .

By conditions f1 and f3 we have

Ω

F x, ηt x, ηt x dx

Btr0 x0


Br0 x0 \Btr0 x0


≥ wN r0N tN δ0 − M1 1 − tN wN r0N wN r0N δ0 tN − M1 1 − tN .

3.24

12


Hence, for t ∈ t1 , t2 , px qx 1 1 ∇ηt dx ∇ηt dx − λ F x, ηt x, ηt x dx Ω px Ω qx Ω px qx 1 1 ∇ηt dx ∇ηt dx − λwN r N δ0 tN − M1 1 − tN ≤ − 0 p Ω q− Ω p− p q − q ξ0 ξ0 ξ0 ξ0 ≤ max , , , r0 1 − t r0 1 − t r0 1 − t r0 1 − t × wN r0N 1 − tN − λwN r0N δ0 tN − M1 1 − tN wN r0N Kt 1 − tN − λ δ0 tN − M1 1 − tN ,

ϕ ηt , ηt

3.25

so that ϕηt , ηt < 0 whenever λ > λ∗ . Step 4. We will check the C-condition in the following. 1,px 1,qx Suppose {un , vn }n≥1 ⊆ W0 Ω × W0 Ω such that ϕun , vn → c and 1 un , vn mun , vn → 0. Let u∗n , vn∗ ∈ ∂ϕun , vn be such that mun , vn

u∗n , vn∗ W 1,px Ω×W 1,qx Ω∗ . The interpretation of u∗n , vn∗ ∈ ∂ϕun , vn is that u∗n ∈ 0 0 ∂u ϕun , vn and vn∗ ∈ ∂v ϕun , vn . We know that u∗n −Δpx un − λwn1 ,

3.26

vn∗ −Δqx vn − λwn2

with wn1 ∈ ∂u Ψun , vn and wn2 ∈ ∂v Ψun , vn . From Chang 23 we know that wn1 ∈ Lα x Ω and wn2 ∈ Lβ x Ω, where α x αx/αx − 1, β x βx/βx − 1. Since ϕ is coercive, {un }n≥1 , {vn }n≥1 are bounded and passed to a subsequence, still 1,px

1,qx

denoting {un }n≥1 and {vn }n≥1 , we may assume that there exist u ∈ W0

Ω, v ∈ W0

such that un u weakly in that

Next we will prove

1,px W0 Ω

1,px

un → u in W0 1,px

and vn v weakly in

Ω,

1,qx W0 Ω.

1,qx

vn → v in W0

Ω.

Ω,

3.27

1,qx

Ω → Lpx Ω, W0 Ω → Lqx Ω, we have un → u in Lpx Ω By W0 qx and vn → v in L Ω. Moreover, since u∗n , vn∗ W 1,px Ω×W 1,qx Ω∗ → 0, we get 0 0

u∗n W 1,px Ω∗ → 0, vn∗ W 1,qx Ω∗ → 0, so |u∗n , un | ≤ εn , |vn∗ , vn | ≤ εn . 0 0 From 3.26, we have

−Δpx un , un − u −

−Δqx vn , vn − v −

Ω

Ω

wn1 un − udx ≤ εn ,

∀n ≥ 1.

− vdx ≤ εn ,

∀n ≥ 1.

wn2 vn

3.28


13

Moreover, Ω wn1 un − udx → 0 and Ω wn2 vn − vdx → 0, since un → u in Lpx Ω, vn → v in Lqx Ω, {wn1 }n≥1 in Lp x Ω and {wn2 }n≥1 in Lq x Ω are bounded, where 1/px 1/p x 1, 1/qx 1/q x 1. Therefore, lim sup −Δpx un , un − u ≤ 0, n→∞

lim sup −Δqx vn , vn − v ≤ 0. n→∞

3.29

From Lemma 2.5, we have un → u, vn → v as n → ∞. Thus ϕ satisfies the nonsmooth C-condition. Step 5. We will show that there exists another nontrivial weak solution of problem P . From Lebourg Mean Value Theorem, we obtain

Fx, t, s − Fx, 0, s w1 , t, Fx, 0, s − Fx, 0, 0 w2 , s

3.30

for some w1 ∈ ∂t Fx, ϑt, s, w2 ∈ ∂s Fx, 0, τs, and 0 < ϑ, τ < 1. Thus by the condition f3 , there exists β ∈ 0, 1 such that

|Fx, t, s| ≤ |w1 , t| |w2 , s| ≤ μ1 x|t|γ1 x μ2 x|s|γ2 x ,

3.31

for all |t|, |s| < β and a.e. x ∈ Ω. It follows from the conditions f1 , f2 1 < α− ≤ α < p− ≤ p < γ1 x < p∗ x and − 1 < β ≤ β < q− ≤ q < γ2 x < p∗ x that for all |t| > β, |s| > β and a.e. x ∈ Ω, |Fx, t, s| ≤ c1 |t| 2|t|αx |s|βx c2 |s| 2|s|βx |t|αx ! 2c2 c2 c1 ≤ |t|γ1 x βγ1 x−1 βγ1 x−αx βγ1 x−βx ! c2 2c2 c1 |t|γ2 x βγ2 x−1 βγ2 x−βx βγ1 x−αx ! c1 2c2 c2 ≤ − γ −β− |t|γ1 x βγ1 −1 βγ1 −α β1 ! c2 2c2 c1 − γ −α− |t|γ2 x , βγ2 −1 βγ2 −β β1

3.32

14


this together with 3.31 yields that, for all t ∈ R and a.e. x ∈ Ω, μ1 x

|Fx, t, s| ≤

≤ c3 |t|

c1

μ2 x γ1 x

βγ1 −1 c2 β

γ2 −1

c4 |s|

γ2 x

2c2

−

βγ1 −α

2c2 β

γ2 −β−

c2

βγ1 −β

! |t|γ1 x

−

!

c1 β

γ1 −α−

3.33

|t|γ2 x

,

for positive constants c3 , c4 . Note that p < γ1 x < p∗ x, q < γ2 x < q∗ x, then, by Lemma 2.1, we have 1,px 1,qx Ω → Lγ1 x Ω and W0 Ω → Lγ2 x Ω. Furthermore, there exist c5 , c6 such that W0 1,px

|u|γ1 x ≤ c5 u ,

∀u ∈ W0

Ω,

|v|γ2 x ≤ c6 u ,

1,qx

∀v ∈ W0

Ω.

For all λ > λ∗ , u, v < 1, |u|γ1 x < 1 and |v|γ2 x < 1, from 3.33 we have 1 1 Fx, ux, vxdx ϕu, v |∇u|px dx |∇v|qx dx − λ px qx Ω Ω Ω 1 1 ≥ |∇u|px dx |∇v|qx dx − λc3 |ux|γ1 x dx Ω px Ω qx Ω − λc4 |vx|γ2 x dx

3.34

3.35

Ω

γ− − − 1 1 γ− ≥ u p v q − λ c3 c51 u γ1 c4 c62 u γ2 . p q So, for ρ > 0 small enough, there exists a ν > 0 such that ϕu, v > ν,

for u, v ρ

3.36

and u0 , v0 > ρ. So by the Nonsmooth Mountain Pass Theorem cf. Theorem 2.7, we can 1,px 1,qx Ω × W0 Ω which satisfies get u1 , v1 ∈ W0 ϕu1 , v1 c > 0,

mu1 , v1 0.

3.37

Therefore, u1 , v1 is another nontrivial solution of problem P . Remark 3.3. Let p− > α , q− > β and consider the following nonsmooth locally Lipschitz function: ⎧ ⎪ t ∈ 0, δ or s ∈ 0, δ, tγ1 x sγ2 x , ⎪ ⎪ ⎪ ⎪ ⎨max |t − δ|θx , |t − δ|αx |δ|γ1 x Fx, t, s 3.38 ⎪ max |s − δ|θx , |s − δ|βx |δ|γ2 x , t ∈ δ, ∞ or s ∈ δ, ∞, ⎪ ⎪ ⎪ ⎪ ⎩0, t ∈ −∞, 0 or s ∈ −∞, 0, where 0 < δ < 1, θ− > 1, θ < α− and θ < β− .


15

Obvious, t → Fx, t, s and s → Fx, t, s are locally Lipschitz. Then, ⎧ ⎪ γ1 xtγ1 x−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θxt − δθx−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪αxt − δαx−1 , ⎨ ∂t Fx, t, s ⎪ 0, γ xδγ1 x−1 , ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θx, αx, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0, ⎧ ⎪ γ2 xsγ2 x−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪θxs − δθx−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨βxt − δβx−1 , ∂s Fx, t, s ⎪ 0, γ xδγ1 x−1 , ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θx, βx , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0,

t ∈ 0, δ, t ∈ δ, 1 δ, t ∈ 1 δ, ∞, t δ, t 1 δ, t ∈ −∞, 0, 3.39 s ∈ 0, δ, s ∈ δ, 1 δ, s ∈ 1 δ, ∞, s δ, s 1 δ, s ∈ −∞, 0.

Hence, for any w1 ∈ ∂t Fx, t, s and w2 ∈ ∂s Fx, t, s, we have ⎧ ⎪ γ1 xtαx−1 tγ1 x−αx ≤ γ1 |t|αx−1 , ⎪ ⎪ ⎪ ⎪ α −1 ⎪ ⎪ ⎪ θx−1 ⎪ 1 ⎪ θxt − δ < θ < θ |t|αx−1 , ⎪ ⎪ δ ⎪ ⎪ ⎪ ⎨αxt − δαx−1 ≤ α |t|αx−1 , |w1 | ≤ ⎪ ⎪ ⎪ γ1 xδγ1 x−1 γ1 xδαx−1 δγ1 x−αx ≤ γ δαx−1 , ⎪ ⎪ ⎪ ⎪ ⎪ αx−1 ⎪ ⎪ , ⎪θx, αx ≤ α 1 δ ⎪ ⎪ ⎪ ⎪ ⎩0, ⎧ ⎪ γ2 xsβx−1 sγ2 x−βx ≤ γ2 |s|βx−1 , ⎪ ⎪ ⎪ ⎪ β −1 ⎪ ⎪ ⎪ ⎪θxs − δθx−1 < θ < θ 1 ⎪ |s|βx−1 , ⎪ ⎪ δ ⎪ ⎪ ⎪ ⎨βxs − δβx−1 ≤ β |s|βx−1 , ≤ |w2 | ⎪ ⎪γ xδγ2 x−1 γ xδβx−1 δγ2 x−βx ≤ γ δβx−1 , ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ θx, βx ≤ β 1 δβx−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0,

t ∈ 0, δ, t ∈ δ, 1 δ, t ∈ 1 δ, ∞, t δ, t 1 δ, t ∈ −∞, 0, 3.40 s ∈ 0, δ, s ∈ δ, 1 δ, s ∈ 1 δ, ∞, s δ, s 1 δ, s ∈ −∞, 0.

16

Abstract and Applied Analysis Therefore,

α −1 ! 1 α θ |w1 | ≤ |t|αx−1 , δ β −1 ! 1 |w2 | ≤ γ2 β θ |t|βx−1 , δ ⎧ γ xtγ1 x < w1 , t > ⎨lim 1 lim sup γ x t → 0 tγ1 x , 1 ⎩ t → 0,s → 0 |t| 0, ⎧ γ xsγ2 x < w2 , s > ⎨ lim 2 s → 0 sγ2 x , lim sup γ2 x ⎩ t → 0,s → 0 |s| 0, γ1

∀w1 ∈ ∂t Fx, t, s, ∀w2 ∈ ∂s Fx, t, s, t>0

⎫ ⎬

≤ γ1 x,

3.41

⎭ ⎫ ⎬ s > 0 ≤ γ x, 2 ⎭ s≤0

t≤0

uniformly for almost all x ∈ Ω, all w1 ∈ ∂t Fx, t, s and w2 ∈ ∂s Fx, t, s. Thus far the results involved potential functions exhibiting px-sublinear. The next theorem concerns problems where the potential function is px-superlinear. Theorem 3.4. Suppose that HF, f1 with α− > p , f2 with β− > q , f3 , f4 and the following condition f5 hold.

f5 For almost all x ∈ Ω and all t, s ∈ R, one has Fx, t, s ≤ κx γ3 x < min{p− , q− }.

with κ ∈ Lγ3 x Ω, 1 ≤

Then there exists a λ∗ > 0 such that, for each λ > λ∗ , the problem P has at least two nontrivial solutions. Proof. The steps are similar to those of Theorem 3.2. In fact,we only need to modify Step 1 and Step 5 as follows: Step 6 shows, that ϕ is coercive under the condition f5 ; Step 7 shows, that there exists second nontrivial solution under the conditions f1 , f2 , and f3 . Then from Steps 6, 2, 3, 4, and 7 above, the problem P has at least two nontrivial solutions. 1,px

Step 6. By f5 , for all u, v ∈ W0

1,qx

Ω × W0

Ω, u, v > 1, we have

1 1 px qx ϕu, v Fx, ux, vxdx |∇u| dx |∇v| dx − λ Ω px Ω qx Ω 1 1 ≥ u p v q − λ κxdx −→ ∞ −→ ∞, as u, v → ∞. p q Ω

3.42


17

Step 7. Because of hypothesis f1 , f2 and mean value theorem for locally Lipschitz functions, we have Fx, t, s ≤ c1 |t| 2|t|αx |s|βx c2 |s| 2|s|βx |t|αx αx−1 t αx βx |s| ≤ c1 |t| 2|t| β βx−1 s βx αx c2 |t| |s| 2|s| β α −1 1 αx αx βx ≤ c1 2|t| |s| |t| β β −1 1 βx βx αx c2 2|s| |t| |t| β

3.43

c7 |t|αx c8 |t|βx for a.e. x ∈ Ω, all |t| ≥ β, |s| ≥ β with c7 , c8 > 0. Combining 3.31 and 3.43, it follows that |Fx, t, s| ≤ μ1 x|t|γ1 x c7 |t|αx μ2 x|s|γ2 x c8 |s|αx

3.44

for a.e. x ∈ Ω and all t, s ∈ R. Thus, for all λ > λ∗ , u, v < 1, |u|γ1 x < 1, |u|αx < 1|v|γ2 x < 1 and |v|βx < 1, we have 1 1 px qx Fx, ux.vxdx ϕu, v |∇u| dx |∇v| dx − λ Ω px Ω qx Ω 1 1 ≥ u p v q − λ μ1 x|u|γ1 x dx − λc7 |u|αx dx p q Ω Ω γ2 x βx −λ μ2 x|v| dx − λc8 |v| dx

Ω

3.45

Ω

− − − − 1 1 ≥ u p v q − λc9 u γ1 − λc7 u α − λc10 v γ2 − λc8 v β . p q

So, for ρ > 0 small enough, there exists a ν > 0 such that ϕu, v > ν,

for u, v ρ

3.46

and u0 , v0 > ρ. Arguing as in proof of Step 4 of Theorem 3.2, we conclude that ϕ satisfies the nonsmooth C-condition. So by the Nonsmooth Mountain Pass Theorem cf. Theorem 2.7, 1,px 1,qx Ω × W0 Ω which satisfies we can get u1 , v1 ∈ W0 ϕu1 , v1 c > 0,

mu1 , v1 0.

3.47

18

Abstract and Applied Analysis Therefore, u1 , v1 is second nontrivial of problem P .

Remark 3.5. Consider the following nonsmooth locally Lipschitz function: Fx, t, s

−|t|γ1 x − |s|γ2 x , cosπ|t| cosπ|s|,

|t| ≤ 1 or |s| ≤ 1, |t| > 1; or |s| > 1,

3.48

In the following, we will show that Fx, t, s satisfies hypotheses HF and f1 –f5 . It is clear that Fx, 0, 0 0 for a.e. x ∈ Ω,then hypotheses HF is satisfied. A direct verification shows that conditions f4 and f5 are satisfied. Note that ⎧ ⎪ −γ1 xtγ1 x−1 , ⎪ ⎪ ⎪ ⎪ γ1 x−1 ⎪ , ⎪ ⎨γ1 x−t ∂t Fx, t, s −γ1 x, 0 , ⎪ ⎪ ⎪ ⎪ 0, γ1 x , ⎪ ⎪ ⎪ ⎩{−π sinπt}, ⎧ ⎪ −γ2 xsγ2 x−1 , ⎪ ⎪ ⎪ ⎪ γ1 x−1 ⎪ , ⎪ ⎨γ2 x−s ∂s Fx, t, s −γ2 x, 0 , ⎪ ⎪ ⎪ ⎪ 0, γ , x 2 ⎪ ⎪ ⎪ ⎩{−π sinπs},

0 ≤ t < 1, −1 < t ≤ 0, t 1, t −1, |t| > 1, 3.49 0 ≤ s < 1, −1 < s ≤ 0, s 1, s −1, |s| > 1.

So, |w1 | ≤ γ1 x π |t|γ1 x−1 , ∀w1 ∈ ∂t Fx, t, s, |w1 | ≤ γ2 x π |s|γ2 x−1 , ∀w2 ∈ ∂s Fx, t, s, ⎧ ⎫ −γ1 xtγ1 x ⎪ ⎪ ⎪ ⎪ lim ⎪ , t > 0⎪ ⎬ γ x 1 < w1 , t > ⎨ t → 0 t lim sup −γ1 x ≤ γ1 x, γ1 x γ1 x −γ1 x−t ⎪ ⎪ ⎪ ⎪ t → 0,s → 0 |t| ⎪ ⎪ lim , t ≤ 0 ⎩t → 0 ⎭ −tγ1 x ⎫ ⎧ −γ2 xsγ2 x ⎪ ⎪ ⎪ ⎪ ⎪ lim , s > 0⎪ ⎬ γ x 2 < w2 , s > ⎨ t → 0 s lim sup −γ2 x ≤ γ2 x γ2 x γ2 x −γ2 x−s ⎪ ⎪ ⎪ ⎪ t → 0,s → 0 |s| ⎪ ⎪ , s ≤ 0 lim ⎭ ⎩t → 0 −sγ2 x

3.50

which shows that assumptions f1 , f2 , and f3 are fulfilled.

Acknowledgments This paper was supported by the National Natural Science Foundation of China nos. 11126286, 11001063, 10971043, the Fundamental Research Funds for the Central Universities


19

no. 2012, China Postdoctoral Science Foundation Funded Project no. 20110491032, and the National Natural Science Foundation of Heilongjiang Province no. A200803.

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