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Three solutions to a class of $p(x)$-Laplace equation based on N...

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基于Nehari流形方法的一类p(x)-Laplace方程的三解问题，刘都超，，这篇文章中，我们证明了在光滑区域上具有非线性边值或者Dirichlet边值条件下，光滑区域上p(x)-Laplacian方程至少具有三个解。我们在这里�

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Three solutions to a class of p(x)-Laplace

equation based on Nehari skill

∗

Duchao Liu

†

School of Mathematics and Statistics, Lanzhou University

Lanzhou 730000, P. R. China

Abstract In this paper, we show the existence of at least three non-

trivial solutions to the quasilinear elliptic equation

−∆

p(x)

u + |u|

p(x)−2

u = f(x, u)

in a smooth bounded domain Ω ⊂ R

with nonlinear boundary condi-

tion |∇u|

p(x)−2

∂u

∂ν

= g(x, u) or Dirichlet boundary condition u = 0 on

∂Ω. The methods we used here is based on Nehari skills on three sub-

manifolds of the space W

1,p(x)

(Ω).

Keywords Critical points; p(x)-Laplacian; Integral functionals; Gener-

alized Lebesgue-Sobolev spaces

Mathematics Subject Classiﬁcation (2000) 35B38 35D05 35J20

1. Introduction

The study of variational problems with nonstandard growth conditions is

an interesting topic in recent years. p(x)−growth conditions can be regarded

as an important case of nonstandard (p, q)−growth conditions. Many results

have been obtained on this kind of problems, for example [1, 2, 8, 9, 12]. We

refer to the overview papers [10, 16] for advances and references in this area.

In this paper, we consider the inhomogeneous and nonlinear Neumann

∗

Research supported by the National Natural Science Foundation of China (10671084).

†

Email address: liudch06@lzu.cn.

http://www.paper.edu.cn

boundary value problem (P ) and Dirichlet problem (P

(P )







−∆

p(x)

u + |u|

p(x)−2

u = f(x, u), x ∈ Ω,

|∇u|

p(x)−2

∂u

∂ν

= g(x, u), x ∈ ∂Ω;

)

(

−∆

p(x)

u = f(x, u), x ∈ Ω,

u = 0, x ∈ ∂Ω,

where Ω ⊂ R

is a bounded domain with Lipschitz boundary ∂Ω,

∂

∂ν

is the

outer unit normal derivative, p(x) ∈ C(Ω), inf

x∈Ω

p(x) > 1.

The operator −∆

p(x)

u := −div(|∇u|

p(x)−2

∇u) is called p(x)-Laplacian,

which becomes p-Laplacian when p(x) ≡ p (a constant). It possesses more

complicated nonlinearities than the p-Laplacian. For related results involving

the Laplace operator, see [1, 10, 16].

In this paper, we construct three sub-manifolds of the space W

1,p(x)

(Ω)

based on Naheri skill. And under some assumptions, there exist three diﬀer-

ent, nontrivial solutions of (P ) or (P

) on sub-manifold respectively. More-

over these solutions are, one positive, one negative and the other one has

non-constant sign. This result extends the conclusions of [15]. And in [3],

the author also do some research with the similar method.

Throughout this paper, by (weak) solutions of (P ) or (P

) we under-

stand critical points of the associated energy functional Φ or Ψ acting on the

Sobolev space W

1,p(x)

(Ω):

Φ(v) =

Ω

p(x)

(|∇v|

p(x)

+ |v|

p(x)

)dx −

Ω

F (x, v)dx −

∂Ω

G(x, v)dS;

Ψ(v) =

Ω

p(x)

|∇v|

p(x)

dx −

Ω

F (x, v)dx,

where F (x, u) =

f(x, z)dz, G(x, u) =

g(x, z)dz and dS is the surface

measure. We also denote F(v) =

Ω

F (x, v)dx and G(v) =

∂Ω

G(x, v)dS.

For a reﬂexive manifold M, T

M denotes the tangential space at a point

u ∈ M, T

∗

M denotes the co-tangential space, and ∗ : T

∗

M → T

∗∗

∼

is the duality mapping.

In this paper span{v

, ..., v

} denotes the vector space generated by the

vectors v

, ..., v

2. The space W

1,p(x)

(Ω)

In order to discuss problem (P ) or (P

), we need some theories on space

1,p(x)

(Ω) which we call variable exponent Sobolev space. Firstly, we state

http://www.paper.edu.cn

some basic properties of the space W

1,p(x)

(Ω) which will be used later (for

details, see [5 - 7]).

Let Ω be a bounded domain of R

, denote:

(Ω) = {h|h ∈ C(Ω), h(x) > 1, ∀x ∈ Ω},

= max

x∈Ω

h(x), h

−

= min

x∈Ω

h(x), ∀h ∈ C(Ω),

p(x)

(Ω) = {u|u is a measurable real-valued function,

Ω

|u|

p(x)

dx < ∞}.

We can introduce the norm on L

p(x)

(Ω) by

|u|

p(x)

= inf

λ > 0

Ω

u(x)

p(x)

dx 6 1

and (L

p(x)

(Ω), | · |

p(x)

) becomes a Banach space, we call it a generalized

Lebesgue space.

Proposition 2.1 (see [8] and [18]) (1)The space (L

p(x)

(Ω), | · |

p(x)

) is a

separable, uniform convex Banach space, and its conjugate space is L

q(x)

(Ω),

where 1/q(x) + 1/p(x) = 1. For any u ∈ L

p(x)

(Ω) and v ∈ L

q(x)

(Ω), we have

Ω

uvdx

6 (

−

)|u|

p(x)

|v|

q(x)

;

(2)If p

, p

∈ C

(Ω), p

(x) 6 p

(x), for any x ∈ Ω, then L

(x)

(Ω) →

(x)

(Ω), and the imbedding is continuous.

Proposition2.2 (see [8] and [18]) If f : Ω × R → R is a Caratheodory

function and satisﬁes

|f(x, s)| 6 a(x) + b|s|

(x)

, for any x ∈ Ω, x ∈ R,

where p

, p

∈ C

(

Ω), a(x) ∈ L

p(x)

(Ω), a(x) > 0 and b(x ) > 0 is a con-

stant, then the Nemytsky operator from L

(x)

(Ω) to L

(x)

(Ω) deﬁned by

(u))(x) = f(x, u(x)) is a continuous and bounded operator.

Proposition2.3 (see [8] and [19]) If we denote

ρ(u) =

Ω

|u|

p(x)

dx, ∀u ∈ L

p(x)

(Ω),

then

(1) |u(x)|

p(x)

< 1(= 1; > 1) ⇔ ρ(u) < 1(= 1; > 1);

http://www.paper.edu.cn

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