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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Multiplicity of Solutions of Weighted

(p, q)-Laplacian with

Small Sources

Huijuan Song

1∗

, Jingxue Yin

, Zejia Wang

Department of Mathematics, Jilin University, Changchun 130012

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

College of Mathematics and Informational Science,

Jiangxi Normal University, Nanchang 330022

Abstract: In this paper we study the existence of inﬁnitely many solutions to the degenerate

quasilinear elliptic system

−div(h

(x)|∇u|

p−2

∇u) = d(x)|u|

r−2

u + G

(x, u, v) in Ω,

−div(h

(x)|∇v|

q−2

∇v) = f (x)|v|

s−2

v + G

(x, u, v) in Ω,

u = v = 0 on ∂Ω,

where Ω is a bounded domain in R

with smooth boundary ∂Ω, N ≥ 2, 1 < r < p < N, 1 < s < q < N,

(x), h

(x) are allowed to have “essential” zeroes at some points in Ω, d(x)|u|

r−2

u and f (x)|v|

s−2

v are

small sources with G

(x, u, v), G

(x, u, v) being their high-order perturbations with respect to (u, v) near

the origin respectively.

Key words: Weighted (p, q)-Laplacian; Small sources; Multiplicity

1 Introduction

The purpose of this paper is to study the multiplicity of solutions for the system with small sources:











−div(h

(x)|∇u|

p−2

∇u) = d(x)|u|

r−2

u + G

(x, u, v) in Ω,

−div(h

(x)|∇v|

q−2

∇v) = f (x)|v|

s−2

v + G

(x, u, v) in Ω,

u = v = 0 on ∂Ω,

(1)

Huijuan Song (1985-), female, Ph. D. student, major research direction: nonlinear diﬀusion equations. Jingxue Yin (1956-), male, professor,

major research direction: nonlinear diﬀusion equations. Correspondence author: Zejia Wang (1979-), female, associate professor, major research

direction: nonlinear diﬀusion equations.

∗

Supported by the Graduate Innovation Fund of Jilin University (No. 20121031).

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

where Ω ⊂ R

is a bounded domain with smooth boundary ∂Ω, N ≥ 2, 1 < r < p < N, 1 < s < q < N,

(x), h

(x) are allowed to have “essential” zeroes at some points in Ω, d(x) and f (x) can be very small,

in particular, small supports and sign changing for d(x) and f (x) are permitted, the terms G

(x, u, v) and

(x, u, v) will be considered as high-order perturbations of the small sources d(x)|u|

r−2

u and f (x)|v|

s−2

with respect to (u, v) near the origin respectively.

The present paper is a continuation of our previous paper [1], in which we investigated the follow-

ing quasilinear elliptic system











−div(h

(x)|∇u|

p−2

∇u) = ζa(x)|u|

r−2

u + λµ(x)F

(x, u, v) in Ω,

−div(h

(x)|∇v|

q−2

∇v) = σb(x)|v|

s−2

v + λµ(x)F

(x, u, v) in Ω,

u = v = 0 on ∂Ω,

(2)

where F(x, u, v) satisﬁes certain global structure conditions which can be divided into the (p, q)-sublinear

case, the (p, q)-linear case and the (p, q)-suplinear case with diﬀerent features of multiplicity of solu-

tions according to the parameters ζ, σ, λ. The relationship between these two papers will be described

in Section 2.

For the case of single equation, it was Wang [2] who ﬁrst considered a similar problem with small

sources











−div(|∇u|

p−2

∇u) = d(x)|u|

r−2

u + g(x, u) in Ω,

u = 0 on ∂Ω,

(3)

where 1 < r < p < ∞, d(x) ≡ λ with λ being a positive constant, g(x, u) ∈ C(Ω × (−δ, δ), R) for some

δ > 0, is odd in u and g(x, u) = o(|u|

r−1

) as u → 0 uniformly for x ∈ Ω. By using a variational principle

by Clark [3], Wang proved that the problem (3) has a sequence of solutions with negative energy such

that the sequence of solutions converges to 0 in the L

∞

norm. That is to say, the existence of such a

sequence of solutions relies only on local behaviors of the equations, and assumptions on g only for

small u are required. Later, this result was generalized by Guo [4] to the problem (3) where the weight

function d(x) satisﬁes Ω

= {x ∈ Ω; d(x) > 0} , ∅ and g(x, u) = o(|u|

p−1

), as |u| → 0 uniformly for

x ∈ Ω. It is worthy of pointing out that the assumption on d in [2] has been relaxed in [4], but the

assumption on g with respect to u near the origin in [4] is stronger than that in [2]. It should be noticed

that positivity b(x) ≡ λ was used in an essential way in [2]. Recently, Chung [5] further extended the

result to the system (1) with linear principal parts, that is

p = q = 2,

where the diﬀusion coeﬃcients h

, h

are from the space (H )

with σ ∈ [0, +∞), consisting of all

functions h : Ω → [0, +∞) such that h ∈ L

loc

(Ω) and lim inf

x→z

|x − z|

−σ

h(x) > 0 holds for all z ∈ Ω,

r = s, Ω

∩ Ω

, ∅ and G(x, u, v) ∈ C

(Ω × (−ρ

, ρ

) × (−ρ

, ρ

), R) for small positive constants ρ

, ρ

is even in (u, v) and

lim

|u|→0

(x, u, v)

|u|

|v|

τ+1

= 0, lim

|v|→0

(x, u, v)

|u|

θ+1

|v|

= 0

- 2 -

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