Research Article
Received 13 December 2012 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2805
MOS subject classification: 35D05; 35J25; 58E05
Three solutions for a class of quasilinear
elliptic equation involving the
p q-Laplace operator
Honghui Yin
a
*
†
and Jing Wen
b,c
Communicated by P. Colli
The existence of at least three weak solutions is established for a class of quasilinear elliptic equation involving the p q-
Laplace operator with Dirichlet boundary condition. The technical approach is mainly on the basis of a three critical points
theorem due to Ricceri. Copyright © 2013 John Wiley & Sons, Ltd.
Keywords: p q-Laplacian; Sobolev space; three critical points theorem
1. Introduction
In this paper, we consider the problem of the type
4
p
u 4
q
u D f .x, u/ C g.x, u/, x 2 ,
u D 0, x 2 @,
(1)
where R
N
.N 1/ is a bounded domain with boundary of class C
1
. , 0 are real numbers, N < q < p. 4
s
u D div.jruj
s2
ru/ is
the usual s-Laplacian operator. f : R 7! R is a continuous function, and g : R 7! R is a Carathéodory function.
Problem (1) comes, for example, from a general reaction–diffusion system
u
t
D divŒH.u/ru C c.x, u/, (2)
where H.u/ D jruj
p2
Cjruj
q2
. This system has a wide range of applications in physics and related science such as biophysics, plasma
physics, and chemical reaction design. In such applications, the function u describes a concentration, the first term on the right-hand
side of (2) corresponds to the diffusion with a diffusion coefficient H.u/; whereas the second one is the reaction and relates to sources
and loss processes. Typically, in chemical and biological applications, the reaction term c.x, u/ has a polynomial form with respect to
the concentration u.
Recently, the existence and multiplicity of the stationary solutions of (2) were studied by many authors, that is, many works
considered the solutions of the following problem
divŒH.u/ru D c.x, u/,
where c.x, u/ are varies functions, see [1–5] and the references therein for details.
For the special case, p D q, (1) becomes the well known p-Laplacian problem. There have been many papers in which the technical
approach adopted is based on the three critical points theorem due to Ricceri [6], for example.
In [7], Bonanno obtained three solutions of the two-point boundary value
u
00
C f .u/ D 0,
u.0/ D u.1/ D 0,
a
School of Mathematical Sciences, Huaiyin Normal University, Jiangsu Huaian 223001, China
b
School of Computer Science and Technology, Huaiyin Normal University, Jiangsu Huaian 223001, China
c
College of Computer Science and Technology, Nanjing University of Aeronautics & Astronautics, Jiangsu Nanjing 210016, China
*Correspondence to: Honghui Yin, School of Mathematical Sciences, Huaiyin Normal University, Jiangsu Huaian 223001, China.
†
E-mail: yinhh@hytc.edu.cn
Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2013