Research Article
Received 9 May 2013 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.3051
MOS subject classification: 35J60; 35J70; 58E05
On a bi-nonlocal p.x /-Kirchhoff equation via
Krasnoselskii’s genus
Francisco Julio S.A. Corrêa
a
*
†
and Augusto César dos Reis Costa
b
Communicated by S. Chen
We study existence and multiplicity of solutions to the following bi-nonlocal p.x/-Kirchhoff equation via Krasnoselskii’s
genus, on the Sobolev space with variable exponent,
M
Z
1
p.x/
jr uj
p.x/
p.x/
u D f .x, u/
Z
F.x, u/
r
in , u D 0 on @,
where is a bounded smooth domain of IR
N
,1 < p.x/< N, M and f are continuous functions, f is an odd function,
F.x, u/ D
Z
u
0
f .x, /d and r > 0 is a real parameter. Copyright © 2014 John Wiley & Sons, Ltd.
Keywords: p.x/-Laplace operator; Sobolev space with variable exponent; Krasnoselskii’s genus
1. Introduction
In this paper, we are going to study the existence and multiplicity of solutions of the p.x/- Kirchhoff equation, with an additional
nonlocal term,
8
<
:
M
Z
1
p.x/
jruj
p.x/
p.x/
u D f .x, u/
Z
F.x, u/
r
in ,
u D 0 on @,
(1.1)
where IR
N
is a bounded smooth domain, f : IR ! IR and M : IR
C
! IR
C
are continuous functions that satisfy conditions,
which will be stated later, F.x, u/ D
Z
u
0
f .x, /d , r > 0 is a real parameter and
p.x/
is the p.x/-Laplacian operator, that is,
p.x/
u D
N
X
iD1
@
@x
i
jruj
p.x/2
@u
@x
i
, 1 < p.x/< N.
We assume the following hypotheses on M and f : there are positive constants A
0
, A, B
0
, B, Q
1
, Q
2
, and functions ˛.x/, ˇ.x/, .x/,
q.x/ 2 C
C
. /, see Section 2, such that
A
0
C At
˛.x/
M.t/ B
0
C Bt
ˇ.x/
, (1.2)
and
Q
1
t
.x/1
f .x, t/ Q
2
t
q.x/1
, (1.3)
a
Centro de Ciências e Tecnologia, Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, CEP:58.109-970, Campina Grande - PB -
Brazil
b
Instituto de Ciências Exatas e Naturais, Faculdade de Matemática Universidade Federal do Pará CEP:66075-110 Belém - PA - Brazil.
* Correspondence to: Francisco Julio S.A. Corrêa, Centro de Ciências e Tecnologia, Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina
Grande, CEP:58.109-970, Campina Grande - PB - Brazil.
†
E-mail: fjsacorrea@gmail.com
Copyright © 2014 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014