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Multiplicity of solutions for a class of fractional ▫$p(x, \cdot)$▫Kirchhofftype problems without the AmbrosettiRabinowitz condition
Hamdani, Mohamed Karim
(
Author
),
Zuo, Jiabin
(
Author
),
Chung, Nguyen Thanh
(
Author
),
Repovš, Dušan
(
Author
)
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Abstract
We are interested in the existence of solutions for the following fractional ▫$p(x,\cdot)$▫Kirchhofftype problem: ▫$$\textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)u(y) \vert ^{p(x,y)}}{p(x,y) \vert xy \vert ^{N+p(x,y)s}}} \,dx \,dy )(\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases}$$▫ where ▫$\Omega \subset \mathbb{R}^{N}$▫,▫$ N\geq 2$▫ is a bounded smooth domain, ▫$s\in (0,1)$▫, ▫$p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$▫, ▫$(\Delta )^{s}_{p(x,\cdot)}$▫ denotes the ▫$p(x,\cdot )$▫fractional Laplace operator, ▫$M: [0,\infty ) \to [0, \infty )$▫, and ▫$f: \Omega \times \mathbb{R} \to \mathbb{R}$▫ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to BartoloBenciFortunato (Nonlinear Anal. 7(9):9811012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the AmbrosettiRabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.
Language:
English
Keywords:
fractional ▫$p(x,\cdot)$▫Kirchhofftype problems
,
▫$p(x,\cdot)$▫fractional Laplace operator
,
AmbrosettiRabinowitz type conditions
,
symmetric mountain pass theorem
,
Cerami compactness condition
,
fractional Sobolev spaces with variable exponent
,
multiplicity of solutions
Work type:
Article (dk_c)
Tipology:
1.01  Original Scientific Article
Organization:
PEF  Faculty of Education
Year:
2020
Number of pages:
art. 150, str. 116
Numbering:
Vol. 2020, iss. 1
UDC:
517.956
ISSN on article:
16872770
DOI:
10.1186/s13661020014479
COBISS.SIID:
28792835
Views:
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Record is a part of a journal
Title:
Boundary value problems
Shortened title:
Bound. value probl.
Publisher:
Springer
ISSN:
16872770
COBISS.SIID:
62113025
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