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Multiplicity of solutions for a class of fractional ▫$p(x, \cdot)$▫-Kirchhoff-type problems without the Ambrosetti-Rabinowitz 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 ▫$p(x,\cdot)$▫-Kirchhoff-type problem: ▫$$\textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases}$$▫ 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 Bartolo-Benci-Fortunato (Nonlinear Anal. 7(9):981-1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti-Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

Language:English
Keywords:fractional ▫$p(x,\cdot)$▫-Kirchhoff-type problems, ▫$p(x,\cdot)$▫-fractional Laplace operator, Ambrosetti-Rabinowitz 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. 1-16
Numbering:Vol. 2020, iss. 1
UDC:517.956
ISSN on article:1687-2770
DOI:10.1186/s13661-020-01447-9 This link opens in a new window
COBISS.SI-ID:28792835 This link opens in a new window
Views:134
Downloads:70
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Record is a part of a journal

Title:Boundary value problems
Shortened title:Bound. value probl.
Publisher:Springer
ISSN:1687-2770
COBISS.SI-ID:62113025 This link opens in a new window

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