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)

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
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: art. 150, str. 1-16
• Numbering: Vol. 2020, iss. 1
• PID: 20.500.12556/RUL-120370
• UDC: 517.956
• ISSN on article: 1687-2770
• DOI: 10.1186/s13661-020-01447-9
• COBISS.SI-ID: 28792835
• Publication date in RUL: 18.09.2020

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