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)

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 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 Article (dk_c) 1.01 - Original Scientific Article PEF - Faculty of Education 2020 art. 150, str. 1-16 Vol. 2020, iss. 1 517.956 1687-2770 10.1186/s13661-020-01447-9 28792835 134 70 (0 votes) Voting is allowed only to logged in users. AddThis uses cookies that require your consent. Edit consent...

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Title: Boundary value problems Bound. value probl. Springer 1687-2770 62113025