Multiplicity of solutions for a class of fractional p(x, [cdot])-Kirchhoff-type problems without the Ambrosetti-Rabinowitz condition

Mohamed Karim Hamdani (Author), Jiabin Zuo (Author), Nguyen Thanh Chung (Author), Dušan Repovš (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.

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;

Data

Language:	English
Year of publishing:	2020
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.956
COBISS:	28792835
ISSN:	1687-2770
Views:	418
Downloads:	122
Average score:	0 (0 votes)
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Other data

Type (COBISS):	Article
Pages:	art. 150, str. 1-16
Volume:	ǂVol. ǂ2020
Issue:	ǂiss. ǂ1
Chronology:	Dec. 2020
DOI:	10.1186/s13661-020-01447-9
ID:	12042780