Existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems

Abstract

In this paper, we deal with the existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems as follows: $ \left\{ \begin{array}{l}M\Big(\int_Q\frac{1}{p(x, y)}\frac{| v(x)-v(y)|^{p(x, y)}}{| x-y|^{d+sp(x, y)}}dxdy\Big)(-\Delta_{p(x)})^s v(x)\ \, \, \, \, \, \, \,\\ = \lambda| v(x)|^{r(x)-2}v(x), \;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \, \;\; \;\;\;\, \, \, \, \, \, \, \, \, \text{in}\;\;\Omega, \\ v = 0, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \, \, \, \, \, \text{in}\;\mathbb{R}^d\backslash\Omega, \end{array}\right. $ where $ (-\triangle_{p(x)})^s $ is the fractional $ p(x) $-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of $ M $ and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.

Keywords

• the symmetryic mountain pass theorem
• kirchhoff-type problem
• fractional $ p(x) $-laplacian
• fractional sobolev space with variable exponents