Solutions for fractional p ( x , ⋅ ) $p(x,\cdot )$ -Kirchhoff-type equations in R N $\mathbb{R}^{N}$

Abstract

Abstract In this paper, we discuss the fractional p ( x , ⋅ ) $p(x,\cdot )$ -Kirchhoff-type equations M ( ∫ R N × R N 1 p ( x , y ) | u ( x ) − u ( y ) | p ( x , y ) | x − y | N + s p ( x , y ) d x d y ) ( − Δ p ( x , . ) ) s u + | u | p ¯ ( x ) − 2 u = f ( x , u ) . $$ M\left (\int _{\mathbb{R}^{N}\times \mathbb{R}^{N}} \frac{1}{p(x,y)} \frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}dxdy\right )(-\Delta _{p(x,.)})^{s} u+|u|^{\bar{p}(x)-2}u=f(x,u).$$ We weaken the conditions on the nonlinear term f and get the existence and multiplicity of solutions via variational methods, which improves some previous results.

Keywords

• Fractional p ( x , ⋅ ) $p(x,\cdot )$ -Kirchhoff-type equations
• Mountain pass lemma
• Symmetric mountain pass lemma