Existence results for a Kirchhoff-type equations involving the fractional $$p_{1}(x)$$ & $$p_{2}(x)$$-Laplace operator

(p(x),q(x))-Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

“…dx, and has a wide range of applications and research in physical systems, such as non-homogeneous Kirchhoff-type equations in R N [2], nonlocal Kirchhoff equations of elliptic type [3], Kirchhoff-Schrödinger type equations [4], p(x)-Laplacian Dirichlet problem [5,6], Kirchhoff-Choquard equations involving variable-order [7,8], fractional p(•)-Kirchhoff type problem in R N [9], Kirchhofftype equations involving the fractional p 1 (x)&p 2 (x)-Laplace operator [10], fractional p(x, •)-Kirchhoff-type problems in R N [11], and fractional Sobolev space and applications to nonlocal variational problems [12]. For more Kirchhoff-type problems, we also mention that [13] studied a class of Kirchhoff nonlocal fractional equations and obtained the existence of three solutions, Ref.…”

“…where M([η] p s,p ) = [η] p(θ−1) s,p and h(x) is a sign-changing function. The existence of least energy solutions (10) was obtained by utilizing the Nehari manifold method.…”