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

Abstract

We are interested in the existence of solutions for the following fractional $p(x,\cdot )$ p ( x , ⋅ ) -Kirchhoff-type problem: $$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $$ { M ( ∫ Ω × Ω | u ( x ) − u ( y ) | p ( x , y ) p ( x , y ) | x − y | N + p ( x , y ) s d x d y ) ( − Δ ) p ( x , ⋅ ) s u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N , $N\geq 2$ N ≥ 2 is a bounded smooth domain, $s\in (0,1)$ s ∈ ( 0 , 1 ) , $p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$ p : Ω ‾ × Ω ‾ → ( 1 , ∞ ) , $(-\Delta )^{s}_{p(x,\cdot )}$ ( − Δ ) p ( x , ⋅ ) s denotes the $p(x,\cdot )$ p ( x , ⋅ ) -fractional Laplace operator, $M: [0,\infty ) \to [0, \infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) , and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ f : Ω × R → 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.

Existence and multiplicity solutions for a singular elliptic p(x)-Laplacian equation

“…Proposition 2.1. [10,13]. L s(x) (Ω), ∥u∥ s(x) is separable, uniformly convex, reflexive and its conjugate space is L s ′ (x) (Ω), ∥u∥ s ′ (x) , where…”

New class of sixth-order nonhomogeneous $p(x)$-Kirchhoff problems with sign-changing weight functions

“…On the other hand, similar variational methods are also used to study the p(x)-biharmonic operator. For example, see [2,3,4,12,32,33,44] and [14,23,26] for a general Kirchhoff problems with or without the (AR) condition. However, in literature the only result involving a sixth-order problem like (1.1) by using variational methods can be found in [38].…”

Multiplicity of solutions for a higher $ m $-polyharmonic Kirchhoff type equation on unbounded domains

“…For this reason, recently several authors studied problems like (2) trying to drop the condition (AR) γ . We refer the reader to [4,17,22,24,28,31,32] for an overview of references on this subject. Motivated by the works above, we shall assume K is a positive weight function and satisfies the following condition:…”