Existence and multiplicity results for a new p(x)-Kirchhoff problem. (January 2020) - Hamdani, Mohamed Karim; Harrabi, Abdellaziz; Mtiri, Foued; Repovš, Dušan D.
Abstract: In this work, we study the existence and multiplicity results for the following nonlocal p ( x ) -Kirchhoff problem: (0.1) − a − b ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x d i v ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | p ( x ) − 2 u + g ( x, u )  in  Ω, u = 0,  on  ∂ Ω, where a ≥ b > 0 are constants, Ω ⊂ R N is a bounded smooth domain, p ∈ C ( Ω ¯ ) with N > p ( x ) > 1, λ is a real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.