p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators

Abstract

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M ( σ ) ( Δ p ( x ) 2 u - Δ p ( x ) u ) = λ ϑ ( x ) | u | q ( x ) - 2 u ( ∫ Ω ϑ ( x ) q ( x ) | u | q ( x ) d x ) r in Ω , u ∈ W 2 , p ( . ) ( Ω ) ∩ W 0 1 , p ( . ) ( Ω ) , \left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right. under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ : = ∫ Ω | Δ u | p ( x ) p ( x ) + | ∇ u | p ( x ) p ( x ) d x . \sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.} Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.