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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)(Δp(x)2u-Δp(x)u)=λϑ(x)|u|q(x)-2u(∫Ωϑ(x)q(x)|u|q(x)dx)r in Ω,u∈W2,p(.)(Ω)∩W01,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|Δu|p(x)p(x)+|∇u|p(x)p(x)dx.\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.

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form {Δpu=|u|p-2uin Ω,|∇u|p-2∂u∂v=λ‖u‖q,∂Ωp-q|u|q-2uon ∂Ω,\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right. where Ω is a smooth bounded domain in ℝN(N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W1,p(Ω). Using the Ljusterneck-Shnirelmann theory on C1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λk)k≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated.

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

In this paper, we consider the nonlinear eigenvalue problem: