We study existence and multiplicity of solutions to the following bi-nonlocal p( x)-Kirchhoff equation via Krasnoselskii's genus, on the Sobolev space with variable exponent [ABSTRACT FROM AUTHOR]
Abstract In this paper, we consider the following nonlinear waveequation with variable exponents: u t t − div (| ∇ u | r (⋅) − 2 ∇ u) + | u t | m (⋅) − 2 u t = 0. By using a lemma by Komornik, we prove the decay estimates for the solution under suitable assumptions on the variable exponents m , r and the initial data. We also give two numerical applications to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]
A higher order Kirchhoff type equation ∫ R 2 m ( ∣ ∇ m u ∣ 2 + ∑ γ = 0 m − 1 a γ ( x ) ∣ ∇ γ u ∣ 2 ) d x ( ( − Δ ) m u + ∑ γ = 0 m − 1 ( − 1 ) γ ∇ γ ⋅ ( a γ ( x ) ∇ γ u ) ) = f ( x , u ) ∣ x ∣ β + ϵ h ( x ) in R 2 m is considered in this paper. We assume that the nonlinearity of the equation has exponential critical growth and prove that, for a positive ϵ which is small enough, there are two distinct nontrivial solutions to the equation. When ϵ = 0 , we also prove that the equation has a nontrivial mountain-pass type solution. [ABSTRACT FROM AUTHOR]