On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Abstract

In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent ∆ 2 p(x) u − M Ω 1 p(x) |∇u| p(x) dx ∆ p(x) u = λf (x, u) in Ω, u = ∆u = 0 on ∂Ω, where Ω ⊂ R N , N ≥ 3, is a smooth bounded domain, M (t) = a + bt κ , a, κ > 0, b ≥ 0, λ is a positive parameter, ∆ 2 p(x) u = ∆(|∆u| p(x)−2 ∆u) is the operator of fourth order called the p(x)-biharmonic operator, ∆ p(x) u = div |∇u| p(x)−2 ∇u is the p(x)-Laplacian, p : Ω → R is a log-Hölder continuous function and f : Ω × R → R is a continuous function satisfying some certain conditions. Using Ekeland's variational principle combined with variational techniques, an existence result is established in an appropriate function space.

Existence and Multiplicity of Solutions for a Biharmonic Equation with p(x)-Kirchhoff Type

“…is model can describe the physical phenomenon of "point by point anisotropy." Various methods have extensively studied the problem of differential operators with variable exponents, as shown in [1][2][3][4][5][6]. In [7,8], the authors used changeable nonlinear terms to outline the boundary of the real image, eliminate the possible noise, and solve the problems in image processing.…”