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Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

  • Nguyen Thanh Chung EMAIL logo and Zohreh Naghizadeh
Published/Copyright: December 10, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 6

Abstract

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Δp(x)2u−M(∫Ω1p(x)|∇u|p(x)dx)Δp(x)u+|u|p(x)−2u=λf(x,u)+μg(x,u) in Ω,u=Δu=0 on ∂Ω,

where p−:=infx∈Ω¯p(x)>max1,N2,λ>0 and μ ≥ 0 are real numbers, Ω ⊂ ℝN (N ≥ 1) is a smooth bounded domain, Δp(x)2u=Δ(|Δu|p(x)−2Δu) is the operator of fourth order called the p(x)-biharmonic operator, Δp(x)u = div(|∇u|p(x)–2∇u) is the p(x)-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f, g : Ω × ℝ → ℝ are two L1-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.

Keywords: Fourth order elliptic equations; Kirchhoff type problems; variable exponents; three critical point theorem; variational methods
MSC 2010: Primary 35J35; Secondary 35G30; 46E35

  1. Communicated by Alberto Lastra

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Received: 2020-05-14
Accepted: 2021-01-28
Published Online: 2021-12-10
Published in Print: 2021-12-20

© 2021 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2021-0063
Keywords for this article
Fourth order elliptic equations; Kirchhoff type problems; variable exponents; three critical point theorem; variational methods

Articles in the same Issue

  1. Regular Papers
  2. Semidistributivity and Whitman Property in implication zroupoids
  3. Composition of binary quadratic forms over number fields
  4. On Z-Symmetric Rings
  5. On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
  6. Coefficient estimates for Libera type bi-close-to-convex functions
  7. Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
  8. On rapidly oscillating solutions of a nonlinear elliptic equation
  9. Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
  10. Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
  11. On absolute double summability methods with high indices
  12. On the continuity of lattice isomorphisms on C(X, I)
  13. New fixed point theorems for countably condensing maps with an application to functional integral inclusions
  14. Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
  15. The form of locally defined operators in waterman spaces
  16. Conformal vector fields on almost co-Kähler manifolds
  17. A certain η-parallelism on real hypersurfaces in a nonflat complex space form
  18. On log-bimodal alpha-power distributions with application to nickel contents and erosion data
  19. Univariate and bivariate extensions of the generalized exponential distributions
  20. Pellian equations of special type
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