New class of sixth-order nonhomogeneous p ( x )-Kirchhoff problems with sign-changing weight functions

Advances in Nonlinear Analysis

article

Mohamed Karim Hamdani¹  Nguyen Thanh Chung³  Dušan D. Repovš⁴ 

[1] Science and Technology for Defense Laboratory LR19DN01, Military Research Center, Tunisia Military School of Aeronautical Specialities;Mathematics Department, University of Sfax, Faculty of Science of Sfax;Department of Mathematics, Quang Binh University;Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana;Institute of Mathematics

关键词: Variable exponents;    Kirchhoff type problems;    p(x)-triharmonic operator;    Sign-changing functions;    Concave-convex terms;    Ekeland’s variational principle;    Multiple solutions;   

DOI  :  10.1515/anona-2020-0172

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

In this paper, we prove the existence of multiple solutions for the following sixth-order p ( x )-Kirchhoff-type problem − M∫ Ω 1p(x)|∇ Δ u|p(x)dxΔ p(x)3u=λ f(x)|u|q(x)− 2u+g(x)|u|r(x)− 2u+h(x)inΩ ,u=Δ u=Δ 2u=0,on∂ Ω , $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\[0.3em] u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ N is a smooth bounded domain, N> 3,Δ p(x)3u:=div⁡ (Δ (|∇ Δ u|p(x)− 2∇ Δ u)) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ is the p ( x )-triharmonic operator, p , q , r ∈ C ( Ω ), 1 0, λ > 0, g : Ω × ℝ → ℝ is a nonnegative continuous function while f , h : Ω × ℝ → ℝ are sign-changing continuous functions in Ω . To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order p ( x )-Kirchhoff type problems with sign changing Kirchhoff functions.

【 授权许可】

CC BY