Abstract
This paper analyzes the nonlocal elliptic system involving the p(x)-biharmonic operator. We give the corresponding variational structure of the problem, and then by means of Ricceri’s Variational theorem and the definition of general Lebesgue-Sobolev space, we obtain sufficient conditions for the infinite solutions to this problem.
Highlights
This article analyzes the system R∆ p(x) u ( x ) − M Ω|∇u(x)|p(x) dx p(x)∆p(x) u(x) + ρ(x)|u|p(x)−2 u(x) = λ f (x, u) in Ω, u = ∆u = 0 on ∂Ω, (1)where Ω ⊂ RN (N ≥ 2) with a smooth boundary. p(x) ∈ C(Ω), λ > 0, ρ(x) ∈ L∞ (Ω),∆2p(x) (u) is the operator defined as ∆(|∆u|p(x)−2 ∆u)
In [1], under appropriate conditions and Ricceri’s three critical point theory, Li &Tang researched a class of p-biharmonic problems, and three solutions were obtained under the navier boundary value
The study of a nonlocal type problem involving p-biharmonic operator has been extended to the p(x)-biharmonic operator and reached more general conclusions
Summary
Where Ω ⊂ RN (N ≥ 2) with a smooth boundary. p(x) ∈ C(Ω), λ > 0, ρ(x) ∈ L∞ (Ω),∆2p(x) (u) is the operator defined as ∆(|∆u|p(x)−2 ∆u). In [1], under appropriate conditions and Ricceri’s three critical point theory, Li &Tang researched a class of p-biharmonic problems, and three solutions were obtained under the navier boundary value. In [3], when the nonlinear term f (x, u) satisfying the (AR) condition, using the mountain pass theorem and local minimum theorem, two non-trivial solutions of the p-biharmonic system have been obtained. The authors in [4] researched the same problem in [3] and obtained multiple solutions according to Ricceri’s variational Principle. -Rabinowitz condition, there are multiple solutions to the problem (4) using the Fountain theorem. Based on the Ricceri’s variational principle, Miao [18] studied the (p1 (x), · · · , pn (x)) problems with a Kirchhoff operator. The chief aim of this article is to research the system (1) under appropriate conditions using Ricceri’s variational principle
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