Asymptotic Analysis - Volume 138, issue 4

Authors: Ganguly, Debdip | Gupta, Diksha | Sreenadh, K.

Article Type: Research Article

Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in  B N , u ∈ H 1 ( B N ) , where B N …denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞ , if N = 2 , λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ) . We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1 . Next, we prove that for a ( x ) ⩾ 1 , there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space. Show more

Keywords: Hyperbolic space, hyperbolic bubbles, Palais–Smale decomposition, semilinear elliptic problem

DOI: 10.3233/ASY-241895

Citation: Asymptotic Analysis, vol. 138, no. 4, pp. 225-253, 2024

Authors: Chakrabortty, Amartya | Griso, Georges | Orlik, Julia

Article Type: Research Article

Abstract: This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic perforated plate made of stiff material with holes filled by a soft matrix material. The structure is clamped on a cylindrical part. Two cases of asymptotic analysis are considered: one without pre-strain and the other with matrix pre-strain. In both cases, the total elastic energy is in the von-Kármán (vK) regime (ε 5 ). A new splitting of the displacements is introduced to …analyze the asymptotic behavior. The displacements are decomposed using the Kirchhoff–Love (KL) plate displacement decomposition. The use of a re-scaling unfolding operator allows for deriving the asymptotic behavior of the Green St. Venant’s strain tensor in terms of displacements. The limit homogenized energy is shown to be of vK type with linear elastic cell problems, established using the Γ-convergence. Additionally, it is shown that for isotropic homogenized material, our limit vK plate is orthotropic. The derived results have practical applications in the design and analysis of composite structures. Show more

Keywords: Homogenization, dimension reduction, unfolding operators, Γ-convergence, non-linear elasticity, von-Kármán plate, pre-strain

DOI: 10.3233/ASY-241896

Citation: Asymptotic Analysis, vol. 138, no. 4, pp. 255-310, 2024