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A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

For n = 2 m 4 , let Ω n be a bounded smooth domain and 𝒩 L a compact smooth Riemannian manifold without boundary. Suppose that { u k } W m , 2 ( Ω , 𝒩 ) is a sequence of weak solutions in the critical dimension to the perturbed m -polyharmonic maps d d t | t = 0 E m ( Π ( u + t ξ ) ) = 0 with Φ k 0 in ( W m , 2 ( Ω , 𝒩 ) ) * and u k u weakly in W m , 2 ( Ω , 𝒩 ) . Then u is an m -polyharmonic map. In particular, the space of m -polyharmonic maps is sequentially compact for the weak- W m , 2 topology.

A Weighted Eigenvalue Problems Driven by both p ( · ) -Harmonic and p ( · ) -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u W 2 , p ( · ) ( Ω ) W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

Equations de von Kármán. I. Résultat d'existence pour les problèmes aux limites non homogènes.

Július Cibula (1984)

Aplikace matematiky

Dans l'article, on a défini une équation d'operateur équivalent à la formulation variationnelle du problème. Les solutions de cette équation sont des points critiques de la fonctionnelle qu'elle porte le nom d'énergie totale de déformation. La fonctionnelle est coercive et faiblement séquentiellement semi-continue inférieure. Par le théorème de l'analyse fonctionnelle, on a obtenu le résultat d'existence pour le problème.

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ - d i v ( | x | - a p | u | p - 2 u ) + h ( x ) | u | p - 2 u = g ( x ) | u | r - 2 u , x ∈ Ω, ⎨ ⎩ | x | - a p | u | p - 2 u / ν = λ f ( x ) | u | q - 2 u , x ∈ ∂Ω, where Ω is an exterior domain in N , that is, Ω = N D , where D is a bounded domain in N with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method,...

Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces

Nguyen Thanh Chung (2015)

Annales Polonici Mathematici

We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where Ω N , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, ρ ( u ) = Ω ( Φ ( | u | ) + Φ ( | u | ) ) d x , M: [0,∞) → ℝ is a continuous function, K L ( Ω ) , and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ Δ ( | Δ u | p 2 Δ u ) d i v ( | u | p 2 u ) = λ f ( x , u ) + μ g ( x , u ) in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where Ω N (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

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