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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.

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.

Faisceaux d'espaces de Sobolev et principes du minimum

Denis Feyel, A. de La Pradelle (1975)

Annales de l'institut Fourier

On montre que le faisceau des sursolutions locales dans W loc 2 d’un certain opérateur elliptique L est maximal pour un principe du minimum adapté aux espaces de Sobolev. La continuité de la réduite variationnelle des éléments continus permet alors d’étudier des représentants s.c.i.

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