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On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

M. Prizzi, K. P. Rybakowski (2003)

Studia Mathematica

We study a family of semilinear reaction-diffusion equations on spatial domains Ω ε , ε > 0, in l lying close to a k-dimensional submanifold ℳ of l . As ε → 0⁺, the domains collapse onto (a subset of) ℳ. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by H ¹ s ( Ω ) . The definition of H ¹ s ( Ω ) , given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations...

On Kelvin type transformation for Weinstein operator

Martina Šimůnková (2001)

Commentationes Mathematicae Universitatis Carolinae

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator W k : = Δ + k x n x n on n is proved. In this note there is shown that in the cases k 0 , k 2 no other transforms of this kind exist and for case k = 2 , all such transforms are described.

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ - ( a + b Ω | u | ² d x ) Δ u = u + λ u q - 1 / | x | β in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

On Korn's second inequality

J. A. Nitsche (1981)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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