Nondegeneracy of Nodal Solutions to the Critical Yamabe Problem

AbstractWe prove that the fractional Yamabe equation ${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$ on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where ${\rm {\cal L}}_\gamma $ denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and $\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).

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Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2020123 ◽

2020 ◽

Vol 40 (4) ◽

pp. 2495-2514

Author(s):

Juan Carlos Fernández ◽

◽

Oscar Palmas ◽

Jimmy Petean ◽

Keyword(s):

Elliptic Problems ◽

Projective Spaces ◽

Yamabe Problem ◽

Round Sphere ◽

Nodal Solutions

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Multiple minimal nodal solutions for a quasilinear Schrödinger equation with symmetric potential

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2004.09.012 ◽

2005 ◽

Vol 304 (1) ◽

pp. 170-188 ◽

Cited By ~ 3

Author(s):

Marcelo F. Furtado

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Nodal Solutions ◽

Quasilinear Schrödinger Equation ◽

Symmetric Potential

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Finite Element Analysis of a Polymer- Polymer Sliding Contact for Schallamach Wave and Wear

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.348-349.633 ◽

2007 ◽

Vol 348-349 ◽

pp. 633-636 ◽

Cited By ~ 1

Author(s):

Muhammad Azeem Ashraf ◽

Bijan Sobhi-Najafabadi ◽

Özdemir Göl ◽

D. Sugumar

Keyword(s):

Finite Element ◽

Polymer Surface ◽

Sliding Contact ◽

Quasi Steady State ◽

Nodal Solutions ◽

Element Analysis ◽

Steady State Analysis ◽

Initiation And Propagation ◽

Wear Law ◽

Nodal Level

Sliding polymer-polymer surface contacts, due to their inherent elastic properties, exhibit detachment waves also termed as Schallamach waves. Such waves effect the initiation and propagation of wear along the sliding contacts. This paper presents quasi steady-state analysis of such a sliding contact using finite element. The contact is modeled and nodal solutions for pressure are obtained for small sliding steps. Analysis of orthogonal pressure components at the contact nodes reveals the formation of Schallamach wave phenomenon. Further, appropriate wear law is used for calculation of wear at nodal level.

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