scholarly journals Nodal solutions for the fractional Yamabe problem on Heisenberg groups

2019 ◽  
Vol 150 (2) ◽  
pp. 771-788 ◽  
Author(s):  
Alexandru Kristály
Keyword(s):  
Weak Solutions ◽  
Unitary Group ◽  
Original Problem ◽  
Laplacian Operator ◽  
Yamabe Problem ◽  
Heisenberg Groups ◽  
Nodal Solutions ◽  
Compactness Result ◽  
Homogeneous Dimension ◽  
Nodal Properties

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).

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

2020 ◽  
Vol 57 (4) ◽  
pp. 465-507
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Heat Semigroup ◽  
Schrodinger Operator ◽  
Heisenberg Groups ◽  
Area Integral ◽  
Weak Type Estimates ◽  
Homogeneous Dimension ◽  
Lusin Area Integral ◽  
Reverse Hölder

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

scholarly journals A non-compactness result on the fractional Yamabe problem in large dimensions

2017 ◽  
Vol 273 (12) ◽  
pp. 3759-3830 ◽  
Author(s):  
Seunghyeok Kim ◽  
Monica Musso ◽  
Juncheng Wei
Keyword(s):  
Yamabe Problem ◽  
Compactness Result

WEAK SOLUTIONS OF QUASILINEAR BIHARMONIC PROBLEMS WITH POSITIVE, INCREASING AND CONVEX NONLINEARITIES

2008 ◽  
Vol 06 (03) ◽  
pp. 213-227 ◽  
Author(s):  
I. ABID ◽  
M. JLELI ◽  
N. TRABELSI
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Weak Solutions ◽  
Source Term ◽  
Semilinear Elliptic Equation ◽  
Laplacian Operator ◽  
Navier Boundary Conditions ◽  
Semilinear Elliptic ◽  
Biharmonic Problems

We study the existence of positive weak solutions to a fourth-order semilinear elliptic equation with Navier boundary conditions and a positive, increasing and convex source term. We also prove the uniqueness of extremal solutions. In particular, we generalize results of Mironescu and Rădulescu for the bi-Laplacian operator.

Nondegeneracy of Nodal Solutions to the Critical Yamabe Problem

2015 ◽  
Vol 340 (3) ◽  
pp. 1049-1107 ◽  
Author(s):  
Monica Musso ◽  
Juncheng Wei
Keyword(s):  
Yamabe Problem ◽  
Nodal Solutions

Low-dimensional compact embeddings of symmetric Sobolev spaces with applications

2011 ◽  
Vol 141 (2) ◽  
pp. 383-395 ◽  
Author(s):  
Francesca Faraci ◽  
Antonio Iannizzotto ◽  
Alexandru Kristály
Keyword(s):  
Sobolev Space ◽  
Neumann Problem ◽  
Unbounded Domain ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Symmetric Functions ◽  
Laplacian Operator ◽  
Compact Embeddings ◽  
Cylindrically Symmetric ◽  
Low Dimensional

If Ω is an unbounded domain in ℝN and p > N, the Sobolev space W1,p(Ω) is not compactly embedded into L∈(Ω). Nevertheless, we prove that if Ω is a strip-like domain, then the subspace of W1,p(Ω) consisting of the cylindrically symmetric functions is compactly embedded into L∈(Ω). As an application, we study a Neumann problem involving the p-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many weak solutions. Analogous results are obtained for the case of partial symmetry.

scholarly journals Existence of three weak solutions for a class of discrete problems driven by p-Laplacian operator

2021 ◽  
Vol 22 (1) ◽  
pp. 231-240
Author(s):  
Maurizio Imbesi ◽  
Rahmatollah Lashkaripour ◽  
Zahra Ahmadi
Keyword(s):  
Weak Solutions ◽  
Laplacian Operator ◽  
Discrete Problems

Weak Solutions of the Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Unbounded Domain ◽  
Mathematical Analysis ◽  
Weak Solutions ◽  
Spectral Properties ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Navier Stokes ◽  
Bounded Domains ◽  
Navier Stokes Equations ◽  
Compactness Result

The mathematical analysis of the incompressible Stokes and Navier–Stokes equations in a possibly unbounded domain Ω of Rd (d = 2 or 3) is the purpose of this chapter. Notice that no regularity assumptions will be required on the domain Ω. Because of the compactness result stated in Theorem 1.3, page 27, the case of bounded domains will be different (in fact slightly simpler) than the case of general domains. The study of the spectral properties of the Stokes operator previously defined relies on the study of its inverse, which is in fact much easier. We shall restrict ourselves here to the case of the homogeneous Stokes operator which is adapted to the case of a bounded domain.

scholarly journals Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Boundary Problem ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Ekeland’S Variational Principle ◽  
Laplacian Operator ◽  
Ekeland's Variational Principle ◽  
Existence Of Weak Solutions

We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator:  -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in  Ω,∂u/∂ν=0,on  ∂Ω. Under some appropriate conditions on the functionsp,  e,  a, and  f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.

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