scholarly journals Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities

2013 ◽  
Vol 33 (2) ◽  
pp. 237 ◽  
Author(s):  
Jianqing Chen ◽  
Eugénio M. Rocha
Keyword(s):  
Heisenberg Group ◽  
Elliptic Equations ◽  
Existence Of Solution ◽  
Critical Growth

On existence of solution of variational multivalued elliptic equations with critical growth via the Ekeland principle

2015 ◽  
Vol 17 (06) ◽  
pp. 1450038 ◽  
Author(s):  
Claudianor O. Alves ◽  
Marcos L. M. Carvalho ◽  
José V. A. Gonçalves
Keyword(s):  
Variational Principle ◽  
Continuous Function ◽  
Elliptic Equations ◽  
Energy Functional ◽  
Existence Of Solution ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Bounded Smooth Domain ◽  
Locally Lipschitz ◽  
Bounded Smooth

We study the existence and regularity of the solution to the multivalued equation -ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We apply the Ekeland Variational Principle to an associated locally Lipschitz energy functional. A major point in our study is that in order to deal with the obtained Ekeland sequence we developed a generalized version for the framework of Orlicz–Sobolev spaces of a well-known Brézis–Lieb lemma which was employed together with a variant of the Lions concentration-compactness theory to get a solution of the equation.

Nodal solutions for fractional elliptic equations involving exponential critical growth

2020 ◽  
Vol 43 (6) ◽  
pp. 3650-3672
Author(s):  
Manassés Souza ◽  
Uberlandio Batista Severo ◽  
Thiago Luiz do Rêgo
Keyword(s):  
Elliptic Equations ◽  
Critical Growth ◽  
Nodal Solutions ◽  
Exponential Critical Growth

scholarly journals On a class of nonlinear elliptic equations with fast increasing weight and critical growth

2010 ◽  
Vol 249 (5) ◽  
pp. 1035-1055 ◽  
Author(s):  
Marcelo F. Furtado ◽  
Olímpio H. Myiagaki ◽  
João Pablo P. da Silva
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Critical Growth ◽  
Nonlinear Elliptic

scholarly journals Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and Data

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Hicham Redwane
Keyword(s):  
Elliptic Equations ◽  
Existence Result ◽  
Nonlinear Elliptic Equations ◽  
Existence Of Solution ◽  
Unbounded Coefficients ◽  
Renormalized Solution ◽  
Nonlinear Elliptic

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functions may not be in for a finite value of the unknown and the data belong to .

The polynomial growth solutions to some sub-elliptic equations on the Heisenberg group

2019 ◽  
Vol 21 (01) ◽  
pp. 1750069 ◽  
Author(s):  
Hairong Liu ◽  
Tian Long ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Heisenberg Group ◽  
Elliptic Equations ◽  
Polynomial Growth ◽  
Bounded Solution ◽  
Type Theorem ◽  
Divergence Form ◽  
Liouville Type ◽  
Periodic Coefficients ◽  
Liouville Type Theorem

We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [Formula: see text]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [Formula: see text]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant.

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