On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Heisenberg Group ◽  
Weak Solutions ◽  
Critical Growth ◽  
Subelliptic Equations ◽  
Minimax Methods

AbstractLet ℍwhere f : Ω × ℝ → ℝ behaves like exp ( α |u|we will apply minimax methods to obtain multiplicity of weak solutions.

An Elliptic Equation Involving Exponential Critical Growth in ℝ2

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Francisco S. B. Albuquerque ◽  
Everaldo S. Medeiros
Keyword(s):  
Elliptic Equation ◽  
Weak Solutions ◽  
Functional Space ◽  
Critical Growth ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Existence And Multiplicity ◽  
Minimax Methods ◽  
Exponential Critical Growth

AbstractWe study the following class of nonhomogeneous Schrödinger equations−Δu + V(|x|)u = Q(|x|) f(u) + h(x) in ℝwhere V and Q are unbounded or decaying radial potentials, the nonlinearity f (s) has exponential critical growth and the nonhomogeneous term h belongs to the dual of an appropriate functional space. By combining minimax methods and a version of the Trudinger-Moser inequality, we establish the existence and multiplicity of weak solutions for this class of equations.

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.

A result of multiplicity of solutions for a class of quasilinear equations

2012 ◽  
Vol 55 (2) ◽  
pp. 291-309 ◽  
Author(s):  
Claudianor O. Alves ◽  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Category Theory ◽  
Nonlinear Term ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Subcritical Growth ◽  
Minimax Methods ◽  
Multiplicity Of Positive Solutions

AbstractWe establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem−Lpu+ |u|p−2u=h(u)in Ωλ,u= 0 on ∂Ωλ, where Ωλ= λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter,Lpu≐ Δpu+ Δp(u2)uand the nonlinear termh(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

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