scholarly journals The Solvability of Fractional Elliptic Equation with the Hardy Potential

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Siyu Gao ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Zhan-Ping Ma
Keyword(s):  
Elliptic Equation ◽  
Lipschitz Domain ◽  
Sobolev Inequality ◽  
Elliptic Equations ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Bounded Lipschitz Domain ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Fractional Laplace Operator

In this paper, we study the existence and nonexistence of solutions to fractional elliptic equations with the Hardy potential −Δsu−λu/x2s=ur−1+δgu,in Ω,ux>0,in Ω,ux=0,in ℝN∖Ω, where Ω⊂ℝN is a bounded Lipschitz domain with 0∈Ω, −Δs is a fractional Laplace operator, s∈0,1, N>2s, δ is a positive number, 2<r<rλ,s≡N+2s−2αλ/N−2s−2αλ+1, αλ∈0,N−2s/2 is a parameter depending on λ, 0<λ<ΛN,s, and ΛN,s=22sΓ2N+2s/4/Γ2N−2s/4 is the sharp constant of the Hardy–Sobolev inequality.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR QUADRATIC GROWTH TERMS

2011 ◽  
Vol 13 (04) ◽  
pp. 607-642 ◽  
Author(s):  
LUCIO BOCCARDO ◽  
TOMMASO LEONORI ◽  
LUIGI ORSINA ◽  
FRANCESCO PETITTA
Keyword(s):  
Positive Solutions ◽  
Lebesgue Space ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Quadratic Growth ◽  
Open Set ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Quasilinear Problems ◽  
Quasilinear Elliptic

In this paper, we deal with positive solutions for singular quasilinear problems whose model is [Formula: see text] where Ω is a bounded open set of ℝN, g ≥ 0 is a function in some Lebesgue space, and γ > 0. We prove both existence and nonexistence of solutions depending on the value of γ and on the size of g.

scholarly journals Analysis of the controllability from the exterior of strong damping nonlocal wave equations

2020 ◽  
Vol 26 ◽  
pp. 42 ◽  
Author(s):  
Mahamadi Warma ◽  
Sebastián Zamorano
Keyword(s):  
Wave Equation ◽  
Lipschitz Domain ◽  
Wave Equations ◽  
Control Function ◽  
Complete Analysis ◽  
Strong Damping ◽  
Open Set ◽  
Bounded Lipschitz Domain ◽  
Approximately Controllable ◽  
Fractional Laplace Operator

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 <s< 1, Ω ⊂ ℝN(N≥ 1) is a bounded Lipschitz domain and the parameterδ> 0, then there is no control functiongsuch that the following system\begin{align} u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^++ \delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^+ t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{39}\\ u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^- +\delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^- t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{40} \end{align}is exact or null controllable at timeT> 0. In the second part, we prove that for everyδ≥ 0 and 0 <s< 1, the system is indeed approximately controllable for anyT> 0 andg∈D(O× (0,T)), whereO⊂ ℝN\ Ω is any non-empty open set.

scholarly journals FOUNTAIN-LIKE SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATIONS WITH CRITICAL GROWTH AND HARDY POTENTIAL

2005 ◽  
Vol 07 (06) ◽  
pp. 867-904 ◽  
Author(s):  
VERONICA FELLI ◽  
SUSANNA TERRACINI
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equation ◽  
Blow Up ◽  
Nonlinear Elliptic Equations ◽  
Critical Growth ◽  
Hardy Potential ◽  
Nonlinear Elliptic ◽  
Hardy Type ◽  
Type Potential

We prove the existence of fountain-like solutions, obtained by superposition of bubbles of different blow-up orders, for a nonlinear elliptic equation with critical growth and Hardy-type potential.

Sign in / Sign up

Export Citation Format

Share Document