Solutions for a nonlocal problem involving a Hardy potential and critical growth

2021 ◽  
Author(s):  
Chunhua Wang ◽  
Jing Yang ◽  
Jing Zhou
Keyword(s):  
Nonlocal Problem ◽  
Critical Growth ◽  
Hardy Potential

scholarly journals Fractional KPZ equations with critical growth in the gradient respect to Hardy potential

2020 ◽  
Vol 201 ◽  
pp. 111942
Author(s):  
Boumediene Abdellaoui ◽  
Ireneo Peral ◽  
Ana Primo ◽  
Fernando Soria
Keyword(s):  
Critical Growth ◽  
Hardy Potential

scholarly journals MULTIPLE POSITIVE SOLUTIONS FOR A CRITICAL GROWTH PROBLEM WITH HARDY POTENTIAL

2006 ◽  
Vol 49 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Pigong Han
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Variational Approach ◽  
Critical Growth ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Unique Positive Solution ◽  
Existence And Nonexistence ◽  
Growth Problem

AbstractIn this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

scholarly journals Three solutions for a nonlocal problem with critical growth

2019 ◽  
Vol 469 (2) ◽  
pp. 841-851
Author(s):  
Natalí Ailín Cantizano ◽  
Analía Silva
Keyword(s):  
Nonlocal Problem ◽  
Critical Growth ◽  
Three Solutions

scholarly journals Optimal solvability for a nonlocal problem at critical growth

2018 ◽  
Vol 264 (3) ◽  
pp. 2242-2269 ◽  
Author(s):  
Lorenzo Brasco ◽  
Marco Squassina
Keyword(s):  
Nonlocal Problem ◽  
Critical Growth

scholarly journals The Nehari manifold for indefinite Kirchhoff problem with Caffarelli-Kohn-Nirenberg type critical growth

2021 ◽  
pp. 1-30
Author(s):  
David G. Costa ◽  
João Marcos Do Ó ◽  
Pawan K. Mishra
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Nonlocal Problem ◽  
Nehari Manifold ◽  
Critical Growth ◽  
Positive Real ◽  
Alpha Beta ◽  
The Nehari Manifold ◽  
Kirchhoff Model ◽  
Kirchhoff Problem

In this paper we study the following class of nonlocal problem involving Caffarelli-Kohn-Nirenberg type critical growth $$ L(u)-\lambda h(x)|x|^{-2(1+a)}u=\mu f(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\quad \text{in } \mathbb R^N, $$% where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $\lambda, \mu$ are positive real parameters and $1< q< 2< 4< p=2N/[N+2(b-a)-2]$, $0\leq a< b< a+1< N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div} \big(|x|^{-2a}\nabla u\big) $$ and the function $M\colon \mathbb R^+_0\to\mathbb R^+_0$ is exactly the Kirchhoff model, given by $M(t)=\alpha+\beta t$, $\alpha, \beta> 0$. The above problem has a double lack of compactness, firstly because of the non-compactness of Caffarelli-Kohn-Nirenberg embedding and secondly due to the non-compactness of the inclusion map $$u\mapsto \int_{\mathbb R^N}h(x)|x|^{-2(a+1)}|u|^2dx,$$ as the domain of the problem in consideration is unbounded. Deriving these crucial compactness results combined with constrained minimization argument based on Nehari manifold technique, we prove the existence of at least two positive solutions for suitable choices of parameters $\lambda$ and $\mu$.

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