scholarly journals Positive solutions for a semipositone problem involving nonlocal operator

2014 ◽  
Vol 132 ◽  
pp. 25-32 ◽  
Author(s):  
Ghasem Afrouzi ◽  
N.T. Chung ◽  
S. Shakeri
Keyword(s):  
Positive Solutions ◽  
Nonlocal Operator ◽  
Semipositone Problem

scholarly journals On the Fractional NLS Equation and the Effects of the Potential Well’s Topology

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Silvia Cingolani ◽  
Marco Gallo
Keyword(s):  
Positive Solutions ◽  
Variational Approach ◽  
Center Of Mass ◽  
Nonlocal Operator ◽  
Potential Well ◽  
Local Minima ◽  
Nls Equation ◽  
Positive Potential ◽  
Regularity Results ◽  
Fractional Nonlinear Schrödinger Equation

AbstractIn this paper we consider the fractional nonlinear Schrödinger equation\varepsilon^{2s}(-\Delta)^{s}v+V(x)v=f(v),\quad x\in\mathbb{R}^{N},where {s\in(0,1)}, {N\geq 2}, f is a nonlinearity satisfying Berestycki–Lions type conditions and {V\in C(\mathbb{R}^{N},\mathbb{R})} is a positive potential. For {\varepsilon>0} small, we prove the existence of at least {{\rm cupl}(K)+1} positive solutions, where K is a set of local minima in a bounded potential well and {{\rm cupl}(K)} denotes the cup-length of K. By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of K for ε small.

Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method

2020 ◽  
Vol 23 (3) ◽  
pp. 837-860 ◽  
Author(s):  
Adel Daoues ◽  
Amani Hammami ◽  
Kamel Saoudi
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
Singular Term ◽  
Nonlocal Problem ◽  
Nonlocal Operator ◽  
Approximation Methods ◽  
Multiple Positive Solutions ◽  
Smooth Functions ◽  
Singular Nonlinearities ◽  
Sobolev Exponent

AbstractIn this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < $\begin{array}{} \displaystyle 2_s^* \end{array}$, where $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).

scholarly journals Existence of positive solutions to BVPs of higher order delay differential equations

2009 ◽  
Vol 42 (1) ◽  
Author(s):  
Jianhua Shen ◽  
Jing Dong
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Positive Solutions ◽  
Delay Differential Equations ◽  
Nonlinear Eigenvalue Problem ◽  
Higher Order ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
Semipositone Problem ◽  
Delay Differential

AbstractThe paper is concerned with the existence of positive solutions for the nonlinear eigenvalue problem with singularity and the superlinear semipositone problem of higher order delay differential equations. The main results are obtained by using Guo-Krasnoselskii’s fixed point theorem in cones. These results extend some of the existing literature.

A direct blowing-up and rescaling argument on nonlocal elliptic equations

2016 ◽  
Vol 27 (08) ◽  
pp. 1650064 ◽  
Author(s):  
Wenxiong Chen ◽  
Congming Li ◽  
Yan Li
Keyword(s):  
Positive Solutions ◽  
Fractional Laplacian ◽  
A Priori Estimates ◽  
A Priori ◽  
Degree Theory ◽  
Nonlocal Operator ◽  
Nonlocal Operators ◽  
Priori Estimates ◽  
Blowing Up ◽  
Existence Of Positive Solutions

In this paper, we develop a direct blowing-up and rescaling argument for nonlinear equations involving nonlocal elliptic operators including the fractional Laplacian. Instead of using the conventional extension method introduced by Caffarelli and Silvestre to localize the problem, we work directly on the nonlocal operator. Using the defining integral, by an elementary approach, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Based on this estimate and the Leray–Schauder degree theory, we establish the existence of positive solutions. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.

scholarly journals Existence of Two Positive Solutions for Two Kinds of Fractional p -Laplacian Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yong Wu ◽  
Said Taarabti
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Multiple Solutions ◽  
Nehari Manifold ◽  
Nonlocal Operator ◽  
Laplacian Equations

The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L K p . Specifically, we get multiple solutions to the following fractional p -Laplacian equations with the help of fibering maps and Nehari manifold. − Δ p s u x = λ u q + u r , u > 0   in   Ω , u = 0 , in   ℝ N \ Ω . . Our results extend the previous results in some respects.

Existence of positive solutions for a class of semipositone problem in whole ℝN

2019 ◽  
Vol 150 (5) ◽  
pp. 2349-2367
Author(s):  
Claudianor O. Alves ◽  
Angelo R. F. de Holanda ◽  
Jefferson A. dos Santos
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Main Tool ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Subcritical Growth ◽  
Existence Of Positive Solutions ◽  
Semipositone Problem ◽  
Image Position

In this paper we show the existence of solution for the following class of semipositone problem P$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$ where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System
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