SMOOTH POSITIVE SOLUTIONS TO AN ELLIPTIC MODEL WITH $C^{2}$ FUNCTIONS

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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On existence, multiplicity, uniqueness and stability of positive solutions of a Leslie–Gower type diffusive predator–prey system

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2014.4.11 ◽

2014 ◽

Vol 19 (4) ◽

pp. 669-688

Author(s):

Jun Zhou ◽

Keyword(s):

Positive Solutions ◽

Predator Prey ◽

Prey System

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Existence of Multiple Positive Solutions for Singular Semipositone Boundary Value Problem of Impulsive Differential Equations with the First Derivative in the Nonlinear Term

Acta Analysis Functionalis Applicata ◽

10.3724/sp.j.1160.2012.00298 ◽

2012 ◽

Vol 14 (3) ◽

pp. 298

Author(s):

Mingming HUANG ◽

Huiqin LU

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Positive Solutions ◽

Nonlinear Term ◽

Boundary Value ◽

Impulsive Differential Equations ◽

Multiple Positive Solutions ◽

First Derivative

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Positive Solutions of Singular Superlinear (k,n-k) Multi-Point Boundary Value Problems

Acta Analysis Functionalis Applicata ◽

10.3724/sp.j.1160.2010.00017 ◽

2010 ◽

Vol 12 (1) ◽

pp. 17-20

Author(s):

Yujun CUI ◽

Xingdong TANG ◽

Feng WANG

Keyword(s):

Boundary Value Problems ◽

Positive Solutions ◽

Boundary Value ◽

Point Boundary

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Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

Communications in Contemporary Mathematics ◽

10.1142/s021919972050008x ◽

2020 ◽

pp. 2050008

Author(s):

Shaya Shakerian

Keyword(s):

Variational Methods ◽

Bounded Domain ◽

Positive Solutions ◽

Variational Approach ◽

Nehari Manifold ◽

Multiple Positive Solutions ◽

Hardy Potential ◽

Smooth Bounded Domain ◽

Existence And Multiplicity ◽

The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

Abstract and Applied Analysis ◽

10.1155/2014/602604 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Hongjie Liu ◽

Xiao Fu ◽

Liangping Qi

Keyword(s):

Fixed Point ◽

Green Function ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Positive Solutions ◽

Boundary Value ◽

Fractional Boundary Value Problem ◽

Existence Of Positive Solutions ◽

Liouville Fractional Derivative ◽

Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

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Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽

10.3390/sym13010107 ◽

2021 ◽

Vol 13 (1) ◽

pp. 107

Author(s):

Daliang Zhao ◽

Juan Mao

Keyword(s):

Positive Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Main Tool ◽

Integral Boundary ◽

Differential Systems ◽

Fractional Differential Systems ◽

Existence And Multiplicity ◽

Fractional Differential ◽

Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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