Multiple positive solutions of elliptic systems in exterior domains

2018 ◽  
Vol 20 (06) ◽  
pp. 1750063 ◽  
Author(s):  
Haidong Liu ◽  
Zhaoli Liu
Keyword(s):  
Positive Solutions ◽  
Exterior Domain ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Multiple Positive Solutions ◽  
Exterior Domains ◽  
Existence And Multiplicity ◽  
Large Ball ◽  
Multiplicity Of Positive Solutions ◽  
Positive Functions

In this paper, existence and multiplicity of positive solutions of the elliptic system [Formula: see text] is proved, where [Formula: see text] is an exterior domain in [Formula: see text] such that [Formula: see text] is far away from the origin and contains a sufficiently large ball, [Formula: see text], and the coefficients [Formula: see text] are continuous functions on [Formula: see text] which tend to positive constants at infinity. We do not assume [Formula: see text] to be positive functions.

scholarly journals Existence of multiple positive solutions for semipositone fractional boundary value problems

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 749-759 ◽  
Author(s):  
Şerife Ege ◽  
Fatma Topal
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions ◽  
Fractional Boundary Value Problems

In this paper, we study the existence and multiplicity of positive solutions to the four-point boundary value problems of nonlinear semipositone fractional differential equations. Our results extend some recent works in the literature.

scholarly journals Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

scholarly journals Connected sets of positive solutions of elliptic systems in exterior domains

2019 ◽  
Vol 191 (4) ◽  
pp. 761-778
Author(s):  
Aleksandra Orpel
Keyword(s):  
Elliptic System ◽  
Positive Solutions ◽  
Exterior Domain ◽  
Main Tool ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Exterior Domains ◽  
Asymptotic Decay ◽  
Semilinear Equations ◽  
Connected Sets

Abstract The existence of infinitely many connected sets of positive solutions for a certain elliptic system is investigated in this paper. We consider semilinear equations with perturbed Laplace operators described in an exterior domain. We show that each of these solutions $$\mathbf {u}=( u_{1},u_{2})$$u=(u1,u2) has the minimal asymptotic decay, namely $$ u_{i}(x)=O(||x||^{2-n})$$ui(x)=O(||x||2-n) as $$||x||\rightarrow \infty ,$$||x||→∞,$$i=1,2,$$i=1,2, and finite energy in a neighborhood of infinity. Our main tool is the sub and super-solutions theorem which is based on the Sattinger’s iteration procedure. We do not need any growth assumptions concerning nonlinearities.

Non-homogeneous elliptic equations in exterior domains

2006 ◽  
Vol 136 (1) ◽  
pp. 139-147 ◽  
Author(s):  
J. do Ó ◽  
S. Lorca ◽  
J. Sánchez ◽  
P. Ubilla
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domains ◽  
Existence And Multiplicity ◽  
Negative Function ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Typical Model ◽  
Homogeneous Elliptic Equation

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

scholarly journals EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL HARDY–SOBOLEV EXPONENTS AND SIGN-CHANGING WEIGHT FUNCTION

2012 ◽  
Vol 17 (3) ◽  
pp. 330-350 ◽  
Author(s):  
Nemat Nyamoradi
Keyword(s):  
Weight Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Sobolev Exponent ◽  
Quasilinear Elliptic

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

scholarly journals Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoxiao Su ◽  
Ruyun Ma
Keyword(s):  
Dirichlet Problem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Second Order ◽  
Topological Method ◽  
Multiple Positive Solutions ◽  
Nonlinear Difference Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractWe consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , where $\lambda ,\mu \geq 0$ λ , μ ≥ 0 , $\mathbb{T}=\{2,\ldots ,N-1\}$ T = { 2 , … , N − 1 } with $N>3$ N > 3 , $\phi (s)=s/\sqrt{1-s^{2}}$ ϕ ( s ) = s / 1 − s 2 . The function $f:=\lambda a(t,s)+\mu b(t,s)$ f : = λ a ( t , s ) + μ b ( t , s ) is either sublinear, or superlinear, or sub-superlinear near $s=0$ s = 0 . Applying the topological method, we prove the existence of either one or two, or three positive solutions.

Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents

2010 ◽  
Vol 140 (3) ◽  
pp. 617-633 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Multiple Positive Solutions ◽  
Singular Elliptic Equations ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , $1\le q<2$, $0\le\mu<\bar{\mu}=(N-2)^2/4$, 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on $\bar{\varOmega}$, that change sign on Ω.

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