scholarly journals Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2073-2088 ◽  
Author(s):  
Mouna Kratou
Keyword(s):  
Fractional Derivative ◽  
Variational Methods ◽  
Positive Solutions ◽  
Ground State ◽  
Multiplicity Of Solutions ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Multiplicity Of Positive Solutions ◽  
Kirchhoff Problem

The purpose of this paper is to study the existence and multiplicity of solutions to the following Kirchhoff equation with singular nonlinearity and Riemann-Liouville Fractional Derivative: (P?){a+b ?T0|0D?t(u(t))|pdt)p-1 tD?T (?p(0D?tu(t)) = ?g(t)/u?(t) + f(t, u(t)), t ? (0,T); u(0)=u(T)=0, where a ? 1, b, ? > 0, p > 1 are constants, 1/p < ? ? 1, 0 < ? < 1, g ? C([0,1]) and f ? C1([0,T] x R,R). Under appropriate assumptions on the function f, we employ variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter ?.

Positive solutions of fractional elliptic equation with critical and singular nonlinearity

2017 ◽  
Vol 6 (3) ◽  
pp. 327-354 ◽  
Author(s):  
Jacques Giacomoni ◽  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Content Type ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractIn this article, we study the following fractional elliptic equation with critical growth and singular nonlinearity:(-\Delta)^{s}u=u^{-q}+\lambda u^{{2^{*}_{s}}-1},\qquad u>0\quad\text{in }% \Omega,\qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega,where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega}, {n>2s}, {s\in(0,1)}, {\lambda>0}, {q>0} and {2^{*}_{s}=\frac{2n}{n-2s}}. We use variational methods to show the existence and multiplicity of positive solutions with respect to the parameter λ.

scholarly journals Positive Solutions for Elliptic Problems with the Nonlinearity Containing Singularity and Hardy-Sobolev Exponents

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yong-Yi Lan ◽  
Xian Hu ◽  
Bi-Yun Tang
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Elliptic Problems ◽  
Semilinear Elliptic Equations ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

In this paper, we study multiplicity of positive solutions for a class of semilinear elliptic equations with the nonlinearity containing singularity and Hardy-Sobolev exponents. Using variational methods, we establish the existence and multiplicity of positive solutions for the problem.

scholarly journals Existence and Multiplicity of Positive Solutions for Schrödinger-Kirchhoff-Poisson System with Singularity

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Mengjun Mu ◽  
Huiqin Lu
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Nehari Manifold ◽  
Poisson System ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold

We study a singular Schrödinger-Kirchhoff-Poisson system by the variational methods and the Nehari manifold and we prove the existence, uniqueness, and multiplicity of positive solutions of the problem under different conditions.

scholarly journals On Dirichlet problem for fractional p-Laplacian with singular non-linearity

2016 ◽  
Vol 8 (1) ◽  
pp. 52-72 ◽  
Author(s):  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega} , {n>sp} , {s\in(0,1)} , {\lambda>0} , {0<q\leq 1} and {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

Multiplicity of Positive Solutions for Critical Singular Elliptic Systems With Concave-Convex Nonlinearities

2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractIn this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

scholarly journals Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-21
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Ground State ◽  
Unbounded Domains ◽  
Ground State Solution ◽  
Multiple Positive Solutions ◽  
Dirichlet Problems ◽  
Existence And Multiplicity ◽  
Sobolev Constant ◽  
Multiplicity Of Positive Solutions

We consider the elliptic problem−Δu+u=b(x)|u|p−2u+h(x)inΩ,u∈H01(Ω), where2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ωis a smooth unbounded domain inℝN, b(x)∈C(Ω), andh(x)∈H−1(Ω). We use the shape of domainΩto prove that the above elliptic problem has a ground-state solution if the coefficientb(x)satisfiesb(x)→b∞>0as|x|→∞andb(x)≥cfor some suitable constantsc∈(0,b∞), andh(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficientb(x)also satisfies the above conditions,h(x)≥0and0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), whereS(Ω)is the best Sobolev constant of subcritical operator inH01(Ω)andbsup=supx∈Ωb(x).

scholarly journals Multiplicity of Weak Positive Solutions for Fractional p & q Laplacian Problem with Singular Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Nehari Manifold ◽  
Analysis Techniques ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractional p & q Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold.

scholarly journals Existence and multiplicity of solutions to fractional p-Laplacian systems with concave–convex nonlinearities

2020 ◽  
Vol 10 (01) ◽  
pp. 2050007
Author(s):  
Hamed Alsulami ◽  
Mokhtar Kirane ◽  
Shabab Alhodily ◽  
Tareq Saeed ◽  
Nemat Nyamoradi
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Nehari Manifold ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

This paper is concerned with a fractional [Formula: see text]-Laplacian system with both concave–convex nonlinearities. The existence and multiplicity results of positive solutions are obtained by variational methods and the Nehari manifold.

scholarly journals Positive Solutions for Multipoint Boundary Value Problem of Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Wenyong Zhong
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions ◽  
Multipoint Boundary Value Problem

We study the existence and multiplicity of positive solutions for the fractionalm-point boundary value problemD0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=u'(0)=0,u'(1)=∑i=1m-2aiu'(ξi), where2<α<3,D0+αis the standard Riemann-Liouville fractional derivative, andf:[0,1]×[0,∞)↦[0,∞)is continuous. Here,ai⩾0fori=1,…,m-2,0<ξ1<ξ2<⋯<ξm-2<1, andρ=∑i=1m-2aiξiα-2withρ<1. In light of some fixed point theorems, some existence and multiplicity results of positive solutions are obtained.

Multiplicity of positive solutions for a class of problems with critical growth in ℝN

2009 ◽  
Vol 52 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Claudianor O. Alves ◽  
Daniel C. de Morais Filho ◽  
Marco A. S. Souto
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Continuous Functions ◽  
Critical Growth ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractUsing variational methods, we establish the existence and multiplicity of positive solutions for the following class of problems:where λ,β∈(0,∞), q∈(1,2*−1), 2*=2N/(N−2), N≥3, V,Z:ℝN→ℝ are continuous functions verifying some hypotheses.

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