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

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].

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.

Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

2014 ◽  
Vol 144 (4) ◽  
pp. 691-709 ◽  
Author(s):  
Ching-yu Chen ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problems ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
The Nehari Manifold ◽  
Indefinite Elliptic Problem

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

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 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 Nehari manifold and multiplicity result for elliptic equation involving p-laplacian problems

2018 ◽  
Vol 36 (4) ◽  
pp. 197-208
Author(s):  
Khaled Ben Ali ◽  
Abdeljabbar Ghanmi
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Smooth Function ◽  
Smooth Boundary ◽  
Nehari Manifold ◽  
Multiplicity Result ◽  
Existence And Multiplicity ◽  
Open Set ◽  
Multiplicity Of Positive Solutions

This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem $$\displaystyle -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad \mbox{for } x\in\partial\Omega$$ where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

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.

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