The Nehari manifold approach for N-Laplace equation with singular and exponential nonlinearities in ℝN

2015 ◽  
Vol 17 (03) ◽  
pp. 1450011 ◽  
Author(s):  
Sarika Goyal ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Laplace Equation ◽  
Nehari Manifold ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this article, we study the existence and multiplicity of solutions of the singular N-Laplacian equation: [Formula: see text] where N ≥ 2, 0 ≤ q < N - 1 < p + 1, β ∈ [0, N), λ > 0, and h ≥ 0 in ℝN. Using the nature of the Nehari manifold and fibering maps associated with the Euler functional, we prove that there exists λ0such that for λ ∈ (0, λ0), the problem admits at least two positive solutions. We also show that when h(x) > 0, there exists λ0such that (Pλ) has no solution for λ > λ0.

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 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 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 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 Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

2020 ◽  
Vol 10 (1) ◽  
pp. 636-658
Author(s):  
Fuliang Wang ◽  
Die Hu ◽  
Mingqi Xiang
Keyword(s):  
Sobolev Inequality ◽  
Nehari Manifold ◽  
Combined Effects ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Singular Nonlinearities ◽  
The Nehari Manifold

Abstract The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

scholarly journals The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition

2007 ◽  
Vol 12 (2) ◽  
pp. 143-155 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. Mahdavi ◽  
Z. Naghizadeh
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Laplacian Equation ◽  
The Nehari Manifold ◽  
The Relationship

The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)|p−2 + b(x)|u(x)|γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function

2015 ◽  
Vol 4 (3) ◽  
pp. 177-200 ◽  
Author(s):  
Sarika Goyal ◽  
Konijeti Sreenadh
Keyword(s):  
Weight Function ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Nehari Manifold ◽  
Polyharmonic Equation ◽  
Unbounded Function ◽  
Multiplicity Of Solutions ◽  
The Real ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

AbstractIn this article, we consider the following quasilinear polyharmonic equation: Δn/mmu = λh(x)|u|q-1u + u|u|pe|u|β in Ω, u = ∇u = ⋯ = ∇m-1u = 0 on ∂Ω, where Ω ⊂ ℝn, n ≥ 2m ≥ 2, is a bounded domain with smooth boundary. The real-valued function h is a sign-changing and unbounded function. The exponents p, q and β satisfy 0 < q < n/(m-1) < p+1, β ∈ (1,n/(n-m)] and λ > 0. Using the Nehari manifold and fibering maps, we show the existence and multiplicity of solutions.

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 for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

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