scholarly journals THE NEHARI MANIFOLD APPROACH FOR DIRICHLET PROBLEM INVOLVING THE p(x)-LAPLACIAN EQUATION

2010 ◽  
Vol 47 (4) ◽  
pp. 845-860 ◽  
Author(s):  
Rabil A. Mashiyev ◽  
Sezai Ogras ◽  
Zehra Yucedag ◽  
Mustafa Avci
Keyword(s):  
Dirichlet Problem ◽  
Nehari Manifold ◽  
Laplacian Equation ◽  
The Nehari Manifold

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 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.

scholarly journals The Nehari manifold method for discrete fractional p-Laplacian equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xuewei Ju ◽  
Hu Die ◽  
Mingqi Xiang
Keyword(s):  
Difference Equation ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Fractional Difference ◽  
Laplacian Equation ◽  
Variational Framework ◽  
Fractional Difference Equation ◽  
The Nehari Manifold ◽  
Laplacian Equations

Abstract The aim of this paper is to investigate the multiplicity of homoclinic solutions for a discrete fractional difference equation. First, we give a variational framework to a discrete fractional p-Laplacian equation. Then two nontrivial and nonnegative homoclinic solutions are obtained by using the Nehari manifold method.

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 A Multiplicity Theorem for Superlinear Double Phase Problems

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1556
Author(s):  
Beata Derȩgowska ◽  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Nehari Manifold ◽  
Bounded Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Double Phase ◽  
Multiplicity Theorem ◽  
The Nehari Manifold ◽  
Superlinear Reaction

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at +∞ and −∞ also negative.

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].

The Nehari manifold for a ψ-Hilfer fractional p-Laplacian

2021 ◽  
pp. 1-31
Author(s):  
J. Vanterler da C. Sousa ◽  
Jiabin Zuo ◽  
Donal O'Regan
Keyword(s):  
Nehari Manifold ◽  
The Nehari Manifold
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