scholarly journals Regularity and multiplicity results for fractional (p,q)-Laplacian equations

2019 ◽  
Vol 22 (08) ◽  
pp. 1950065 ◽  
Author(s):  
Divya Goel ◽  
Deepak Kumar ◽  
K. Sreenadh
Keyword(s):  
Weak Solutions ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Subcritical Case ◽  
Existence Of A Solution ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold ◽  
Weak Harnack Inequality ◽  
Laplacian Equations

This paper deals with the study of the following nonlinear doubly nonlocal equation: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are parameters. Here [Formula: see text] and [Formula: see text] are sign-changing functions. We prove [Formula: see text] estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case [Formula: see text] Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of [Formula: see text], we show the existence of a solution.

scholarly journals Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations withp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Guowei Sun ◽  
Ali Mai
Keyword(s):  
Difference Equations ◽  
Critical Point Theory ◽  
Point Theory ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Superlinear Nonlinearity ◽  
Laplacian Equations

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discretep-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

scholarly journals MULTIPLE SOLUTIONS FOR A FRACTIONAL LAPLACIAN SYSTEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND HOMOGENEOUS TERM

2020 ◽  
Vol 25 (1) ◽  
pp. 1-20
Author(s):  
Jinguo Zhang ◽  
Tsing-San Hsu
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we deal with a class of fractional Laplacian system with critical Sobolev-Hardy exponents and sign-changing weight functions in a bounded domain. By exploiting the Nehari manifold and variational methods, some new existence and multiplicity results are obtain.

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 Multiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractional p-Laplacian of Order s

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
M. Khiddi
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nehari Manifold ◽  
Lusternik Schnirelmann ◽  
The Nehari Manifold

In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional p-Laplacian of order s. We show the above result when λ>0 is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.

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 Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results

2007 ◽  
Vol 2007 ◽  
pp. 1-23
Author(s):  
Francesca Papalini
Keyword(s):  
Existence Theorems ◽  
Point Theory ◽  
Energy Functional ◽  
Periodic Systems ◽  
Multiplicity Result ◽  
Nonsmooth Critical Point Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz

We study second-order nonlinear periodic systems driven by the vectorp-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

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.

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 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.

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