A singular elliptic system involving the p(x)-Laplacian and generalized Lebesgue–Sobolev spaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950064
Author(s):  
Kamel Saoudi
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Compact Support ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Variational Techniques ◽  
Multiplicity Of Positive Solutions ◽  
Two Parameters ◽  
The Nehari Manifold

The purpose of this work is to study a class of singular elliptic system involving the [Formula: see text]-Laplace operator of the form [Formula: see text] where [Formula: see text] [Formula: see text] is a bounded domain with [Formula: see text] boundary, [Formula: see text] are two parameters, [Formula: see text] are non-negative weight functions with compact support in [Formula: see text] and [Formula: see text] are assumed to satisfy the assumptions (A0)–(A2) in Sec. 1. We employ the Nehari manifold approach combined with some variational techniques in order to show the existence and the multiplicity of positive solutions.

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 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.

Multiple solutions for a critical fractional elliptic system involving concave–convex nonlinearities

2016 ◽  
Vol 146 (6) ◽  
pp. 1167-1193 ◽  
Author(s):  
Wenjing Chen ◽  
Shengbing Deng
Keyword(s):  
Elliptic System ◽  
Multiple Solutions ◽  
Critical Sobolev Exponent ◽  
Nehari Manifold ◽  
Dirichlet Boundary Conditions ◽  
Bounded Set ◽  
Sobolev Exponent ◽  
Image Position ◽  
Two Parameters ◽  
The Nehari Manifold

We are concerned with the multiplicity of solutions to the system driven by a fractional operator with homogeneous Dirichlet boundary conditions. Namely, using fibering maps and the Nehari manifold, we obtain multiple solutions to the following fractional elliptic system:where Ω is a smooth bounded set in ℝn, n > 2s, with s ∈ (0, 1); (–Δ)s is the fractional Laplace operator;, λ, μ > 0 are two parameters; the exponent n/(n – 2s) ⩽ q < 2; α > 1, β > 1 satisfy is the fractional critical Sobolev exponent.

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.

On a class of weighted Sobolev spaces and the minimization of quadratic forms

1978 ◽  
Vol 81 (1-2) ◽  
pp. 119-130
Author(s):  
Frans Penning ◽  
Niko Sauer
Keyword(s):  
Weight Function ◽  
Sobolev Spaces ◽  
Compact Support ◽  
Quadratic Forms ◽  
Weighted Sobolev Spaces ◽  
Weight Functions ◽  
Compact Embedding ◽  
Embedding Theorems ◽  
Smooth Functions ◽  
Square Integrability

SynopsisIn this paper a class of weighted Sobolev spaces defined in terms of square integrability of the gradient multiplied by a weight function, is studied. The domain of integration is either the spaceRnor a half-space ofRn. Conditions on the weight functions that will ensure density of classes of smooth functions or functions with compact support, and compact embedding theorems, are derived. Finally the results are applied to a class of isoperimetrical problems in the calculus of variations in which the domain of integration is unbounded.

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