scholarly journals Existence of Two Positive Solutions for Two Kinds of Fractional p -Laplacian Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yong Wu ◽  
Said Taarabti
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Multiple Solutions ◽  
Nehari Manifold ◽  
Nonlocal Operator ◽  
Laplacian Equations

The aim of this paper is to investigate the existence of two positive solutions to subcritical and critical fractional integro-differential equations driven by a nonlocal operator L K p . Specifically, we get multiple solutions to the following fractional p -Laplacian equations with the help of fibering maps and Nehari manifold. − Δ p s u x = λ u q + u r , u > 0   in   Ω , u = 0 , in   ℝ N \ Ω . . Our results extend the previous results in some respects.

Fractional elliptic equations with critical exponential nonlinearity

2016 ◽  
Vol 5 (1) ◽  
pp. 57-74 ◽  
Author(s):  
Jacques Giacomoni ◽  
Pawan Kumar Mishra ◽  
K. Sreenadh
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Exponential Nonlinearity ◽  
Existence Of Positive Solutions ◽  
Fractional Laplacian Operator

AbstractWe study the existence of positive solutions for fractional elliptic equations of the type (-Δ)1/2u = h(u), u > 0 in (-1,1), u = 0 in ℝ∖(-1,1) where h is a real valued function that behaves like eu2 as u → ∞ . Here (-Δ)1/2 is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near t = 0. In case h is concave near t = 0, we show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold.

scholarly journals Existence of Multiple Solutions for a -Kirchhoff Problem with Nonlinear Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Zonghu Xiu ◽  
Caisheng Chen
Keyword(s):  
Variational Method ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Nehari Manifold ◽  
The Nehari Manifold ◽  
Kirchhoff Problem

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem , ,   = , on , where , . By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.

Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent

2018 ◽  
Vol 61 (2) ◽  
pp. 353-369 ◽  
Author(s):  
Dongdong Qin ◽  
Yubo He ◽  
Xianhua Tang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Ground State ◽  
Multiple Solutions ◽  
Type Equation ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Kirchhoff Type ◽  
The Nehari Manifold

AbstractIn this paper, we consider the following critical Kirchhoff type equation:By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when 3 < q < 5. The relation between the number of maxima of Q and the number of positive solutions for the problem is also investigated.

Existence of Multiple Solutions for a p-Laplacian System in ℝN with Sign-changing Weight Functions

2016 ◽  
Vol 59 (2) ◽  
pp. 417-434 ◽  
Author(s):  
Hongxue Song ◽  
Caisheng Chen ◽  
Qinglun Yan
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Image Position ◽  
The Nehari Manifold ◽  
Linear Elliptic Problem

AbstractIn this paper, we consider the quasi-linear elliptic problemwhere and the weight H(x); h1(x); h2(x) are continuous functions that change sign in ℝN. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

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].

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