scholarly journals New multiplicity of positive solutions for some class of nonlocal problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhigao Shi ◽  
Xiaotao Qian
Keyword(s):  
Asymptotic Behavior ◽  
Variational Method ◽  
Positive Solutions ◽  
Blow Up ◽  
Nonlocal Problem ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Nonlocal Problems ◽  
Smooth Bounded Domain ◽  
Multiplicity Of Positive Solutions

AbstractIn this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | 2 d x ) Δ u = λ | u | q − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ R N with $N\ge 3$ N ≥ 3 , $a,b>0$ a , b > 0 , $1< q<2$ 1 < q < 2 and $\lambda >0$ λ > 0 is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ b ↘ 0 .

scholarly journals Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

2021 ◽  
pp. 1-15
Author(s):  
Linyan Peng ◽  
Hongmin Suo ◽  
Deke Wu ◽  
Hongxi Feng ◽  
Chunyu Lei
Keyword(s):  
Variational Method ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Multiple Positive Solutions ◽  
Poisson System ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n   Ω , − Δ ϕ = u 2 , i n   Ω , u = ϕ = 0 , o n   ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

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 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.

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 Multiplicity of Weak Positive Solutions for Fractional p & q Laplacian Problem with Singular Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Nehari Manifold ◽  
Analysis Techniques ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this paper, we prove the existence and multiplicity of positive solutions for a class of fractional p & q Laplacian problem with singular nonlinearity. Our approach relies on the variational method, some analysis techniques, and the method of Nehari manifold.

scholarly journals Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

2021 ◽  
Vol 11 (1) ◽  
pp. 598-619
Author(s):  
Guofeng Che ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Positive Solutions ◽  
Type Equation ◽  
Nehari Manifold ◽  
Sobolev Embedding ◽  
Multiple Positive Solutions ◽  
Kirchhoff Type ◽  
Indefinite Nonlinearities ◽  
The Nehari Manifold ◽  
Constraint Method

Abstract We study the following Kirchhoff type equation: − a + b ∫ R N | ∇ u | 2 d x Δ u + u = k ( x ) | u | p − 2 u + m ( x ) | u | q − 2 u     in     R N , $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N=3, a , b > 0 $ a,b \gt 0 $ , 1 < q < 2 < p < min { 4 , 2 ∗ } $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2N/(N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p/(p−q)(ℝ N ). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) ↪ L r ( R N ) ( 2 ≤ r < 2 ∗ ) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

Multiple Positive Solutions to a Kirchhoff Type Problem Involving a Critical Nonlinearity in ℝ3

2017 ◽  
Vol 17 (4) ◽  
pp. 661-676 ◽  
Author(s):  
Xiao-Jing Zhong ◽  
Chun-Lei Tang
Keyword(s):  
Positive Solutions ◽  
Principal Eigenvalue ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearity ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Kirchhoff Type Problem ◽  
The Nehari Manifold ◽  
Kirchhoff Type Problems

AbstractIn this paper, we investigate a class of Kirchhoff type problems in {\mathbb{R}^{3}} involving a critical nonlinearity, namely,-\biggl{(}1+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,dx\biggr{)}\triangle u=% \lambda f(x)u+|u|^{4}u,\quad u\in D^{1,2}(\mathbb{R}^{3}),where {b>0}, {\lambda>\lambda_{1}} and {\lambda_{1}} is the principal eigenvalue of {-\triangle u=\lambda f(x)u}, {u\in D^{1,2}(\mathbb{R}^{3})}. We prove that there exists {\delta>0} such that the above problem has at least two positive solutions for {\lambda_{1}<\lambda<\lambda_{1}+\delta}. Furthermore, we obtain the existence of ground state solutions. Our tools are the Nehari manifold and the concentration compactness principle. This paper can be regarded as an extension of Naimen’s work [21].

scholarly journals FULLY DISCRETE ADAPTIVE METHODS FOR A BLOW-UP PROBLEM

2004 ◽  
Vol 14 (10) ◽  
pp. 1425-1450 ◽  
Author(s):  
CRISTINA BRÄNDLE ◽  
PABLO GROISMAN ◽  
JULIO D. ROSSI
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Blow Up ◽  
Numerical Study ◽  
Adaptive Methods ◽  
Fully Discrete ◽  
Adaptive Procedures ◽  
Exact Asymptotic Behavior ◽  
Blow Up Rate ◽  
Continuous Problem

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem: [Formula: see text] with p,m>0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate and the blow-up set) of the continuous problem.

scholarly journals Multiple Solutions for the Asymptotically Linear Kirchhoff Type Equations onRN

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Yu Duan ◽  
Chun-Lei Tang
Keyword(s):  
Variational Method ◽  
Positive Solutions ◽  
Multiple Solutions ◽  
Asymptotically Linear ◽  
Kirchhoff Type ◽  
Multiplicity Of Positive Solutions

The multiplicity of positive solutions for Kirchhoff type equations depending on a nonnegative parameterλonRNis proved by using variational method. We will show that if the nonlinearities are asymptotically linear at infinity andλ>0is sufficiently small, the Kirchhoff type equations have at least two positive solutions. For the perturbed problem, we give the result of existence of three positive solutions.

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.

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