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 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 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.

scholarly journals Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Junjun Zhou ◽  
Xiangyun Hu ◽  
Tiaojie Xiao
Keyword(s):  
Critical Exponent ◽  
Positive Solutions ◽  
Mountain Pass Theorem ◽  
Critical Sobolev Exponent ◽  
Mountain Pass ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions ◽  
Kirchhoff Type Problems

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu>0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.

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 Existence of multiple positive solutions for semipositone fractional boundary value problems

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 749-759 ◽  
Author(s):  
Şerife Ege ◽  
Fatma Topal
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions ◽  
Fractional Boundary Value Problems

In this paper, we study the existence and multiplicity of positive solutions to the four-point boundary value problems of nonlinear semipositone fractional differential equations. Our results extend some recent works in the literature.

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