Some remarks on uniqueness of positive solutions to Kirchhoff type equations

2022 ◽  
Vol 124 ◽  
pp. 107642
Author(s):  
Ke Wu ◽  
Fen Zhou ◽  
Guangze Gu
Keyword(s):  
Positive Solutions ◽  
Kirchhoff Type

scholarly journals The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

2018 ◽  
Vol 26 (1) ◽  
pp. 5-41 ◽  
Author(s):  
Baoqiang Yan ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Existence Of A Solution ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Global Bifurcation Theory ◽  
Necessary And Sufficient ◽  
Kirchhoff Type Problems

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

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].

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