Multiple solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian and concave-convex nonlinearities

2017 ◽  
Vol 47 (6) ◽  
pp. 1803-1823 ◽  
Author(s):  
Chang-Mu Chu ◽  
Jiao-Jiao Sun ◽  
Zhi-Peng Cai
Keyword(s):  
Multiple Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

Multiplicity results for elliptic Kirchhoff-type problems

2017 ◽  
Vol 6 (1) ◽  
pp. 85-93 ◽  
Author(s):  
Sami Baraket ◽  
Giovanni Molica Bisci
Keyword(s):  
Weak Solutions ◽  
Multiple Solutions ◽  
Fractional Operators ◽  
Type Problem ◽  
Small Perturbations ◽  
Multiplicity Results ◽  
Nonlinear Part ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Type Problems

AbstractThe aim of this paper is to establish the existence of multiple solutions for a perturbed Kirchhoff-type problem depending on two real parameters. More precisely, we show that an appropriate oscillating behaviour of the nonlinear part, even under small perturbations, ensures the existence of at least three nontrivial weak solutions. Our approach combines variational methods with properties of nonlocal fractional operators.

scholarly journals Multiple solutions of the Kirchhoff-type problem in RN

2016 ◽  
Vol 1 (1) ◽  
pp. 229-238 ◽  
Author(s):  
Jiahua Jin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Variational Principle ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Nonlinear Partial Differential Equation ◽  
Point Theory ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

AbstractIn this paper, we concern with a class of quasilinear Kirchhoff-type problem. By using the Ekeland’s Variational Principle and Mountain Pass Theorem, the existence of multiple solutions is obtained. Besides, we also take this problem as an example to give the main frame of using critical point theory to find the weak solutions of nonlinear partial differential equation.

scholarly journals Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Zhou ◽  
Yunshun Wu
Keyword(s):  
Morse Theory ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Indefinite Potential

AbstractIn this paper, we consider the existence of solutions of the following Kirchhoff-type problem: $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in  R 3 , u ∈ H 1 ( R 3 ) , where $a,b>0$ a , b > 0 are constants, and the potential $V(x)$ V ( x ) is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.

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