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 Multiplicity results for the non-homogeneous fractional p -Kirchhoff equations with concave–convex nonlinearities

2015 ◽  
Vol 471 (2177) ◽  
pp. 20150034 ◽  
Author(s):  
Mingqi Xiang ◽  
Binlin Zhang ◽  
Massimiliano Ferrara
Keyword(s):  
Variational Principle ◽  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Type Problem ◽  
Entire Solutions ◽  
Kirchhoff Equations ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Non Local

In this paper, we are interested in the multiplicity of solutions for a non-homogeneous p -Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities: a + b ∬ R 2 N | u ( x ) − u ( y ) | p | x − y | N + s p   d x   d y θ − 1 ( − Δ ) p s u = λ ω 1 ( x ) | u | q − 2 u + ω 2 ( x ) | u | r − 2 u + h ( x ) in   R N , where ( − Δ ) p s is the fractional p -Laplace operator, a + b >0 with a , b ∈ R 0 + , λ>0 is a real parameter, 0 < s < 1 < p < ∞ with sp < N , 1< q < p ≤ θp < r < Np /( N − sp ), ω 1 , ω 2 , h are functions which may change sign in R N . Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that a may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.

Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains

2016 ◽  
Vol 146 (2) ◽  
pp. 435-448 ◽  
Author(s):  
Juntao Sun ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Bounded Domains ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

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.

Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Moloud Makvand Chaharlang ◽  
Abdolrahman Razani
Keyword(s):  
Boundary Condition ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Mountain Pass Theorem ◽  
Minimum Principle ◽  
Neumann Boundary ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

AbstractIn this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.

scholarly journals On a nonlinear PDE involving weighted $p$-Laplacian

2019 ◽  
Vol 38 (5) ◽  
pp. 131-145
Author(s):  
A. El Khalil ◽  
M. D. Morchid Alaoui ◽  
Mohamed Laghzal ◽  
A. Touzani
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Variational Method ◽  
Nonlinear Partial Differential Equation ◽  
Weighted Sobolev Spaces ◽  
Point Theory ◽  
Energy Functional ◽  
Nonlinear Pde ◽  
Laplacian Operator ◽  
Uniqueness Of Solutions

In the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator\begin{gather*}- \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}},\end{gather*}on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate conditionson the functions $f, w$ and the nonlinearity $\frac{1}{(1-u)^{2}}$, we prove the existence and the uniqueness of solutions of the above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces.

scholarly journals Multiple Solutions for a Nonlocal Elliptic Problem Involving p x , q x -Biharmonic Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Qi Zhang ◽  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Multiple Solutions ◽  
Sufficient Conditions ◽  
Biharmonic Operator ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

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