scholarly journals Infinitely Many Solutions for a Superlinear Fractional p-Kirchhoff-Type Problem without the (AR) Condition

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Xiangsheng Ren ◽  
Jiabin Zuo ◽  
Zhenhua Qiao ◽  
Lisa Zhu
Keyword(s):  
Dirichlet Boundary ◽  
Mountain Pass Theorem ◽  
Dirichlet Boundary Conditions ◽  
Infinitely Many Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Symmetric Mountain Pass Theorem ◽  
Integrodifferential Operator ◽  
Kirchhoff Type Problem ◽  
Homogeneous Dirichlet Boundary Conditions

In this paper, we investigate the existence of infinitely many solutions to a fractional p-Kirchhoff-type problem satisfying superlinearity with homogeneous Dirichlet boundary conditions as follows: [a+b(∫R2Nux-uypKx-ydxdy)]Lpsu-λ|u|p-2u=gx,u, in  Ω, u=0, in  RN∖Ω, where Lps is a nonlocal integrodifferential operator with a singular kernel K. We only consider the non-Ambrosetti-Rabinowitz condition to prove our results by using the symmetric mountain pass theorem.

scholarly journals Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuping Cao ◽  
Chuanzhi Bai
Keyword(s):  
Dirichlet Boundary ◽  
Main Tool ◽  
Dirichlet Boundary Conditions ◽  
Type Problem ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Integrodifferential Operators ◽  
Homogeneous Dirichlet Boundary Conditions

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.

Infinitely many solutions for nonlocal elliptic systems in Orlicz–Sobolev spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samira Heidari ◽  
Abdolrahman Razani
Keyword(s):  
Sobolev Spaces ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Elliptic Systems ◽  
Type Systems ◽  
Dirichlet Boundary Conditions ◽  
Infinitely Many Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem

Abstract Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory.

scholarly journals Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

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.

Infinitely Many Solutions for Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations

2017 ◽  
Vol 60 (4) ◽  
pp. 1003-1020 ◽  
Author(s):  
Hongxue Song ◽  
Caisheng Chen
Keyword(s):  
Fourth Order ◽  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Biharmonic Operator ◽  
Kirchhoff Type ◽  
Symmetric Mountain Pass Theorem ◽  
Image Position ◽  
Fourth Order Elliptic Equations ◽  
Biharmonic Problems

AbstractThis paper deals with the class of Schrödinger–Kirchhoff-type biharmonic problemswhere Δ2 denotes the biharmonic operator, and f ∈ C(ℝN × ℝ, ℝ) satisfies the Ambrosetti–Rabinowitz-type conditions. Under appropriate assumptions on V and f, the existence of infinitely many solutions is proved by using the symmetric mountain pass theorem.

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.

scholarly journals Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff-type problems without the Ambrosetti–Rabinowitz condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
M. K. Hamdani ◽  
J. Zuo ◽  
N. T. Chung ◽  
D. D. Repovš
Keyword(s):  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Continuous Functions ◽  
Infinitely Many Solutions ◽  
Kirchhoff Type ◽  
Nonlinear Anal ◽  
Kirchhoff Type Problem ◽  
Bounded Smooth ◽  
Kirchhoff Type Problems ◽  
Fractional Laplace Operator

Abstract We are interested in the existence of solutions for the following fractional $p(x,\cdot )$ p ( x , ⋅ ) -Kirchhoff-type problem: $$ \textstyle\begin{cases} M ( \int _{\Omega \times \Omega } {\frac{ \vert u(x)-u(y) \vert ^{p(x,y)}}{p(x,y) \vert x-y \vert ^{N+p(x,y)s}}} \,dx \,dy )(-\Delta )^{s}_{p(x,\cdot )}u = f(x,u), \quad x\in \Omega , \\ u= 0, \quad x\in \partial \Omega , \end{cases} $$ { M ( ∫ Ω × Ω | u ( x ) − u ( y ) | p ( x , y ) p ( x , y ) | x − y | N + p ( x , y ) s d x d y ) ( − Δ ) p ( x , ⋅ ) s u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N , $N\geq 2$ N ≥ 2 is a bounded smooth domain, $s\in (0,1)$ s ∈ ( 0 , 1 ) , $p: \overline{\Omega }\times \overline{\Omega } \rightarrow (1, \infty )$ p : Ω ‾ × Ω ‾ → ( 1 , ∞ ) , $(-\Delta )^{s}_{p(x,\cdot )}$ ( − Δ ) p ( x , ⋅ ) s denotes the $p(x,\cdot )$ p ( x , ⋅ ) -fractional Laplace operator, $M: [0,\infty ) \to [0, \infty )$ M : [ 0 , ∞ ) → [ 0 , ∞ ) , and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ f : Ω × R → R are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo–Benci–Fortunato (Nonlinear Anal. 7(9):981–1012, 1983), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti–Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

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