scholarly journals Multiplicity Results for Variable-Order Nonlinear Fractional Magnetic Schrödinger Equation with Variable Growth

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jianwen Zhou ◽  
Bianxiang Zhou ◽  
Yanning Wang
Keyword(s):  
Magnetic Field ◽  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Limit Problem ◽  
Variable Order ◽  
Infinitely Many Solutions ◽  
Order Elliptic Equation ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Limit Problems

In this paper, we prove the multiplicity of nontrivial solutions for a class of fractional-order elliptic equation with magnetic field. Under appropriate assumptions, firstly, we prove that the system has at least two different solutions by applying the mountain pass theorem and Ekeland’s variational principle. Secondly, we prove that these two solutions converge to the two solutions of the limit problem. Finally, we prove the existence of infinitely many solutions for the system and its limit problems, respectively.

scholarly journals Elliptic problems in variable exponent spaces

2006 ◽  
Vol 74 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Mihai Mihailescu
Keyword(s):  
Variational Principle ◽  
Nonlinear Elliptic Equation ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Cylindrical Symmetry ◽  
Multiplicity Results ◽  
Variable Exponent Spaces ◽  
Nonlinear Elliptic ◽  
Existence And Multiplicity

In this paper we study a nonlinear elliptic equation involving p(x)-growth conditions on a bounded domain having cylindrical symmetry. We establish existence and multiplicity results using as main tools the mountain pass theorem of Ambosetti and Rabinowitz and Ekeland's variational principle.

scholarly journals Multiple solutions for Grushin operator without odd nonlinearity

2019 ◽  
Vol 13 (07) ◽  
pp. 2050131 ◽  
Author(s):  
Mohamed Karim Hamdani
Keyword(s):  
Mountain Pass Theorem ◽  
Infinitely Many Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Grushin Operator ◽  
Clark’S Theorem ◽  
Superlinear Growth ◽  
Degenerate Operator ◽  
Strongly Degenerate ◽  
Odd Nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: [Formula: see text] and [Formula: see text] where [Formula: see text] is the strongly degenerate operator, [Formula: see text] is allowed to be sign-changing, [Formula: see text], [Formula: see text] is a perturbation and the nonlinearity [Formula: see text] is a continuous function does not satisfy the Ambrosetti–Rabinowitz superquadratic condition ((AR) for short). First, via the mountain pass theorem and the Ekeland’s variational principle, existence of two different solutions for [Formula: see text] are obtained when [Formula: see text] satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for [Formula: see text] if [Formula: see text] is odd in [Formula: see text] thanks an extension of Clark’s theorem near the origin. So, our main results considerably improve results appearing in the literature.

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.

scholarly journals Existence and multiplicity of nontrivial solutions for nonlinear Schrödinger equations with unbounded potentials

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2465-2481 ◽  
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Bitao Cheng ◽  
Huxiao Luo
Keyword(s):  
Elliptic Equations ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Spaces ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Compact Embedding ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Unbounded Potential ◽  
Superquadratic Growth

We investigate the existence of nontrivial solutions and multiple solutions for the following class of elliptic equations (-?u + V(x)u = K(x)f(u), x ? RN, u ? D1,2(RN), where N ? 3, V(x) and K(x) are both unbounded potential functions and f is a function with a superquadratic growth. Firstly, we prove the existence of infinitely many solutions with compact embedding and by means of symmetric mountain pass theorem. Moreover, we prove the existence of nontrivial solutions without compact embedding in weighted Sobolev spaces and by means of mountain pass theorem. Our results extend and generalize some existing results.

scholarly journals Multiplicity results for nonlocal elliptic transmission problem with variable exponent

2014 ◽  
Vol 33 (2) ◽  
pp. 187-201
Author(s):  
Abdesslem Ayoujil ◽  
Mimoun Moussaoui
Keyword(s):  
Variational Principle ◽  
Nonlinear Equations ◽  
Weak Solutions ◽  
Variable Exponent ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Transmission Problem ◽  
Multiplicity Results ◽  
Nonstandard Growth ◽  
Nonstandard Growth Conditions

In this paper, a transmission problem given by a system of two nonlinear equations of p(x)-Kirchho type with nonstandard growth conditions are studied. Using the mountain pass theorem combined with the Ekeland's variational principle, we obtain at least two distinct, non-trivial weak solutions.

scholarly journals Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Dan Liu ◽  
Xuejun Zhang ◽  
Mingliang Song
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Linking Theorem ◽  
Infinitely Many Solutions ◽  
Nontrivial Solutions ◽  
Symmetric Mountain Pass Theorem ◽  
Sturm Liouville

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x ,     a.e.   t ∈ 0,1 x 0 cos    α − P 0 x ′ 0 sin    α = 0 x 1 cos    β − P 1 x ′ 1 sin    β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

scholarly journals p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

2018 ◽  
Vol 8 (1) ◽  
pp. 1111-1131 ◽  
Author(s):  
Alessio Fiscella ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

scholarly journals The Multiplicity of Nontrivial Solutions for a New p x -Kirchhoff-Type Elliptic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chang-Mu Chu ◽  
Yu-Xia Xiao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Existence Of Weak Solutions ◽  
Kirchhoff Problem

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

scholarly journals Infinitely Many Solutions for Fractional p-Laplacian Schrödinger–Kirchhoff Type Equations with Symmetric Variable-Order

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1393
Author(s):  
Weichun Bu ◽  
Tianqing An ◽  
José Vanteler da C. Sousa ◽  
Yongzhen Yun
Keyword(s):  
Sobolev Spaces ◽  
Lipschitz Domain ◽  
Mountain Pass Theorem ◽  
Variable Order ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Kirchhoff Type ◽  
Symmetric Mountain Pass Theorem ◽  
Bounded Lipschitz Domain ◽  
Symmetric Variable

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1<p<+∞, a,b>0 are constants, s(·):RN×RN→(0,1) is a continuous and symmetric function with N>s(x,y)p for all (x,y)∈Ω×Ω, λ>0 is a parameter, (−Δ)ps(·) is a fractional p-Laplace operator with variable-order, V(x):Ω→R+ is a potential function, and f(x,ξ):Ω×RN→R is a continuous nonlinearity function. Assuming that V and f satisfy some reasonable hypotheses, we obtain the existence of infinitely many solutions for the above problem by using the fountain theorem and symmetric mountain pass theorem without the Ambrosetti–Rabinowitz ((AR) for short) condition.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

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