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 Existence of Multiple Solutions for a Class of Biharmonic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Chunhan Liu ◽  
Jianguo Wang
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Biharmonic Equations ◽  
Symmetric Mountain Pass Theorem ◽  
Type Boundary

By a symmetric Mountain Pass Theorem, a class of biharmonic equations with Navier type boundary value at the resonant and nonresonant case are discussed, and infinitely many solutions of the equations are obtained.

scholarly journals Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zuoshi Xie ◽  
Yuanfeng Jin ◽  
Chengmin Hou
Keyword(s):  
Boundary Value Problem ◽  
Critical Point Theory ◽  
Variational Approach ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Point Theory ◽  
Linking Theorem ◽  
Fractional Difference ◽  
Variational Framework

By establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory, we give the existence of multiple solutions for a fractional difference boundary value problem with parameter. Under some suitable assumptions, we obtain some results which ensure the existence of a well precise interval of parameter for which the problem admits multiple solutions. Some examples are presented to illustrate the main results.

scholarly journals Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Bo Zheng ◽  
Huafeng Xiao
Keyword(s):  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Boundary Value ◽  
Local Minimizer ◽  
Second Order ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Difference Problem ◽  
Order Difference

This paper studies the existence of multiple solutions of the second-order difference boundary value problemΔ2u(n−1)+V′(u(n))=0,n∈ℤ(1,T),u(0)=0=u(T+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalueλk  (k≥2)of linear difference problemΔ2u(n−1)+λu(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)near infinity and the trivial solution of the first equation is a local minimizer under some assumptions onV.

scholarly journals TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

2016 ◽  
Vol 56 (3) ◽  
pp. 245
Author(s):  
Marzena Szajewska ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Boundary Value Problems ◽  
Special Functions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Condition ◽  
Two Dimensional ◽  
Neumann Type ◽  
Neumann Boundary Value Problems ◽  
Real Euclidean Space

Boundary value problems are considered on a simplex <em>F</em> in the real Euclidean space R<sup>2</sup>. The recent discovery of new families of special functions, orthogonal on <em>F</em>, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on <em>F</em>, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of <em>F</em> a Dirichlet condition is fulfilled and on the other Neumann’s works.

Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

2015 ◽  
Vol 4 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Fractional Hamiltonian Systems ◽  
Symmetric Mountain Pass Theorem

AbstractIn this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form ${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$, where ${\alpha \in (\frac{1}{2},1)}$, ${t\in \mathbb {R}}$, ${u\in \mathbb {R}^n}$, ${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in \mathbb {R}}$, ${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ${\nabla W(t,u)}$ is the gradient of ${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants ${0&lt;\tau _1&lt;\tau _2&lt; \infty }$ such that ${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all ${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and ${W(t,u)}$ is of the form ${({a(t)}/({p+1}))|u|^{p+1}}$ such that ${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and ${0&lt;p&lt;1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

scholarly journals Multiple Solutions for Boundary Value Problems of p-Laplacian Difference Equations Containing Both Advance and Retardation

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhenguo Wang ◽  
Zhan Zhou
Keyword(s):  
Boundary Value Problems ◽  
Difference Equations ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Open Interval ◽  
Point Theory ◽  
Infinitely Many Solutions

This paper concerns the existence of solutions for the Dirichlet boundary value problems of p-Laplacian difference equations containing both advance and retardation depending on a parameter λ. Under some suitable assumptions, infinitely many solutions are obtained when λ lies in a given open interval. The approach is based on the critical point theory.

Variational methods to the p-Laplacian type nonlinear fractional order impulsive differential equations with Sturm-Liouville boundary-value problem

2021 ◽  
Vol 24 (4) ◽  
pp. 1069-1093
Author(s):  
Dandan Min ◽  
Fangqi Chen
Keyword(s):  
Differential Equation ◽  
Variational Methods ◽  
Impulsive Differential Equation ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Laplacian Operator ◽  
Infinitely Many Solutions ◽  
Critical Point Theorem ◽  
Sturm Liouville ◽  
New Criteria

Abstract In this paper, we consider a class of nonlinear fractional impulsive differential equation involving Sturm-Liouville boundary-value conditions and p-Laplacian operator. By making use of critical point theorem and variational methods, some new criteria are given to guarantee that the considered problem has infinitely many solutions. Our results extend some recent results and the conditions of assumptions are easily verified. Finally, an example is given as an application of our fundamental results.

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.

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