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 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.

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 ${0<\tau _1<\tau _2< \infty }$ 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 ${0<p<1}$ 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 Infinitely many solutions for the discrete Schrödinger equations with a nonlocal term

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
Qilin Xie ◽  
Huafeng Xiao
Keyword(s):  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
High Energy ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Nonlocal Term ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Symmetric Mountain Pass Theorem

AbstractIn the present paper, we consider the following discrete Schrödinger equations $$ - \biggl(a+b\sum_{k\in \mathbf{Z}} \vert \Delta u_{k-1} \vert ^{2} \biggr) \Delta ^{2} u_{k-1}+ V_{k}u_{k}=f_{k}(u_{k}) \quad k\in \mathbf{Z}, $$ − ( a + b ∑ k ∈ Z | Δ u k − 1 | 2 ) Δ 2 u k − 1 + V k u k = f k ( u k ) k ∈ Z , where a, b are two positive constants and $V=\{V_{k}\}$ V = { V k } is a positive potential. $\Delta u_{k-1}=u_{k}-u_{k-1}$ Δ u k − 1 = u k − u k − 1 and $\Delta ^{2}=\Delta (\Delta )$ Δ 2 = Δ ( Δ ) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities $\{f_{k}\}$ { f k } satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.

scholarly journals Multiple Solutions for a Class of Fractional Schrödinger-Poisson System

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Lizhen Chen ◽  
Anran Li ◽  
Chongqing Wei
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Negative Energy ◽  
Mountain Pass ◽  
Poisson System ◽  
Fountain Theorem ◽  
Symmetric Mountain Pass Theorem ◽  
Dual Fountain Theorem

We investigate a class of fractional Schrödinger-Poisson system via variational methods. By using symmetric mountain pass theorem, we prove the existence of multiple solutions. Moreover, by using dual fountain theorem, we prove the above system has a sequence of negative energy solutions, and the corresponding energy values tend to 0. These results extend some known results in previous papers.

scholarly journals lnfinitely many solutions for fractional Schrödinger equations with perturbation via variational methods

2017 ◽  
Vol 15 (1) ◽  
pp. 578-586
Author(s):  
Peiluan Li ◽  
Youlin Shang
Keyword(s):  
Variational Methods ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Schrodinger Equations ◽  
Infinitely Many Solutions ◽  
Schrödinger Equations ◽  
Symmetric Mountain Pass Theorem ◽  
Fractional Schrödinger Equations ◽  
Existence Criteria

Abstract Using variational methods, we investigate the solutions of a class of fractional Schrödinger equations with perturbation. The existence criteria of infinitely many solutions are established by symmetric mountain pass theorem, which extend the results in the related study. An example is also given to illustrate our results.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

2018 ◽  
Vol 61 (4) ◽  
pp. 943-959 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Reaction Function ◽  
Robin Problem ◽  
Nodal Solutions ◽  
Potential Term ◽  
Symmetric Mountain Pass Theorem ◽  
Indefinite Potential ◽  
Concave Term

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

scholarly journals GENERAL QUASILINEAR PROBLEMS INVOLVING \(p(x)\)-LAPLACIAN WITH ROBIN BOUNDARY CONDITION

2020 ◽  
Vol 6 (1) ◽  
pp. 30
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Mountain Pass Theorem ◽  
Type Equation ◽  
Robin Boundary Condition ◽  
Mountain Pass ◽  
Technical Approach ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Quasilinear Problems ◽  
Robin Boundary ◽  
Laplace Type

This paper deals with the existence and multiplicity of solutions for a class of quasilinear problems involving \(p(x)\)-Laplace type equation, namely $$\left\{\begin{array}{lll}-\mathrm{div}\, (a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u)= \lambda f(x,u)&\text{in}&\Omega,\\n\cdot a(| \nabla u|^{p(x)})| \nabla u|^{p(x)-2} \nabla u +b(x)|u|^{p(x)-2}u=g(x,u) &\text{on}&\partial\Omega.\end{array}\right.$$ Our technical approach is based on variational methods, especially, the mountain pass theorem and the symmetric mountain pass theorem.

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.

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