Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

Ye, Long; Zhang, Huan

doi:10.3934/era.2022137

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…Additionally, via the Morse theory, refs. [19] produced results based on three nontrivial solutions and [20] obtained four nontrivial solutions to Problems (1) and ( 2).…”

Section: Introductionmentioning

confidence: 99%

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

Zhang

2022

Symmetry

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In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems Δ12u(i−1,j)+Δ22u(i,j−1)=−f((i,j),u(i,j)),(i,j)∈Ω,u(i,0)=u(i,T2+1)=0i∈Z(1,T1),u(0,j)=u(T1+1,j)=0j∈Z(1,T2), which have a symmetric structure. When the nonlinearity f(·,u) is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking.

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“…Additionally, via the Morse theory, refs. [19] produced results based on three nontrivial solutions and [20] obtained four nontrivial solutions to Problems (1) and ( 2).…”

Section: Introductionmentioning

confidence: 99%

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

Zhang

2022

Symmetry

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Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems

Ye¹,

Zhang²

2023

CPAA

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Multiple nontrivial periodic solutions to a second-order partial difference equation

2023

era

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<abstract><p>In this article, applying variational technique as well as critical point theory, we establish a series of criteria to ensure the existence and multiplicity of nontrivial periodic solutions to a second-order nonlinear partial difference equation. Our results generalize some known results. Moreover, numerical stimulations are presented to illustrate applications of our major findings.</p></abstract>

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Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

Cited by 4 publications

References 32 publications

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems

Multiple nontrivial periodic solutions to a second-order partial difference equation

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