scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

Multiple Solutions for Perturbed Elliptic Equations in Unbounded Domains

2003 ◽  
Vol 3 (1) ◽  
pp. 1-23 ◽  
Author(s):  
A. Salvatore
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Unbounded Domains ◽  
Dirichlet Boundary Conditions

AbstractWe look for solutions of a nonlinear perturbed Schrödinger equation with nonhomogeneous Dirichlet boundary conditions. By using a perturbation method introduced by Bolle, we prove the existence of multiple solutions in spite of the lack of the symmetry of the problem.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qinqin Zhang
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

scholarly journals Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yu Guo ◽  
Xiao-Bao Shu ◽  
Qianbao Yin
Keyword(s):  
Differential Equations ◽  
Critical Point ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Energy Functional ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Order

<p style='text-indent:20px;'>In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.</p>

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