scholarly journals Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Defang Ma ◽  
Zhan Zhou
Keyword(s):  
Difference Equations ◽  
Mountain Pass Theorem ◽  
Sufficient Conditions ◽  
Homoclinic Solutions ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Special Cases ◽  
Discrete Nonlinear Schrödinger Equations ◽  
The Difference

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

scholarly journals Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations withp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Guowei Sun ◽  
Ali Mai
Keyword(s):  
Difference Equations ◽  
Critical Point Theory ◽  
Point Theory ◽  
Nehari Manifold ◽  
Homoclinic Solutions ◽  
Nonlinear Difference Equations ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Superlinear Nonlinearity ◽  
Laplacian Equations

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discretep-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

scholarly journals Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions

2020 ◽  
Vol 51 (3) ◽  
pp. 187-200
Author(s):  
Nguyen Thanh Chung
Keyword(s):  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism.  Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

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 Existence and Multiplicity of Solutions to a Class of Fractional p-Laplacian Equations of Schrödinger-Type with Concave-Convex Nonlinearities in ℝN

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1792
Author(s):  
Yun-Ho Kim
Keyword(s):  
Potential Function ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Laplacian Operator ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Continuous Potential ◽  
The Given ◽  
Laplacian Equations

We are concerned with the following elliptic equations: (−Δ)psv+V(x)|v|p−2v=λa(x)|v|r−2v+g(x,v)inRN, where (−Δ)ps is the fractional p-Laplacian operator with 0<s<1<r<p<+∞, sp<N, the potential function V:RN→(0,∞) is a continuous potential function, and g:RN×R→R satisfies a Carathéodory condition. By employing the mountain pass theorem and a variant of Ekeland’s variational principle as the major tools, we show that the problem above admits at least two distinct non-trivial solutions for the case of a combined effect of concave–convex nonlinearities. Moreover, we present a result on the existence of multiple solutions to the given problem by utilizing the well-known fountain theorem.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qiongfen Zhang
Keyword(s):  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Impulsive Effects ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Vibration Problems ◽  
Damped Vibration Problems

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

scholarly journals Homoclinic solutions of 2nth-order difference equations containing both advance and retardation

2016 ◽  
Vol 14 (1) ◽  
pp. 520-530 ◽  
Author(s):  
Yuhua Long ◽  
Yuanbiao Zhang ◽  
Haiping Shi
Keyword(s):  
Difference Equation ◽  
Critical Point ◽  
Difference Equations ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonlinear Difference Equation ◽  
Mountain Pass Lemma ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
New Criteria

AbstractBy using the critical point method, some new criteria are obtained for the existence and multiplicity of homoclinic solutions to a 2nth-order nonlinear difference equation. The proof is based on the Mountain Pass Lemma in combination with periodic approximations. Our results extend and improve some known ones.

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 Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Lv ◽  
Chun-Lei Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Second Order Hamiltonian Systems ◽  
Superquadratic Potential

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

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