scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Qiongfen Zhang ◽  
Yuan Li
Keyword(s):  
Continuous Function ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Space ◽  
Point Theory ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonautonomous Systems ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity

This paper is concerned with the following nonlinear second-order nonautonomous problem:ü(t)+q(t)u̇(t)-∇K(t,u(t))+∇W(t,u(t))=0, wheret∈R,u∈RN, andK,W∈C1(R×RN,R)are not periodic intandq:R→Ris a continuous function andQ(t)=∫0t‍q(s)dswithlim|t|→+∞⁡Q(t)=+∞. The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.

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 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 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.

scholarly journals Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jingli Xie ◽  
Zhiguo Luo ◽  
Guoping Chen
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Solution ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Approximation Solution ◽  
Periodic Approximation

This paper is concerned with the existence of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a2kT-periodic approximation solution is a homoclinic solution of our problem.

Existence of homoclinic solutions for a class of second-order non-autonomous Hamiltonian systems

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Qiongfen Zhang ◽  
X. Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Second Order Hamiltonian Systems

AbstractBy using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of second-order Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.

scholarly journals Existence of Homoclinic Orbits for a Class of Asymptoticallyp-Linear Difference Systems withp-Laplacian

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Qiongfen Zhang ◽  
X. H. Tang
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Difference System ◽  
Difference Systems ◽  
Linear Difference Systems

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptoticallyp-linear difference system withp-LaplacianΔ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0, wherep∈(1,+∞),n∈ℤ,u∈ℝN,K,W:ℤ×ℝN→ℝare not periodic inn, and W is asymptoticallyp-linear at infinity.

Existence of infinitely many homoclinic orbits in Hamiltonian systems

2011 ◽  
Vol 141 (5) ◽  
pp. 1103-1119 ◽  
Author(s):  
X. H. Tang ◽  
Xiaoyan Lin
Keyword(s):  
Potential Function ◽  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Symmetric Mountain Pass Theorem ◽  
Second Order Hamiltonian Systems ◽  
Existence Criteria

By using the symmetric mountain pass theorem, we establish some new existence criteria to guarantee that the second-order Hamiltonian systems ü(t) − L(t)u(t) + ∇W(t,u(t)) = 0 have infinitely many homoclinic orbits, where t ∈ ℝ, u ∈ ℝN, L ∈ C(ℝ, ℝN × N) and W ∈ C1(ℝ × ℝN, ℝ) are not periodic in t. Our results generalize and improve some existing results in the literature by relaxing the conditions on the potential function W(t, x).

scholarly journals Existence and multiplicity of solutions for class of Navier boundary $p$-biharmonic problem near resonance

2014 ◽  
Vol 32 (2) ◽  
pp. 83 ◽  
Author(s):  
Mohammed Massar ◽  
EL Miloud Hssini ◽  
Najib Tsouli
Keyword(s):  
Saddle Point ◽  
Point Theorem ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Saddle Point Theorem ◽  
Biharmonic Problem ◽  
Existence And Multiplicity ◽  
Near Resonance

This paper studies the existence and multiplicity of weak solutions for the following elliptic problem\\$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x)|u|^{p-2}u+f(x,u)+h(x)$ in $\Omega,$\\$u=\Delta u=0$ on $\partial\Omega.$By using Ekeland's variationalprinciple, Mountain pass theorem and saddle point theorem, theexistence and multiplicity of weak solutions are established.

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