scholarly journals Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity inR3

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ling Ding ◽  
Lin Li ◽  
Jin-Ling Zhang
Keyword(s):  
Variational Principle ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Kirchhoff Equation ◽  
Mountain Pass ◽  
Asymptotically Linear ◽  
Ekeland's Variational Principle

We study the following nonhomogeneous Kirchhoff equation:-(a+b∫R3‍|∇u|2dx)Δu+u=k(x)f(u)+h(x),  x∈R3,  u∈H1(R3),  u>0,  x∈R3, wherefis asymptotically linear with respect totat infinity. Under appropriate assumptions onk,f, andh, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

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.

scholarly journals On Neumann hemivariational inequalities

2002 ◽  
Vol 7 (2) ◽  
pp. 103-112 ◽  
Author(s):  
Halidias Nikolaos
Keyword(s):  
Critical Point ◽  
Nontrivial Solution ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Mountain Pass ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz

We derive a nontrivial solution for a Neumann noncoercive hemivariational inequality using the critical point theory for locally Lipschitz functionals. We use the Mountain-Pass theorem due to Chang (1981).

TWO POSITIVE SOLUTIONS OF SOME SINGULAR BOUNDARY VALUE PROBLEMS

2010 ◽  
Vol 08 (03) ◽  
pp. 305-314 ◽  
Author(s):  
RADU PRECUP
Keyword(s):  
Boundary Value Problems ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Singular Boundary ◽  
Singular Boundary Value Problems

The existence of two positive solutions for a class of singular boundary value problems is established by means of a combination of the Leray–Schauder principle with techniques from critical point theory.

MULTIPLE HOMOCLINICS IN A HAMILTONIAN SYSTEM WITH ASYMPTOTICALLY OR SUPER LINEAR TERMS

2006 ◽  
Vol 08 (04) ◽  
pp. 453-480 ◽  
Author(s):  
YANHENG DING
Keyword(s):  
Hamiltonian System ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variational Formulation ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Asymptotically Linear ◽  
Recent Information ◽  
Existence And Multiplicity ◽  
The Matrix

This paper is concerned with homoclinic orbits in the Hamiltonian system [Formula: see text] where H is periodic in t with Hz(t, z) = L(t)z + Rz(t, z), Rz(t, z) = o(|z|) as z → 0. We find a condition on the matrix valued function L to describe the spectrum of operator [Formula: see text] so that a proper variational formulation is presented. Supposing Rz is asymptotically linear as |z| → ∞ and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where Rz is super linear as |z| → ∞ with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals 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.

Small perturbations of elliptic problems with variable growth in RN

2021 ◽  
Vol 477 (2249) ◽  
Author(s):  
Bin Ge ◽  
Hai-Cheng Liu ◽  
Bei-Lei Zhang
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Ekeland’S Variational Principle ◽  
Mountain Pass ◽  
Ekeland's Variational Principle ◽  
Small Perturbations ◽  
Whole Space ◽  
Laplacian Equations

In this paper, we study the existence of at least two non-trivial solutions for a class of p ( x )-Laplacian equations with perturbation in the whole space. Using Ekeland’s variational principle and the mountain pass theorem, under appropriate assumptions, we prove the existence of two solutions for the equations.

scholarly journals Nonlocal Kirchhoff Superlinear Equations with Indefinite Nonlinearity and Lack of Compactness

2016 ◽  
Vol 17 (6) ◽  
pp. 325-332 ◽  
Author(s):  
Lin Li ◽  
Vicenţiu Rădulescu ◽  
Dušan Repovš
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Ekeland Variational Principle ◽  
Kirchhoff Equation ◽  
Mountain Pass ◽  
Lack Of Compactness

AbstractWe study the following Kirchhoff equation: (K)$$ - \left({1 + b\int_{{{\mathbb R}^3}} |\nabla u{|^2}dx} \right)\Delta u + V(x)u = f(x, u), \quad x \in {{\mathbb R}^3}. $$A feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and the mountain pass theorem.

scholarly journals On the Struwe—Jeanjean—Toland monotonicity trick

2012 ◽  
Vol 142 (1) ◽  
pp. 155-169 ◽  
Author(s):  
Marco Squassina
Keyword(s):  
Critical Point ◽  
Banach Spaces ◽  
Critical Point Theory ◽  
Point Theory ◽  
Energy Functional ◽  
Real Parameter ◽  
Mountain Pass ◽  
Abstract Version ◽  
Symmetry Assumption ◽  
Continuous Functionals

The abstract version of Struwe's monotonicity trick developed by Jeanjean and Toland for functionals depending on a real parameter is strengthened in the sense that it provides, for almost every value of the parameter, the existence of a bounded almost symmetric Palais–Smale sequence at the mountain-pass level whenever a mild symmetry assumption is set on the energy functional. In addition, the whole theory is extended to the case of continuous functionals on Banach spaces, in the framework of non-smooth critical point theory.

scholarly journals Existence of Solutions for Higher Order Bvp with Parameters via Critical Point Theory

2015 ◽  
Vol 48 (1) ◽  
Author(s):  
Mariusz Jurkiewicz ◽  
Bogdan Przeradzki
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Linear Part ◽  
Point Theory ◽  
Higher Order ◽  
Mountain Pass ◽  
Mountain Pass Lemma

AbstractThis paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters.

scholarly journals Multiple positive solutions for a logarithmic Schrödinger–Poisson system with singular nonelinearity

2021 ◽  
pp. 1-15
Author(s):  
Linyan Peng ◽  
Hongmin Suo ◽  
Deke Wu ◽  
Hongxi Feng ◽  
Chunyu Lei
Keyword(s):  
Variational Method ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Multiple Positive Solutions ◽  
Poisson System ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

In this article, we devote ourselves to investigate the following logarithmic Schrödinger–Poisson systems with singular nonlinearity { − Δ u + ϕ u = | u | p−2 u log ⁡ | u | + λ u γ , i n   Ω , − Δ ϕ = u 2 , i n   Ω , u = ϕ = 0 , o n   ∂ Ω , where Ω is a smooth bounded domain with boundary 0 < γ < 1 , p ∈ ( 4 , 6 ) and λ > 0 is a real parameter. By using the critical point theory for nonsmooth functional and variational method, the existence and multiplicity of positive solutions are established.

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