scholarly journals Existence Results for a Perturbed Problem Involving Fractional Laplacians

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yan Hu
Keyword(s):  
Variational Methods ◽  
Iteration Method ◽  
Existence Results ◽  
Periodic Perturbation ◽  
Monotone Iteration ◽  
Odd Nonlinearity ◽  
Layer Solution

We extend the results of Cabre and Sire (2011) to show the existence of layer solutions of fractional Laplacians with perturbed nonlinearity(-Δ)su=b(x)f(u)inℝwiths∈(0,1). Herebis a positive periodic perturbation forf, and-fis the derivative of a balanced well potentialG. That is,G∈C2,γsatisfiesG(1)=G(-1)<G(τ)  ∀τ∈(-1,1),  G'(1)=G'(-1)=0.First, for odd nonlinearityfand for everys∈(0,1), we prove that there exists a layer solution via the monotone iteration method. Besides, existence results are obtained by variational methods fors∈(1/2,1)and for more general nonlinearities. While the cases≤1/2remains open.

scholarly journals Radially symmetric solutions to the Hénon–Lane–Emden system on the critical hyperbola

2014 ◽  
Vol 16 (03) ◽  
pp. 1350030 ◽  
Author(s):  
Roberta Musina ◽  
K. Sreenadh
Keyword(s):  
Variational Methods ◽  
Existence Results ◽  
Qualitative Properties ◽  
Radially Symmetric Solutions ◽  
Qualitative Properties Of Solutions

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type

2016 ◽  
Vol 23 (3) ◽  
pp. 293-301
Author(s):  
Ghasem A. Afrouzi ◽  
Armin Hadjian
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Existence Results ◽  
Kirchhoff Type ◽  
Nonlocal Elliptic Problem

AbstractExistence results of positive solutions for a nonlocal elliptic problem of Kirchhoff-type are established. The approach is based on variational methods.

scholarly journals Existence of Solutions for Fractional Boundary Value Problems with a Quadratic Growth of Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yaning Li ◽  
Quanguo Zhang ◽  
Baoyan Sun
Keyword(s):  
Fractional Derivative ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Boundary Problem ◽  
Linear Growth ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Existence Results ◽  
Quadratic Growth ◽  
Fractional Boundary Value Problems

In this paper, we deal with two fractional boundary value problems which have linear growth and quadratic growth about the fractional derivative in the nonlinearity term. By using variational methods coupled with the iterative methods, we obtain the existence results of solutions. To the best of the authors’ knowledge, there are no results on the solutions to the fractional boundary problem which have quadratic growth about the fractional derivative in the nonlinearity term.

scholarly journals An Ambrosetti-Prodi-type problem for an elliptic system of equations via monotone iteration method and Leray-Schauder degree theory

1996 ◽  
Vol 1 (2) ◽  
pp. 137-152 ◽  
Author(s):  
D. C. de Morais Filho
Keyword(s):  
Elliptic System ◽  
Iteration Method ◽  
Elliptic Equations ◽  
Degree Theory ◽  
System Of Equations ◽  
Type Problem ◽  
Monotone Iteration

In this paper we employ the Monotone Iteration Method and the Leray-Schauder Degree Theory to study anℝ2-parametrized system of elliptic equations. We obtain a curve dividing the plane into two regions. Depending on which region the parameter is, the system will or will not have solutions. This is an Ambrosetti-Prodi-type problem for a system of equations.

scholarly journals Existence of Standing Waves for a Generalized Davey-Stewartson System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiaoxiao Hu ◽  
Xiao-xun Zhou ◽  
Wu Tunhua ◽  
Min-Bo Yang
Keyword(s):  
Schrödinger Equation ◽  
Variational Methods ◽  
Schrodinger Equation ◽  
Standing Waves ◽  
Existence Results ◽  
Equation Problem

The purpose of this paper is to investigate the existence of standing waves for a generalized Davey-Stewartson system. By reducing the system to a single Schrödinger equation problem, we are able to establish some existence results for the system by variational methods.

A comparative analysis of the monotone iteration method for elliptic problems

2001 ◽  
Vol 36 (2-3) ◽  
pp. 231-248
Author(s):  
Márcia Ap. Gomes-Ruggiero ◽  
Orlando Francisco Lopes ◽  
Véra Lucia Rocha Lopes
Keyword(s):  
Comparative Analysis ◽  
Iteration Method ◽  
Elliptic Problems ◽  
Monotone Iteration

Contact between nonlinearly elastic bodies

2006 ◽  
Vol 136 (6) ◽  
pp. 1239-1266 ◽  
Author(s):  
Daniel Habeck ◽  
Friedemann Schuricht
Keyword(s):  
Variational Methods ◽  
Lagrange Equation ◽  
Existence Results ◽  
Necessary Condition ◽  
Generalized Gradients ◽  
Mechanical Problem ◽  
Elastic Bodies ◽  
Essential Tool ◽  
Nonlinearly Elastic

We study the contact between nonlinearly elastic bodies by variational methods. After the formulation of the mechanical problem, we provide existence results based on polyconvexity and on quasiconvexity. We then derive the Euler—Lagrange equation as a necessary condition for minimizers. Here Clarke's generalized gradients are an essential tool for treating the nonsmooth obstacle condi

scholarly journals Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Rainer Mandel ◽  
Dominic Scheider ◽  
Tolga Yeşil
Keyword(s):  
Helmholtz Equation ◽  
Variational Methods ◽  
Variational Formulation ◽  
Morse Index ◽  
Existence Results ◽  
Sign Changes

AbstractWe prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ - Δ u - k 2 u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p R N with $$k>0,$$ k > 0 , $$N \ge 3$$ N ≥ 3 , $$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$ p ∈ 2 ( N + 1 ) N - 1 , 2 N N - 2 and $$Q \in L^{\infty }({\mathbb {R}}^{N})$$ Q ∈ L ∞ ( R N ) . Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

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