scholarly journals Nontrivial solutions of superlinear nonlocal problems

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Dušan Repovš ◽  
Raffaella Servadei
Keyword(s):  
Weak Solutions ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Results ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Fountain Theorem ◽  
Nonlocal Equations ◽  
Laplacian Equations

AbstractWe study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti–Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

scholarly journals Existence results for fractional p-Laplacian problems via Morse theory

2016 ◽  
Vol 9 (2) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Shibo Liu ◽  
Kanishka Perera ◽  
Marco Squassina
Keyword(s):  
Phase Transition ◽  
Game Theory ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Existence Results ◽  
Nonlocal Problems ◽  
Finite Multiplicity ◽  
Topological Methods ◽  
Multiplicity Results ◽  
Linear Problems

AbstractWe investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

scholarly journals Existence results for a sublinear second order Dirichlet boundary value problem on the half-line

2020 ◽  
Vol 40 (5) ◽  
pp. 537-548
Author(s):  
Dahmane Bouafia ◽  
Toufik Moussaoui
Keyword(s):  
Variational Principle ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Point Theory ◽  
Existence Results ◽  
Half Line ◽  
Dirichlet Boundary Value Problem ◽  
Nontrivial Solutions

In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by using Ekeland's variational principle and critical point theory.

scholarly journals On the spectrum of two different fractional operators

2014 ◽  
Vol 144 (4) ◽  
pp. 831-855 ◽  
Author(s):  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Operators ◽  
Local Operators ◽  
Non Local ◽  
Definition Of ◽  
Image Position ◽  
The Laplace Operator ◽  
Fractional Type

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given bywhere c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is,where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei.The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

scholarly journals Weak Solutions for ap-Laplacian Impulsive Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wei Dong ◽  
Jiafa Xu ◽  
Xiaoyan Zhang
Keyword(s):  
Differential Equation ◽  
Variational Method ◽  
Boundary Conditions ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Impulsive Differential Equation ◽  
Dirichlet Boundary ◽  
Point Theory ◽  
Existence Results ◽  
Dirichlet Boundary Conditions

By the virtue of variational method and critical point theory, we give some existence results of weak solutions for ap-Laplacian impulsive differential equation with Dirichlet boundary conditions.

scholarly journals Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Dorota Bors
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Boundary Data ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Sufficient Condition ◽  
Variational Structure

We consider a class of partial differential equations with the fractional Laplacian and the homogeneous Dirichlet boundary data. Some sufficient condition under which the solutions of the equations considered depend continuously on parameters is stated. The application of the results to some optimal control problem is presented. The methods applied in the paper make use of the variational structure of the problem.

Three Weak Solutions for Nonlocal Fractional Equations

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Bruno Antonio Pansera
Keyword(s):  
Critical Point ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Variational Formulation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Fractional Sobolev Spaces ◽  
Fractional Equations ◽  
Critical Point Result

AbstractThis article concerns a class of nonlocal fractional Laplacian problems depending of three real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci (in order to correctly encode the Dirichlet boundary datum in the variational formulation of our problem) we establish the existence of three weak solutions for fractional equations via a recent abstract critical point result for differentiable and parametric functionals recently proved by Ricceri.

scholarly journals Three nontrivial solutions for nonlinear fractional Laplacian equations

2018 ◽  
Vol 7 (2) ◽  
pp. 211-226 ◽  
Author(s):  
Fatma Gamze Düzgün ◽  
Antonio Iannizzotto
Keyword(s):  
Boundary Value Problem ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Mountain Pass Theorem ◽  
Pseudodifferential Equation ◽  
Nontrivial Solutions ◽  
Reaction Term ◽  
Second Deformation Theorem ◽  
Type Boundary ◽  
Laplacian Equations

AbstractWe study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

Existence theorems for fractional p-Laplacian problems

2016 ◽  
Vol 15 (05) ◽  
pp. 607-640 ◽  
Author(s):  
Paolo Piersanti ◽  
Patrizia Pucci
Keyword(s):  
Weak Solutions ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Positive Function ◽  
Existence Results ◽  
Laplacian Operator ◽  
Bounded Domains ◽  
Lipschitz Boundary ◽  
Nontrivial Solutions ◽  
Existence Of Weak Solutions

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.

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