Preface: Recent progresses in the theory of nonlinear nonlocal problems
Marco Squassina
Discrete & Continuous Dynamical Systems - S 2018, 11(3): i-i doi: 10.3934/dcdss.201803i
Nonlocal Schrödinger-Kirchhoff equations with external magnetic field
Mingqi Xiang Patrizia Pucci Marco Squassina Binlin Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1631-1649 doi: 10.3934/dcds.2017067
The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case
$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$
$s∈ (0,1)$
$a∈ \mathbb{R}^+_0$
$b∈ \mathbb{R}^+_0$
is a magnetic potential,
$V:\mathbb{R}^N\to \mathbb{R}^+$
is an electric potential,
$(-Δ )_A^s$
is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
keywords: Schrödinger-Kirchhoff equation fractional magnetic operators.
Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data
Mónica Clapp Marco Squassina
Communications on Pure & Applied Analysis 2003, 2(2): 171-186 doi: 10.3934/cpaa.2003.2.171
We show the existence of multiple solutions of a perturbed polyharmonic elliptic problem at critical growth with Dirichlet boundary conditions when the domain and the nonhomogenous term are invariant with respect to some group of symmetries.
keywords: Polyharmonic problems symmetric domains multiplicity of solutions.
On symmetry results for elliptic equations with convex nonlinearities
Kanishka Perera Marco Squassina
Communications on Pure & Applied Analysis 2013, 12(6): 3013-3026 doi: 10.3934/cpaa.2013.12.3013
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric. The semi-linear problems are studied in a framework where the associated functional is of class $C^1$ but not of class $C^2$.
keywords: Semi-linear and quasi-linear elliptic equation full and partial symmetry
Bifurcation results for problems with fractional Trudinger-Moser nonlinearity
Kanishka Perera Marco Squassina
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 561-576 doi: 10.3934/dcdss.2018031

By using a suitable topological argument based on cohomological linking and by exploiting a Trudinger-Moser inequality in fractional spaces recently obtained, we prove existence of multiple solutions for a problem involving the nonlinear fractional laplacian and a related critical exponential nonlinearity. This extends the literature for the $N$-Laplacian operator.

keywords: Fractional Trudinger-Moser embedding exponential nonlinearity existence of solutions
Global compactness results for nonlocal problems
Lorenzo Brasco Marco Squassina Yang Yang
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 391-424 doi: 10.3934/dcdss.2018022

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.

keywords: Fractional $p$-Laplacian variational methods global compactness
Ground states for scalar field equations with anisotropic nonlocal nonlinearities
Antonio Iannizzotto Kanishka Perera Marco Squassina
Discrete & Continuous Dynamical Systems - A 2015, 35(12): 5963-5976 doi: 10.3934/dcds.2015.35.5963
We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.
keywords: anisotropic nonlocal nonlinearity variable exponent loss of compactness Scalar field equation existence of ground state.
On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation
Christopher Grumiau Marco Squassina Christophe Troestler
Discrete & Continuous Dynamical Systems - B 2013, 18(5): 1345-1360 doi: 10.3934/dcdsb.2013.18.1345
We discuss the application of the Mountain Pass Algorithm to the so-called quasi-linear Schrödinger equation, which is naturally associated with a class of nonsmooth functionals so that the classical algorithm cannot directly be used. A change of variable allows us to deal with the lack of regularity. We establish the convergence of a mountain pass algorithm in this setting. Some numerical experiments are also performed and lead to some conjectures.
keywords: mountain pass solutions Quasi-linear equations mountain pass algorithm.
Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems
Eugenio Montefusco Benedetta Pellacci Marco Squassina
Communications on Pure & Applied Analysis 2010, 9(4): 867-884 doi: 10.3934/cpaa.2010.9.867
We study the spectral structure of the complex linearized operator for a class of nonlinear Schrödinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable.
keywords: Weakly coupled nonlinear Schrödinger systems orbital stability ground state solutions nondegeneracy of ground states.
The Nehari manifold for fractional systems involving critical nonlinearities
Xiaoming He Marco Squassina Wenming Zou
Communications on Pure & Applied Analysis 2016, 15(4): 1285-1308 doi: 10.3934/cpaa.2016.15.1285
We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $R^2$.
keywords: Nehari manifold. concave-convex nonlinearities Fractional systems

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