On a bifurcation value related to quasi-linear Schrodinger equations
By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the quasi-linear Schrodinger equation.
math.AP↗
arXiv subjects
Marco Squassina
Publications and source records attributed to Marco Squassina.
Finding critical points whose polarization is also a critical point
We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated by the fact that if any polarization of a critical point is also a critical point and the Euler-Lagrange equation is a second-order semi-linear elliptic problem, T. Bartsch, T. Weth and M. Willem (J. Anal. Math., 2005) have proved that the critical point is axially symmetric.
math.AP↗
Uniqueness of ground states for a class of quasi-linear elliptic equations
We prove the uniqueness of positive radial solutions for a class of quasi-linear elliptic problems containing, in particular, the quasi-linear Schrodinger equation.
math.AP↗
On a result by Boccardo-Ferone-Fusco-Orsina
Via a symmetric version of Ekeland's principle recently obtained by the author we improve, in a ball or an annulus, a result of Boccardo-Ferone-Fusco-Orsina on the properties of minimizing sequences of functionals of calculus of variations in the non-convex setting.
math.AP↗
Diffeomorphism-invariant properties for quasi-linear elliptic operators
For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.
math.AP↗
Mountain pass solutions for quasi-linear equations via a monotonicity trick
We obtain the existence of mountain pass solutions for quasi-linear equations without the typical assumptions which guarantee the boundedness of an arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity trick proved by the second author. The main results are new also for the p-Laplacian operator.
math.AP↗
Symmetry in variational principles and applications
We formulate symmetric versions of classical variational principles. Within the framework of non-smooth critical point theory, we detect Palais-Smale sequences with additional second order and symmetry information. We discuss applications to PDEs, fixed point theory and geometric analysis.
math.OC↗
On Ekeland's variational principle
For proper lower semi-continuous functionals bounded below which do not increase upon polarization, an improved version of Ekeland's variational principle can be formulated in Banach spaces, which provides almost symmetric points.
math.FA↗
On Struwe-Jeanjean-Toland monotonicity trick
The abstract version of Struwe's monotonicity trick developed by Jeanjean and Jeanjean-Toland for functionals depending on a real parameter is improved, under suitable assumptions. Besides, all the machinery is extended to the case of continuous functionals on Banach spaces, in the framework of non-smooth critical point theory.
math.AP↗
On the Palais principle for non-smooth functionals
If $G$ is a compact Lie group acting linearly on a Banach space $X$ and $f$ is a $G$-invariant function on $X$, we provide new versions of the so-called Palais' criticality principle for $f:X\to\bar\R$, in the framework of non-smooth critical point theory. We apply the results to a class of PDEs associated with functionals which are merely lower semi-continuous and could not be treated by previous versions of the principle.
math.AP↗
On the symmetry of minimizers in constrained quasi-linear problems
We provide a simple proof of the radial symmetry of any nonnegative minimizer for a general class of quasi-linear minimization problems.
math.AP↗
Generalized Polya-Szego inequality and applications to some quasi-linear elliptic problems
We generalize Polya-Szego inequality to integrands depending on $u$ and its gradient. Under minimal additional assumptions, we establish equality cases in this generalized inequality. We also give relevant applications of our study to a class of quasi-linear elliptic equations and systems.
math.AP↗
Radial symmetry of minimax critical points for nonsmooth functionals
We obtain the existence of radially symmetric and decreasing solutions to a general class of quasi-linear elliptic problems by a nonsmooth version of a symmetric minimax principle recently obtained by Jean Van Schaftingen.
math.AP↗
Semiclassical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities
The semi-classical regime of standing wave solutions of a Schrödinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.
math.AP↗
Soliton dynamics for a general class of Schrödinger equations
The soliton dynamics for a general class of nonlinear focusing Schrödinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.
math.AP↗
Asymptotic symmetry for a class of quasi-linear parabolic problems
We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the $ω$-limit set are nonnegative radially symmetric solutions of the stationary problem.
math.AP↗
Stability and instability results for standing waves of quasi-linear Schrödinger equations
We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. First, using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.
math.AP↗
Numerical computation of soliton dynamics for NLS equations in a driving potential
We provide some numerical computations for the soliton dynamics of the nonlinear Schrödinger equation with an external potential. After computing the ground state solution $r$ of a related elliptic equation we show that, in the semi-classical regime, the center of mass of the solution with initial datum modelled on $r$ is driven by the solution of a Newtonian type law. Finally, we provide some examples and analyze the numerical errors in the two dimensional case when $V$ is an harmonic potential.
math.NA↗