Soliton solutions for quasilinear Schr\"{o}dinger equations involving supercritical exponent in RNR^N

We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\"{o}dinger equations. Schr\"{o}dinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics

Invariance properties of the Monge-Kantorovich mass transport problem

We consider the multidimensional Monge-Kantrovich transport problem in an abstract setting. Our main results state that if a cost function and marginal measures are invariant by a family of transformations, then a solution of the Kantrovich relaxation problem and a solution of its dual can be chosen so that they are invariant under the same family of transformations. This provides a new tool to study and analyze the support of optimal transport plans and consequently to scrutinize the Monge problem. Birkhoff's Ergodic theorem is an essential tool in our analysis

A characterization for solutions of the Monge-Kantorovich mass transport problem

A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of optimal plans for the mass transport problem involving general cost functions. We also establish a criterion for the uniqueness

Solutions to multi-marginal optimal transport problems concentrated on several graphs

We study solutions to the multi-marginal Monge-Kantorovich problem which are concentrated on several graphs over the first marginal. We first present two general conditions on the cost function which ensure, respectively, that any solution must concentrate on either finitely many or countably many graphs. We show that local differential conditions on the cost, known to imply local $d$-rectifiability of the solution, are sufficient to imply a local version of the first of our conditions. We exhibit two examples of cost functions satisfying our conditions, including the Coulomb cost from density functional theory in one dimension. We also prove a number of results relating to the uniqueness and extremality of optimal measures. These include a sufficient condition on a collection of graphs for any competitor in the Monge-Kantorovich problem concentrated on them to be extremal, and a general negative result, which shows that when the problem is symmetric with respect to permutations of the variables, uniqueness cannot occur except under very special circumstances

Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form, \begin{equation}\label{con-c} \left \{ \begin{array}{ll} -\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, & x \in \Omega\\ u=0, & x \in \partial \Omega \end{array} \right. \end{equation} where $\Omega\subset \mathbb{R}^n$ is a bounded domain with $C^2$-boundary and $1<q< 2<p.$ As a consequence of our results we shall show that, for each $p>2$, there exists $\mu^*>0$ such that for each $\mu \in (0, \mu^*)$ this problem has a sequence of solutions with a negative energy. This result was already known for the subcritical values of $p.$ In this paper, we shall extend it to the supercritical values of $p$ as well. Our methodology is based on a new variational principle established by one of the authors that allows one to deal with problems beyond the usual locally compactness structure

On the existence of Hamiltonian paths connecting Lagrangian submanifolds

We use a new variational method --based on the theory of anti-selfdual Lagrangians developed in [2] and [3]-- to establish the existence of solutions of convex Hamiltonian systems that connect two given Lagrangian submanifolds in $\R^{2N}$. We also consider the case where the Hamiltonian is only semi-convex. A variational principle is also used to establish existence for the corresponding Cauchy problem. The case of periodic solutions will be considered in a forthcoming paper [5].Comment: 12 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif

Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems

Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians introduced and developed recently in [3], [4] and [5].Comment: 18 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif

Metric Selfduality and Monotone Vector Fields on Manifolds

We develop a "metrically selfdual" variational calculus for $c$-monotone vector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on $X\times Y$. Remarkably, many of the key properties of classical monotone operators known to hold in a linear context, extend to this non-linear setting. This includes an integral representation of $c$-monotone vector fields in terms of $c$-convex selfdual Lagrangians, their characterization as a partial $c$-gradients of antisymmetric Hamiltonians, as well as the property that these vector fields are generically single-valued. We also use a symmetric Monge-Kantorovich transport to associate to any measurable map its closest possible $c$-monotone "rearrangement". We also explore how this metrically selfdual representation can lead to a global variational approach to the problem of inverting $c$-monotone maps, an approach that has proved efficient for resolving non-linear equations and evolutions driven by monotone vector fields in a Hilbertian setting.Comment: 27 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif

Anti-symmetric Hamiltonians (II): Variational resolutions for Navier-Stokes and other nonlinear evolutions

The nonlinear selfdual variational principle established in a preceeding paper [8] -- though good enough to be readily applicable in many stationary nonlinear partial differential equations -- did not however cover the case of nonlinear evolutions such as the Navier-Stokes equations. One of the reasons is the prohibitive coercivity condition that is not satisfied by the corresponding selfdual functional on the relevant path space. We show here that such a principle still hold for functionals of the form I(u)= \int_0^T \Big [ L (t, u(t),\dot {u}(t)+\Lambda u(t)) +< \Lambda u(t), u(t) > \Big ] dt +\ell (u(0)- u(T), \frac {u(T)+ u(0)}{2}) where $L$ (resp., $\ell$) is an anti-selfdual Lagrangian on state space (resp., boundary space), and $\Lambda$ is an appropriate nonlinear operator on path space. As a consequence, we provide a variational formulation and resolution to evolution equations involving nonlinear operators such as the Navier-Stokes equation (in dimensions 2 and 3) with various boundary conditions. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying $u(0)=\alpha u(T)$ for any given $\alpha$ in $(-1,1)$. Our approach is quite general and does apply to many other situations.Comment: 25 pages. Updated versions --if any-- of this author's papers can be downloaded at http://pims.math.ca/~nassif

Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions

Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as well as certain nonlinear Schrodinger evolutions. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as "periodic orbits up to an isometry" for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, while keeping the system selfdual.Comment: 36 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.birs.ca/~nassi