Abbas Moameni

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 developed recently in Ghoussoub (2007a b c).

On the existence of standing wave solutions to quasilinear Schrödinger equations

The fibreing method is employed to prove the existence of at least one or sometimes two standing wave solutions for quasilinear Schrodinger equations. These equations contain singular nonlinearities which include derivatives of the second order. Such equations have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

On a class of periodic quasilinear Schrodinger equations involving critical growth in R2

Abstract We consider the equation − Δ u + V ( x ) u − k 2 ( Δ ( | u | 2 ) ) u = g ( x , u ) , u > 0 , x ∈ R 2 , where V : R 2 → R and g : R 2 × R → R are two continuous 1-periodic functions and k is a positive constant. Also, we assume g behaves like exp ( β | u | 4 ) as | u | → ∞ . We prove the existence of at least one weak solution u ∈ H 1 ( R 2 ) with u 2 ∈ H 1 ( R 2 ) . The mountain pass in a suitable Orlicz space together with the Trudinger–Moser inequality are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrodinger equations. Schrodinger 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.

Homogenization of Maximal Monotone Vector Fields via Selfdual Variational Calculus

Abstract We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Γ-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.

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, 1

Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in RN

\begin{aligned} \left\{ \begin{array}{ll} -\,\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, &{}\quad x \in \Omega \\ u=0, &{}\quad x \in \partial \Omega \end{array} \right. \end{aligned}

Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

-Δu=|u|p-2u+μ|u|q-2u,x∈Ωu=0,x∈∂Ωwhere