Related papers: A multiplicity result for double singularly pertur…
We prove an Alt-Caffarelli-Friedman montonicity formula for pairs of functions solving elliptic equations driven by different ellipticity matrices in their positivity sets. As application, we derive Liouville-type theorems for subsolutions…
We derive rigorously, for both R^2 and [-L, L]^2, the cubic nonlinear Schrodinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence…
In this work we are concerned with solutions to the linear Schr\"odinger type system with mixed dispersion, the so-called biharmonic Schr\"odinger equation. Precisely, we are able to prove an exact control property for these solutions with…
In this paper we are focusing to the following Schr\"odinger-Maxwell system $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$: \[ \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mathrm{in}\ M -\Delta_{g}\phi+\phi=qu^{2} &…
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first…
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of…
Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without boundary. It is proved that the energy-critical nonlinear Schr\"odinger equation is globally well-posed for small initial data in $H^1(M)$, provided that a certain…
The Schroedinger equation for position-dependent mass singular oscillators is solved by means of the factorization method and point transformations. These systems share their spectrum with the conventional singular oscillator. Ladder…
In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…
In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…
The endpoint of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we…
By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical…
We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which…
In this paper, we study a couple of NLS equations characterized by mixed cubic and superlinear power laws. Classification of the solutions as well as existence and uniqueness of the steady state solutions have been investigated.
For a large class of complete, non-compact Riemannian manifolds, $(M,g)$, with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local…
Let (M,g) be a smooth compact, n dimensional Riemannian manifold, n=3,4 with smooth n-1 dimensional boundary. We search the positive solutions of the singularly perturbed Klein Gordon Maxwell Proca system with homogeneous Neumann boundary…
We prove that the Maxwell-Schr\"odinger system in $\R^{3+1}$ is globally well-posed in the energy space. The key element of the proof is to obtain a short time wave packet parametrix for the magnetic Schr\"odinger equation, which leads to…
We consider a slightly subcritical elliptic system with Dirichlet boundary conditions and a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of…
In this article, we are interested in multi-bump solutions of the singularly perturbed problem \begin{equation*} -\epsilon^2\Delta v+V(x)v=f(v) \ \ \mbox{ in }\ \ \R^N. \end{equation*} Extending previous results \cite{B, DLY,W1}, we prove…
In this paper we consider a competitive weakly coupled elliptic system in which each species is attracted to a small region and repelled from its complement. In this setting, we establish the existence of infinitely many solutions and of a…