Related papers: Geometric optics and boundary layers for Nonlinear…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We find conditions…

We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to…

We study the existence of ground state solutions for a class of non-linear pseudo-relativistic Schr\"odinger equations with critical two-body interactions. Such equations are characterized by a nonlocal pseudo-differential operator closely…

We consider the logarithmic Schr{\"o}dinger equation in a semiclassical scaling, in the presence of a smooth, at most quadratic, external potential. For initial data under the form of a single coherent state, we identify the notion of…

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition…

In this article we prove global propagation of analyticity in finite time for solutions of semilinear Schr\"odinger equations with analytic nonlinearity from a region $\omega$ where the Geometric Control Condition holds. Our approach…

We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with general normal directions that may not be parallel to the…

We establish the existence of solutions to the following semilinear Neumann problem for fractional Laplacian and critical exponent: \begin{align*}\left\{\begin{array}{l l} { (-\Delta)^{s}u+ \lambda u= \abs{u}^{p-1}u } & \text{in $ \Omega,$…

This work studies the initial-boundary value problem of the two-dimensional nonlinear Schr\"odinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It…

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating…

In this work we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a bounded region of the plane that degenerates into a line segment…

We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside…

In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in…

We consider a Schr{\"o}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a…

This paper is concerned with a family of second-order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first-order…

We study a supersymmetric version of the Gardner equation (both focusing and defocusing) using the superbilinear formalism. This equation is new and cannot be obtained from supersymmetric modified Korteweg-de Vries equation with a nonzero…

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as…

By means of non-smooth critical point theory we obtain existence of infinitely many weak solutions of the fractional Schr\"odinger equation with logarithmic nonlinearity. We also investigate the H\"older regularity of the weak solutions.

In this paper, we consider the focusing fractional nonlinear Schr\"{o}dinger equation (FNLS) on the waveguide manifolds $\mathbb{R}^d\times\mathbb{T}^m$ both in the isotropic and anisotropic case. Under different conditions, we establish…

We consider the nonlinear Schrodinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse: there is a loss of regularity,…