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Jonathan Di Cosmo

Publications and source records attributed to Jonathan Di Cosmo.

4 recordsLinked to original sources
arXiv2015

Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

We construct solutions to the nonlinear magnetic Schrödinger equation $$ \left\{ \begin{aligned} - \varepsilon^2 Δ_{A/\varepsilon^2} u + V u &= \lvert u\rvert^{p-2} u & &\text{in}\ Ω,\\ u &= 0 & &\text{on}\ \partialΩ, \end{aligned} \right. $$ in the semiclassical régime with strong magnetic fields. In contrast with the well-studied mild magnetic field régime, the limiting energy depends on the magnetic field allowing to recover the Lorentz force in the semi-classical limit. Our solutions concentrate around global or local minima of a limiting energy that depends on the electric potential and the magnetic field. The results cover unbounded domains, fast-decaying electric potential and unbounded electromagnetic fields. The construction is variational and is based on an asymptotic analysis of solutions to a penalized problem in the spirit of M. del Pino and P. Felmer.

math.AP↗
arXiv2014

Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima

We study positive bound states for the equation $$- ε^2 Δu + Vu = u^p, \qquad \text{in $\mathbf{R}^N$}, $$ where $ε> 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential $V$ without any restriction on the potential.

math.AP↗
arXiv2010

Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres

We study positive bound states for the semiclassical stationary nonlinear Schrödinger equation. We are especially interested in solutions which concentrate on a lower dimensional sphere. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, the potentials might be singular at the origin or vanish superquadratically at infinity.

math.AP↗