arXiv subjects
Publications and source records attributed to Ian Schindler.
The paper studies existence of ground states for the nonlinear Schrödinger equation with a general external magnetic field. In particular, no lattice periodicity or symmetry of the magnetic field, or presence of external electric field is required.
We prove existence results for a stationary Schr{ö}dinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.
The paper studies compactness properties of the affine Sobolev inequality of Gaoyong Zhang et al in the case $p=2$, and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems \[ -\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial x_j}=f \mbox{ in }Ω\subset\mathbb R^N, \] and \[ -\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial x_j}=u^{q-1}\,,\quad u>0,\mbox{ in }Ω\subset\mathbb R^N, \] where $A_{ij}[u]=\int_Ω\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\mathrm{d}x$ and $q\in(2,\frac{2N}{N-2})$.