Ian Schindler

Abstract Concentration Compactness and Elliptic Equations on Unbounded Domains

The notion of concentration compactness, used in numerous applications (cf. P.-L.Lions [4],[5]), was formulated originally in terms of specific functional spaces. In fact, much of the method can be formulated in general terms of a non-compact groupGof bounded operators on a Banach spaceE.We say that a bounded sequence Uk converges to zeroweakly with concentrationif for any sequencegk∈G gkuk converges weakly to zero. IfGis a compact group, the concentrated weak convergence is equivalent to the weak convergence.

Positive solutions for the p-Laplacian with Robin boundary conditions on irregular domains

We consider the Robin boundary conditions on irregular domains where the usual Sobolev embeddings fail. We present a functional framework permitting superhomogeneous growth of the nonlinearity and prove the existence of positive, bounded, and smooth solutions of the p-Laplacian equation.

Compactness properties and ground states for the affine Laplacian

The paper studies compactness properties of the affine Sobolev inequality of Lutwak et al. (J Differ Geom 62:17–38, 2002) and Zhang (J Differ Geom 53:183–202, 1999) in the case

Semilinear elliptic problems on unbounded domains

p=2

A nonlinear Schrödinger equation with external magnetic field

p=2, and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems

Semilinear equations on fractal blowups

\begin{aligned} -\sum _{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial ^2u}{\partial x_i\partial x_j}=f \text{ in } \Omega \subset {\mathbb {R}}^N, \end{aligned}