668 research outputs found
Ground state for the relativistic one electron atom
We study the Dirac-Maxwell system coupled with an external potential of
Coulomb type. We use the Foldy--Wouthuysen (unitary) transformation of the
Dirac operator and its realization as an elliptic problem in the 4-dim half
space with Neumann boundary condition. Using this approach
we study the existence of a "ground state" solution
Ground states for pseudo-relativistic Hartree equations of critical type
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 related to the square-root of the
Laplacian. We investigate such a problem using variational methods after
transforming the problem to an elliptic equation with a nonlinear Neumann
boundary conditions
Minimality properties of set-valued processes and their pullback attractors
We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condition on the solving
operators is assumed. The presentation is complemented with examples and
counterexamples to test the sharpness of the hypotheses involved, including a
reaction-diffusion equation, a discontinuous ordinary differential equation and
an irregular form of the heat equation.Comment: 33 pages. A few typos correcte
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