Abstract
We report on some recent results concerning existence of solutions for nonlinear scalar field equations that lead to semilinear elliptic boundary value problems in ℝN. Such problems arise in a wide variety of contexts in physics (solitons in nonlinear Klein-Gordon or Schrodinger equations, euclidean scalar fields, statistical mechanics, cosmology, nonlinear optics etc…). Existence of a ground state and of infinitely many bound states is proved under assumptions which are “nearly optimal”, using a variational technique. Other methods of resolution are also presented. Some results on bifurcation from the essential spectrum are derived for this class of problems. A generalization of the existence results for systems of equations is also provided here. Lastly, in the appendix, we present some numerical computations emphasizing some qualitative properties of these equations.
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