Minimal support results for Schrödinger equations

Abstract

Abstract We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is -Δ u = V u for u ∈ W 0 1,2 ( D ) with D a bounded domain in ℝ n . We use the Sobolev embedding theorem, and in some cases the Moser–Trudinger inequality and the Hardy–Sobolev inequality, to derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V , the measure of D , and a sharp Sobolev constant. In most cases, these inequalities are best possible.

Locations

• arXiv (Cornell University) - View - PDF
• Forum Mathematicum - View