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Abstract
We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is -Δu = Vu for u ∈ W01,2(D) with D a bounded domain in ℝn. 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.
Received: 2012-7-25
Published Online: 2012-9-28
Published in Print: 2015-1-1
© 2015 by De Gruyter
