Minimal support results for Schr\"odinger equations

We consider a number of linear and non-linear boundary value problems involving generalized Schr"odinger equations. The model case is $-\mathrm{\Delta }u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\mathbf{R}}^{\mathbf{n}}$. 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.