On the Equation $\Delta u+q(x)u=0$

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation $\Delta u+q(x)u=0$, where either $q(x)\ne\mathrm{const}$ or $q(x)=\mathrm{const}=\lambda>0$, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.