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

Sh. M. Nasibov

Institute of Applied Mathematics, Baku State University

Abstract: 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.

Keywords: generalized solution, Dirichlet problem, Sobolev inequality, Steklov inequality, Fourier transform.

UDC: 517.946

Received: 08.01.2016

DOI: 10.4213/mzm11095

 English version: Mathematical Notes, 2017, 101:1, 123–131

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