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Mat. Zametki, 2017, Volume 101, Issue 1, Pages 101–109 (Mi mz11095)  

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.

DOI: https://doi.org/10.4213/mzm11095

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

Bibliographic databases:

UDC: 517.946
Received: 08.01.2016

Citation: Sh. M. Nasibov, “On the Equation $\Delta u+q(x)u=0$”, Mat. Zametki, 101:1 (2017), 101–109; Math. Notes, 101:1 (2017), 123–131

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