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Mat. Zametki, 2012, Volume 92, Issue 6, Pages 928–938 (Mi mz10149)  

Elliptic Equation with a Singular Potential in a Domain with a Conic Point

B. A. Khudaikuliev

Turkmenien State University

Abstract: This paper deals with the behavior of the nonnegative solutions of the problem
$$ -\Delta u=V(x)u,\qquad u|_{\partial\Omega}=\phi(x) $$
in a conical domain $\Omega \subset \mathbb{R}^n$, $n \ge 3$, where $0\le V(x) \in L_1(\Omega)$, $0\le \phi(x) \in L_1(\partial\Omega)$ and $\phi(x)$ is continuous on the boundary $\partial\Omega$. It is proved that there exists a constant $C_\star(n)=(n-2)^2/4$ such that if $V_0(x)=(c+\lambda_1)|x|^{-2}$, then, for $0\le c\le C_\star(n)$ and $V(x) \le V_0(x)$ in the domain $\Omega$, this problem has a nonnegative solution for any nonnegative boundary function $\phi(x) \in L_1(\partial\Omega)$; for $c>C_\star(n)$ and $V(x) \ge V_0(x)$ in $\Omega$, this problem has no nonnegative solutions if $\phi(x)>0$.

Keywords: elliptic equation, singular potential, conic domain, conic point, Laplace operator, Beltrami operator, Dirichlet boundary condition, Cauchy's inequality, Hölder's inequality

DOI: https://doi.org/10.4213/mzm10149

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English version:
Mathematical Notes, 2012, 92:6, 820–829

Bibliographic databases:

UDC: 517.956
Received: 11.12.2009
Revised: 11.07.2011

Citation: B. A. Khudaikuliev, “Elliptic Equation with a Singular Potential in a Domain with a Conic Point”, Mat. Zametki, 92:6 (2012), 928–938; Math. Notes, 92:6 (2012), 820–829

Citation in format AMSBIB
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