Geodesic
Elliptic Equation with a Singular Potential in a Domain with a Conic Point
Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 928-938.

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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 $\phi(x)>0$.
Mots-clés : elliptic equation
Keywords: singular potential, conic domain, conic point, Laplace operator, Beltrami operator, Dirichlet boundary condition, Cauchy's inequality, Hölder's inequality. 
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B. A. Khudaikuliev. Elliptic Equation with a Singular Potential in a Domain with a Conic Point. Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 928-938. http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a12/

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