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 $\phi(x)>0$.
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