Elliptic equation with singular potential

Authors

We consider the following problem of finding a nonnegative function

$u(x)$ in a ball

$B = B(O, R) ⊂ R^n,\; n ≥ 3:$

$−Δu=V(x)u,u|∂B=ϕ(x),$ where

$Δ$ is the Laplace operator,

$x = (x 1, x 2,…, x n )$ , and

$∂B$ is the boundary of the ball

$B$ . It is assumed that

$0 ≤ V(x) ∈ L 1(B), 0 ≤ φ(x) ∈ L 1(∂B)$ , and

$φ(x)$ is continuous on

$∂B$ . We study the behavior of nonnegative solutions of this problem and prove that there exists a constant

$C_{*} (n) = (n − 2)^2/4$ such that if

$V_0 (x) = \frac{c}{|x|^2}, then, for$ 0 ≤ c ≤

$C_{*} (n)$ and

$V(x) ≤ V_0 (x)$ in the ball

$B$ , this problem has a nonnegative solution for any nonnegative continuous boundary function

$φ(x) ∈ L_1(∂B)$ , whereas, for

$c > C_{*} (n)$ and

$V(x) ≥ V_0(x)$ , the ball

$B$ does not contain nonnegative solutions if

$φ(x) > 0$ .