TY - JOUR
AU - B. A. Hudaigulyev
PY - 2010/12/25
Y2 - 2024/02/29
TI - Elliptic equation with singular potential
JF - Ukrains’kyi Matematychnyi Zhurnal
JA - Ukr. Mat. Zhurn.
VL - 62
IS - 12
SE - Short communications
DO -
UR - https://umj.imath.kiev.ua/index.php/umj/article/view/2994
AB - 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 $φ(x) > 0$.
ER -