Elliptic equation with singular potential

  • B. A. Hudaigulyev Туркмен, ун-т, Ашгабат


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 $φ(x) > 0$.
How to Cite
Hudaigulyev, B. A. “Elliptic Equation With Singular Potential”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, no. 12, Dec. 2010, pp. 1715 -, https://umj.imath.kiev.ua/index.php/umj/article/view/2994.
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