Elliptic equation with singular potential - Hudaigulyev, B. A. (Ukrains’kyi Matematychnyi Zhurnal, 2010)
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) = 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.