Ukrainian Mathematical Journal

, Volume 62, Issue 12, pp 1989–1999 | Cite as

Elliptic equation with singular potential

  • B. A. Khudaigulyev
We consider the following problem of finding a nonnegative function u(x) in a ball B = B(O, R) ⊂ R n , n ≥ 3:
$$ - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right. $$
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) , this problem has a nonnegative solution in the ball B 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.


Elliptic Equation Heat Equation Nonnegative Function Mixed Problem Cutoff Function 
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© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • B. A. Khudaigulyev
    • 1
  1. 1.Turkmen State UniversityAshkhabadTurkmenistan

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