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Ukrainian Mathematical Journal

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

Elliptic equation with singular potential

  • B. A. Khudaigulyev
Article
  • 35 Downloads
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.

Keywords

Elliptic Equation Heat Equation Nonnegative Function Mixed Problem Cutoff Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    P. Baras and J. A. Goldstein, “The heat equation with a singular potential,” Trans. Amer. Math. Soc., 284, No. 1, 121–139 (1984).MathSciNetMATHCrossRefGoogle Scholar
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    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

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

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