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Elliptic equation with singular potential

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Ukrainian Mathematical Journal Aims and scope

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.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 12, pp. 1715 – 1723, December, 2010.

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Khudaigulyev, B.A. Elliptic equation with singular potential. Ukr Math J 62, 1989–1999 (2011). https://doi.org/10.1007/s11253-011-0484-1

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  • DOI: https://doi.org/10.1007/s11253-011-0484-1

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