# Elliptic equation with singular potential

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We consider the following problem of finding a nonnegative function where Δ is the Laplace operator,

*u*(*x*) in a ball*B*=*B*(*O, R*) ⊂*R*^{ n },*n*≥ 3:$$ - \Delta u = V(x)u,\,\,\,\,\,u\left| {_{\partial B} = \phi (x),} \right. $$

*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
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## References

- 1.P. Baras and J. A. Goldstein, “The heat equation with a singular potential,”
*Trans. Amer. Math. Soc.,***284**, No. 1, 121–139 (1984).MathSciNetzbMATHCrossRefGoogle Scholar - 2.J. Garsia Azorero and I. Peral, “Hardy inequalities and some critical elliptic and parabolic problems,”
*Different. Equat.,***144**, 441–476 (1998).CrossRefGoogle Scholar - 3.D. Gilbarg and N. S. Trudinger,
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© Springer Science+Business Media, Inc. 2011