Classical and Weak Solutions of a Singular Semilinear Elliptic Problem
The singular semilinear elliptic equation Δu + p(x)f(u) = 0 is shown to have a unique positive classical solution in Rn that decays to zero at ∞ if p(x) is simply a nontrivial nonnegative continuous function satisfying ∫∞0 t max|x| = t p(x) dt < ∞, provided f is a non-increasing continuously differentiable function on (0, ∞). It is also shown that the equation has a unique weakH10-solution on a bounded domain provided ∫ε0 f(s) ds < ∞ and p(x) ∈ L2.
Journal of Mathematical Analysis and Applications
Lair, A. V, & Shaker, A. W. (1997). Classical and Weak Solutions of a Singular Semilinear Elliptic Problem. Journal of Mathematical Analysis and Applications, 211(2), 371–385. https://doi.org/10.1006/jmaa.1997.5470