On a singular nonlinear elliptic boundary-value problem

Abstract

Abstract.We consider the singular boundary-value problem Au+p(x)u~7 = 0 in d, u | 9Í2 = 0 , where y > 0 . Under the assumption p(x) > 0 and certain smoothness assumptions, we show that there exists a solution which is smooth on £2 and continuous on Í2 .

Semilinear elliptic equations with singular nonlinearities

“…The problem was then studied by Crandall, Rabinowitz and Tartar in [9], where they proved existence of solutions, and continuity properties of the solution if g(x, s) does not depend on x. In 1991, a nice paper by Lazer and McKenna (see [14]) dealt with the case g(x, s) = f (x) u γ , with f a continuous function, proving existence and regularity results at the boundary for the solution; for example, they proved that the solution belongs to H 1 0 ( ) if and only if γ < 3, while it is not in C 1 ( ) if γ > 1 (confront these result with Theorem 4.2 below). The results of Lazer and McKenna were then generalized by Lair and Shaker (see [12] and [13], where the function g is of the form f (x) g(u)), and by Zhang and Cheng (see [18], where the function g is of the form f (x) g(u) and f may not even be in L 1 ( )).…”

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

“…is nonnegative continuous and nonincreasing on (0, ∞) and for each c > 0, the function ϕ(., c) is in the classical Kato class K ∞ n (D). In particular, they generalized some existence results which have been already established for the special nonlinearity ϕ(x, t) = p(x)f (t) (see for example [4,[7][8][9] and the references therein).…”

Finite element approximation of a semilinear elliptic problem with a singular nonlinearity

“…We first remark that this result was proved in [11], Theorem 2, under slightly different assumptions. The proof of "=⇒" can easily be adapted.…”

“…The proof of "=⇒" can easily be adapted. However, we believe that the proof of "⇐=" given in [11] is incorrect. We prove "⇐=".…”

“…The kind of singular nonlinearity which occurs in (P) has been considered by several authors (e.g. [4,5,9,11,16] and the references therein) from an analytical point of view.…”