We consider the singular boundary-value problem Au+p(x)uY = 0 in Q, u I aQ = 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 Q2 and continuous on Q .

/pdf/on-a-singular-nonlinear-elliptic-boundary-value-problem-56ukq9tfr6.pdf

On a singular nonlinear elliptic boundary-value problem

We prove existence, regularity and nonexistence results for problems whose model is 

$$-\Delta u = \frac{f(x)}{u^{\gamma}}\quad {{\rm in}\,\Omega},$$

with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of ${\mathbb{R}^{N}}$. Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).

/pdf/semilinear-elliptic-equations-with-singular-nonlinearities-3jf5k6l4lk.pdf

Semilinear elliptic equations with singular nonlinearities

(1989). On a singular nonlinear dirichlet problem. Communications in Partial Differential Equations: Vol. 14, No. 10, pp. 1315-1327.

On a singular nonlinear dirichlet problem

Classical and Weak Solutions of a Singular Semilinear Elliptic Problem

Combined Effects of Singular and Superlinear Nonlinearities in Some Singular Boundary Value Problems

(1977). On a dirichlet problem with a singular nonlinearity. Communications in Partial Differential Equations: Vol. 2, No. 2, pp. 193-222.

L. Tartar

Papers

On a dirichlet problem with a singular nonlinearity