Semilinear elliptic equations with singular nonlinearities

Abstract

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

A semilinear elliptic equation with a mild singularity at u= 0: Existence and homogenization

“…As pointed out in the Introduction, the strong maximum principle is one of the key tools used in the proofs of the results obtained in [2] by L. Boccardo and L. Orsina, results which inspired the present paper.…”

“…Note that, in contrast with the proofs of the results in [2], the proofs of all the results in the present paper do not make use neither of the strong maximum principle nor of the results of Proposition 3.5 and Remark 3.7 below. Proof.…”

“…Crandall, P.H. Rabinowitz and L. Tartar, and to the paper [2] by L. Boccardo and L. Orsina which inspired our work. We also refer to the references quoted in these papers as well as those quoted in the paper [1] by L. Boccardo and J. CasadoDíaz which deals with the homogenization of this problem for a sequence of matrices A ε (x).…”

“…In [2] the authors study the problem (1.1) with F (x, u) = f (x) u γ , γ > 0 and f in Lebesgue spaces. They prove existence, uniqueness and regularity results depending on the values of γ and on the summability of f .…”

Subelliptic equations with singular nonlinearities on the Heisenberg group

“…Boccardo and Orsina [3] derived the existence, regularity, and nonexistence for the Dirichlet problem to the model…”