Semilinear Elliptic Equations with Singular Nonlinearities

Abstract

In this work, we prove the existence of a weak solutions of non-linear singular elliptic problems of the following type: (P)    −div(M(x)∇u) = f (x) u γ in Ω, u > 0 in Ω, u = 0 on ∂Ω, where the function f belongs to L (Ω), The singularity arises due to the term u f γ , which can become unbounded as u → 0. To address this, we approximate the problem by introducing a regularized version, ensuring that the operator remains well-defined. We then derive a priori estimates for the sequence of approximate solutions and pass to the limit,proving the existence of a weak solution for problem (P)