A Type of Brézis–Oswald Problem to the Φ- Laplacian Operator with Very Singular Term

In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in W loc 1 , Φ ( Ω ) to the problem - Δ Φ u = f ( x , u ) in Ω , u > 0 in Ω , u = 0 on ∂ Ω , where f has Φ -sublinear growth. In our main results the function f ( x , u ) may be singular at u = 0 and the nonlinear term f ( x , t ) / t ℓ - 1 , t > 0 is strictly decreasing for a suitable ℓ > 1 . Under different kind of boundary conditions we prove an improvement for the classical Brézis-Oswald and Díaz-Sáa’s results in Orlicz- Sobolev framework for singular nonlinearities as well. Some results discussed here are news even for Laplacian or p -Laplacian operators.