A type of Brezis-Oswald problem to Φ-Laplacian operator with strongly-singular and gradient terms

It is established existence of minimal W-loc(1,Phi)(Omega)-solutions on some appropriated set for the quasilinear elliptic problem [-Delta Phi u = lambda f(x, u) + mu h(x, u, del u) in Omega, u > 0 in Omega, u = 0 on partial derivative Omega, where f may have indefinite sign and behaves in a strongly singular way at u = 0, h has sublinear growth, lambda > 0 and mu >= 0 are real parameters. The main results improve the classical Brezis-Oswald's result to Orlicz-Sobolev setting in the presence of strongly-singular nonlinearities as well as gradient terms. Our approach is based on a new comparison principle for W-loc(1,Phi)(Omega)-sub and super solutions to the Phi-Laplacian operator established here, truncation arguments, L-infinity(Omega) estimates and a generalized Galerkin method.

Automated summary

The existence of minimal W-loc(1,Phi)(Omega)-solutions in an appropriate set for the quasilinear elliptic problem has been established. The study improves classical results in the presence of strongly-singular nonlinearities and gradient terms. The approach is based on a new comparison principle, truncation arguments, estimates and a generalized Galerkin method.