Existence and multiplicity results for singular elliptic problems with logarithmic source terms and Orlicz growth

This paper investigates a class of nonlinear elliptic boundary value problems combining Hardy-type singular potentials, Leray–Lions operators with Orlicz growth, and logarithmic source terms. We study the Dirichlet problem \ aligned -div (A (x, u) ) &+ (x) |u|^{p-2u|x|^{p} \\ &= \, a (x) |u|^p-2u (1+|u|) + \, h (x) |u|^r-2u, in, \\ u&=0 on, aligned. where R^N is bounded, contains the origin, and A exhibits Orlicz growth. The interaction between singular coercivity, nonpolynomial diffusion, and logarithmic reaction yields a nonstandard variational structure. Within an Orlicz–Sobolev framework, we prove the existence of nontrivial weak solutions using Hardy-type inequalities and compact embedding results. A Nehari-manifold approach adapted to the logarithmic nonlinearity is then developed to establish multiplicity of solutions for small values of λ. Finally, a conforming finite element discretization is proposed, and numerical experiments are presented to illustrate the convergence and stability of the scheme in the presence of strong singularities and nonstandard growth.