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.
Salah Boulaaras (Thu,) studied this question.