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 $$ \left \{ \begin{aligned} -\mathrm{div}\big(\mathcal{A}(x,\nabla u)\big) &+\alpha (x) \frac{|u|^{p-2}u}{|x|^{p}} \ &=\lambda \, a(x)|u|^{p-2}u\log \big(1+|u|\big) +\lambda \, h(x)|u|^{r-2}u, \quad \text{in }\Omega , \ u&=0 \quad \text{on }\partial \Omega , \end{aligned} \right . $$ where $\Omega \subset \mathbb{R}^{N}$ is bounded, contains the origin, and $\mathcal{A}$ 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.