{"ID":23664236,"CreatedAt":"2026-09-18T08:14:42.696445972Z","UpdatedAt":"2026-09-18T08:14:42.696445972Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20778","arxiv_id":"2609.20778","title":"Positive normalized solutions for a singular regularized $p(x)$-Laplacian Dirichlet problem","abstract":"We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating functionals. Under the admissibility of signed constrained variations, each minimizer satisfies an approximate Euler--Lagrange equation with a uniquely determined constraint multiplier. An explicit uniform boundary lower barrier and a compatible weight condition, combined with a variable exponent Hardy inequality, provide global dual control of the singular sources and convergence against every zero-trace Sobolev test function. For these approximate minimizing pairs, the singular-limit result gives subsequential strong Sobolev convergence of the states and convergence of their multipliers, while preserving positivity and the prescribed modular. The limiting pair is therefore a positive normalized weak solution of the singular problem, obtained by removing the reaction regularization while keeping the gradient regularization fixed.","short_abstract":"We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular app...","url_abs":"https://arxiv.org/abs/2609.20778","url_pdf":"https://arxiv.org/pdf/2609.20778v1","authors":"[\"Mustafa Avci\"]","published":"2026-09-17T17:49:06Z","proceeding":"math.AP","tasks":"[\"math.AP\"]","methods":"[]","has_code":false}