Regularizing effect due to the interplay between coefficients in some elliptic problems with L1 data

• Mounim El Ouardy ^[1] ; Abdelaaziz Sbai ^[2] ; Youssef El Hadfi ^[1]
  1. [1] Université Sultan Moulay Slimane

     Université Sultan Moulay Slimane

     Alkhalfia, Marruecos

     
  2. [2] Laboratory LIMSIS, Higher School of Technology, Moulay Ismail University
• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol 83, Nº. 1, 2026, págs. 159-176
• Idioma: inglés
• DOI: 10.1007/s40324-025-00378-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper, we study the regularizing effect due to the interaction between the coefficient of the zero-order term and the datum for the following type of elliptic problem u ∈ W1 0,p (Ω) : − div(M(x)|∇u|p−2∇u) + b(x)h(u) = f (x), where Ω is a bounded open subset of RN , N > 2, M is a bounded elliptic matrix, 0 ≤ b(x) ∈ L1(Ω) and h is a continuous odd-increasing function. Even if f(x) only belongs to L1(Ω), the assumption there exists L ∈ (0, lim s→∞ h(s)) such that |f(x)| ≤ Lb(x) implies the existence of a weak solution belonging to W1 0,p(Ω) and to L∞(Ω). Using the strong maximum principle we prove that such a solution u is strictly positive a.e. in the domain Ω. In the second part, we continue to study the previous problem, we add a term having a superlinear growth depending on the gradient of the solution, and we prove that this problem admits a weak bounded solution and from the weak maximum principle we prove that each solution of the problem is positive. Finally, we study the existence and summability of solutions to problems featuring Hardy-type potentials.