Existence and regularity of solutions for a singular anisotropic ( p , q ) -Laplacian with variable exponent

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2025; no. 13; pp. 1 - 28
• Main Authors: Hamidi, Abdellah, El Amrouss, Abdelrachid, Kissi, Fouad
• Format: Journal Article
• Language: English
• Published: University of Szeged 01.01.2025
• Subjects: anisotropic singular $(p; approximation; existence and regularity results; q)$-equations; variational methods
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2025.1.13

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic ( p , q ) -Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on ∂ Ω . The main equation we analyze is − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | p i − 2 ∂ i u ( x ) ) − ∑ i = 1 N ∂ i ( | ∂ i u ( x ) | q i − 2 ∂ i u ( x ) ) = f ( x ) u ( x ) γ ( x ) in  Ω , where Ω is a bounded, regular domain in R N , f is a positive function belonging to a specific Lebesgue space, and γ ( x ) is a positive continuous function on Ω ¯ . In our study, we do not make comparisons between p i and q i , and as a result, we show that the solution belongs to either W 0 1 , p → ( Ω ) ∩ W 0 1 , q → ( Ω ) or W l o c 1 , p → ( Ω ) ∩ W l o c 1 , q → ( Ω ) depending on the summability of f ( x ) and the values of γ ( x ) . The results are achieved using approximation techniques that include truncation, comparison, and variational methods.