Regularizing effect of lower order terms in nonlinear Dirichlet problems with singularity

Abdelaaziz Sbai (), Youssef El Hadfi () and Mounim El Ouardy ()
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Abdelaaziz Sbai: Moulay Ismail University
Youssef El Hadfi: Sultan Moulay Slimane University
Mounim El Ouardy: Sultan Moulay Slimane University

Partial Differential Equations and Applications, 2024, vol. 5, issue 6, 1-19

Abstract: Abstract We study the regularizing effect of two lower order terms to nonlinear Dirichlet problems with singularity. The simplest example model is $$\begin{aligned} {\left\{ \begin{array}{ll} -\operatorname {div}\left( \left( a(x)+|u|^{q}\right) | \nabla u|^{p-2} \nabla u \right) +u^{s}=\frac{f}{u^{\theta }} & \text{ in } \varOmega \\ u>0 & \text{ in } \varOmega \\ u=0 & \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$ - div a ( x ) + | u | q | ∇ u | p - 2 ∇ u + u s = f u θ in Ω u > 0 in Ω u = 0 on ∂ Ω , where $$\varOmega $$ Ω is a bounded open set of $$\mathbb {R}^{N}(N \ge 2),$$ R N ( N ≥ 2 ) , $$1 0,$$ θ > 0 , $$q>0,$$ q > 0 , $$s\ge 1$$ s ≥ 1 and f is a nonnegative function belonging to a suitable Lebesgue space. In particular, we will prove how the presence of lower order terms may lead to an improvement of the summability of the solutions.

Keywords: Singular problems; Nonlinear elliptic problems; Existence; Regularity; Sobolev spaces; Weak solutions; Lower order terms; 3035K55; 35K67; 35B45; 35D (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s42985-024-00305-y

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