A singular perturbation problem governed by the normalized $$p(x)$$ p ( x ) -Laplacian operator

João Vitor Silva () and Víctor A. B. Viloria ()
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João Vitor Silva: Universidade Estadual de Campinas
Víctor A. B. Viloria: Universidad Nacional de Colombia

Partial Differential Equations and Applications, 2025, vol. 6, issue 6, 1-34

Abstract: Abstract In this manuscript, we focus on studying a family of viscosity solutions $$(u_\varepsilon )_{\varepsilon > 0}$$ ( u ε ) ε > 0 for a singular perturbation problem driven by the normalized $$p(x)$$ p ( x ) -Laplacian operator $$ \left\{ \begin{array}{rclcl} \Delta _{p_{\varepsilon } (x)}^{\textrm{N}} u_{\varepsilon }(x) & = & \zeta _{\varepsilon }\left( u_{\varepsilon }\right) + f_{\varepsilon }(x) & {\text {in}} & \Omega , \\ u_{\varepsilon }(x) & = & g(x) & {\text {on}} & \partial \Omega , \end{array} \right. $$ Δ p ε ( x ) N u ε ( x ) = ζ ε u ε + f ε ( x ) in Ω , u ε ( x ) = g ( x ) on ∂ Ω , We establish that the solutions exhibit uniform bounds, local Lipschitz continuity, and non-degeneracy properties in a regular domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n with a sufficiently smooth boundary datum. As a consequence, we demonstrate that, up to a subsequence, $$ \lim _{j \rightarrow \infty } u_{\varepsilon _j} = u_0$$ lim j → ∞ u ε j = u 0 , where $$u_0$$ u 0 is a viscosity solution to a one-phase Bernoulli-type free boundary problem, enjoying uniform and optimal Lipschitz bounds.

Keywords: Singularly perturbed problems; Normalized p(x)-Laplacian; One-phase free boundary problem; Lipschitz regularity estimates; 35B25; 35J60; 35J70; 35R35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00357-8

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