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Juan Pablo Alcon Apaza

Publications and source records attributed to Juan Pablo Alcon Apaza.

7 recordsLinked to original sources
arXiv2026

Asymptotic Behavior of Radial Solutions to Singular $p$-Laplacian Equations as $p\to1$

We study the singular limit $p \downarrow 1$ for positive radial entire solutions $u_p$ of $$ -Δ_p u_p=F_p\left(|x|, u_p,\left|\nabla u_p\right|\right) u_p^{-β_p} \quad \text { in } \mathbb{R}^n, $$ where $n \geq 2,1<p<\min \{2, \sqrt{n}\}$, and $0 \leq β_p \leq p-1$. Assuming that $$ F_p \rightarrow F_1 \quad \text { locally uniformly }, \quad \frac{β_p}{p-1} \rightarrow \mathbf{c} \in[0,1] \quad \text { as } p \downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \rightarrow u$ in $C_{\text {loc }}\left(\mathbb{R}^n\right)$ and $\nabla u_{p_j} \stackrel{*}{\rightharpoonup} \nabla u$ in $L_{\text {loc }}^{\infty}\left(\mathbb{R}^n ; \mathbb{R}^n\right)$, where $u \in W^{1, \infty}\left(\mathbb{R}^n\right)$ is radial, and solves $$ -Δ_1 u=\mathtt{K}(|x|) \quad \text { in } \mathbb{R}^n . $$ On $\left\{x \in \mathbb{R}^n \mid \mathtt{G}(|x|)<1\right\}$, we prove that $\mathtt{K}(|x|)=w^{-\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \rightarrow w$. On the other hand, we show that if $ρ\mapsto F_1(r, t, ρ)$ is affine for every $r$ and $t$, then $\mathtt{K}(|x|)=$ $w^{-\mathbf{c}} F_1(|x|, u(x),|\nabla u(x)|)$ a.e. in $\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \frac{\mathrm{d}}{\mathrm{~d} x}\left(\left|u_p^{\prime}\right|^{p-2} u_p^{\prime}\right)=\mathtt{F}_p\left(|x|, u_p,\left|u_p^{\prime}\right|\right) u_p^{-β_p} \quad \text { in } \mathbb{R} $$ and obtain analogous results as $p \downarrow 1$.

math.AP↗
arXiv2026

Weighted Sobolev Spaces and an Elliptic Eigenvalue Problem with an Indefinite Weight on Domains in Noncompact Riemannian Manifolds

The aim of this work is to establish sufficient conditions ensuring the continuity and compactness of the weighted Sobolev embedding and trace operators $$ W^{1, q}\left(Ω; \mathtt{V}_0, \mathtt{V}_1\right) \rightarrow L^{q_0}\left(Ω; \mathtt{V}_2\right) \quad \text { and } \quad W^{1, q}\left(Ω; \mathtt{V}_0, \mathtt{V}_1\right) \rightarrow L^{q_1}(\partial Ω; \mathtt{W}) . $$ As an application, we study the Neumann eigenvalue problem with an indefinite weight $$ \left\{\begin{aligned} -\operatorname{div}\left(\mathtt{V}_1 \nabla u\right)+\mathtt{V}_0 u & =λ\mathtt{V}_2 τu & & \text { in } Ω, \\ \mathtt{V}_1 \frac{\partial u}{\partial ν} & =0 & & \text { on } \partial Ω. \end{aligned}\right. $$ where $Ω$ is an open subset of a noncompact Riemannian manifold, $λ$ is a real number, $τ$ is a sign-changing function, and $\mathtt{V}_0, \mathtt{V}_1, \mathtt{V}_2$, and $\mathtt{W}$ are weight functions satisfying suitable conditions. We prove that this problem has infinitely many positive and negative eigenvalues. We also establish boundedness of its weak eigenfunctions and, for $u \in W_0^{1,2}\left(\barΩ ; \mathtt{V}_0, \mathtt{V}_1\right)$, the decay property $$ \lim _{m \rightarrow \infty} \underset{Ω\backslash \overline{D_m}}{\operatorname{ess} \sup }|u|=0 $$ where $\left(D_m\right)_{m \in \mathbb{N}}$ is an exhaustion of the manifold by bounded open sets.

math.AP↗
arXiv2026

Pointwise Estimates Near Singular Sets for Quasilinear Elliptic Equations

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $Ω$ of an $n$-dimensional Finsler manifold ( $\mathcal{M}, F, \vartheta$ ). We work with Lipschitz functions $ρ_1$ and $ρ_2$ satisfying distance-type properties; in particular, $F(\cdot, \boldsymbol{\nabla} ρ_1) \leq 1$ and $F(\cdot, \boldsymbol{\nabla} ρ_2) \leq 1$ a.e. in $\mathcal{M}$. The singular set is defined by $Γ=ρ_1^{-1}(\{0\})$. The model problem is $-Δ_{p(x)} u+|u|^{q-1} u=0$ in domains of $\mathbb{R}^n \cong \mathbb{R}^d \times \mathbb{R}^{n-d} \cong ρ_1^{-1}(\{0\}) \times ρ_2^{-1}(\{0\})$, where $ρ_1(x)=|(x_{d+1}, \ldots, x_n)|$ and $ρ_2(x)=|(x_1, \ldots, x_d)|$. The main tool in our analysis is the estimate $$ |u(x)| \leq \mathbf{C} ρ_1(x)^{-τ} $$ near $Γ$ for weak solutions $u \in W_{loc}^{1, p(x)}(\barΩ \backslash(Γ\cup Σ) ; \vartheta) \cap L_{loc}^{\infty}(\barΩ \backslash(Γ\cup Σ))$, where the constants $\mathbf{C}>0$ and $τ>0$ converge to positive values as $p^{+} \rightarrow 1$. This estimate is a key ingredient in proving that the singularity at $Γ$ is removable. Moreover, in a bounded domain $Ω$, using this estimate and assuming that, for every variable exponent satisfying $1<p^{-} \leq p^{+}<\min \{2, q+1\}$, there exists a weak solution $u_p \in W_{loc}^{1, p(x)}(Ω; \vartheta) \cap L_{loc}^{\infty}(Ω)$ of $$ -\operatorname{div}\left(|\boldsymbol{\nabla} u_p|_F^{p-2} \boldsymbol{\nabla} u_p\right)+|u_p|^{q-1} u_p=0 \quad \text { in } Ω, $$ we prove that, for every $U \Subset Ω$, there exists a subsequence $\{u_{p_m}\}$, with $p_m^{+} \rightarrow 1$, that converges to a solution $u \in B V(U ; \vartheta) \cap L^{q+1}(U ; \vartheta)$ of $$ -Δ_1 u+|u|^{q-1} u=0 \quad \text { in } U . $$

math.AP↗
arXiv2026

Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 β}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad Ω, $$ where $Ω\subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-Δ)_{p(\cdot, \cdot), β}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_Ω \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^β|y|^β} \mathrm{d} y $$ Here, $p: Ω\times Ω\rightarrow(1, \infty)$ and $s: Ω\times Ω\rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{Ω\times Ω} p$, and $0 \leq β<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights.

math.AP↗
arXiv2025

Removable set for Hölder continuous solutions of $\mathscr{A}$-harmonic functions on Finsler manifolds

We establish that a closed set $\mathcal{S}$ is removable for $α$-Hölder continuous $\mathscr{A}$-harmonic functions in a reversible Finsler manifold $(Ω, F, \mathtt{V})$ of dimension $n \geq 2$, provided that (under certain conditions on $(Ω, F, \mathtt{V})$ and the variable exponent $p$ ) for each compact subset $K$ of $\mathcal{S}$, the $\mathrm{n}_1-p_K^{+}+α\left(p_K^{+}-1\right)$-Hausdorff measure of $K$ is zero. Here, $p_K^{+}=\sup _K p$ and $\mathrm{n}_1$ is chosen so that $\mathtt{V}(B(x, r)) \leq \mathtt{K} r^{\mathrm{n}_1}$ for every ball. The estimates used to remove the singularities will focus on a family $\left\{u_{\ell}\right\}_{\ell \in \mathcal{J}} \subset W_{\mathrm{loc}}^{1, p(x)}(Ω; \mathtt{V})$ that converges to $u$ in a certain sense. As a second main result of this article, we will also obtain an estimate (when $\lim _{d\left(x, 0_Ω\right) \rightarrow \infty} p=1$ ) for $$ μ_{\ell}(B(x, r)):=\sup \left\{\int_{B(x, r)} \mathscr{A} \left(\cdot, \nabla u_{\ell}\right) \bullet \mathcal{D} ζ\mathrm{dV} \mid 0 \leq ζ\leq 1 \text { and } ζ\in C_0^{\infty}(B(x, r))\right\}, $$ which is related to the measure $μ=\operatorname{div}( \mathscr{A} (\cdot, \nabla u))$.

math.AP↗
arXiv2022

Removable singularities for sub-solutions of elliptic equations involving weighted variable exponent Sobolev spaces

We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces.

math.AP↗