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Manassés de Souza

Publications and source records attributed to Manassés de Souza.

5 recordsLinked to original sources
arXiv2026

Maximizers for the Singular Trudinger-Moser functional beyond the critical regime

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$ F_α(u) := \int_Ω \frac{e^{αu^2}-1}{|x|^{a}} \mathrm{d}x,\;\;\; u \in W_0^{1,2}(Ω), $$ restricted to the manifold $ Σ=:\left\{u\in W_0^{1,2}(Ω):\|\nabla u\|_2=1\right\},$ where $a\in [0, 2)$, $α\ge 0$ and $Ω$ denotes a smooth bounded domain $\mathbb{R}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. \textbf{13}, 2007) showed the following singular Trudinger-Moser type estimate \begin{equation}\nonumber \sup_{u \in W_0^{1,2}(Ω), \, \|\nabla u\|_{L^2} \le 1} F_α(u)<\infty\;\;\;\mbox{iff}\;\;α\leq α_a:=2π(2-a). \end{equation} In particular, the functional $F_α$ is bounded on $ Σ$ whenever $α\le α_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., \textbf{54}, 2015) were able to ensure the existence of maximizers for $F_α$ on $Σ$ when $α\le α_a$. In the supercritical regime $α> α_a$, the functional $F_α$ becomes unbounded on $Σ$. Nevertheless, we prove that $F_α$ still possesses local maximizers on $Σ$ beyond the critical threshold, at least for $α> α_a$ sufficiently close to $α_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter $α_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, \textbf{5}, 1988).

math.AP↗
arXiv2024

Renormalized Solutions for Quasilinear Elliptic Equations with Robin Boundary Conditions, Lower-Order Terms, and $L^1$ Data

In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_α|\nabla u|^{p-2} \nabla u)+\vartheta_γb|\nabla u|^{p-1}+\vartheta_γc|u|^{r-1} u & =f \vartheta_α& & \text { in } Ω, \\ \vartheta_α|\nabla u|^{p-2} \nabla u \cdot ν+\vartheta_β|u|^{p-2} u & =g \vartheta_β& & \text { on } \partial Ω. \end{aligned}\right. \end{equation} Here, $Ω$ is an open subset of $\mathbb{R}^N$ with a Lipschitz boundary, where $N\geq 2$ and $1 < p < N$. We define $\vartheta_a(x) = (1 + |x|)^a$ for $a \in (-N, (p-1)N)$, and the constants $α, β, γ, r$ satisfy suitable conditions. Additionally, $f$ and $g$ are measurable functions, while $b$ and $c$ belong to a Lorentz space. Our approach also allows us to establish stability results for renormalized solutions.

math.AP↗
arXiv2023

Removable singularities in the boundary for quasilinear elliptic equations

In this work, we are interested in to study removability of a singular set in the boundary for some classes of quasilinear elliptic equations. We will approach this question in two different ways: through an asymptotic behavior at the infinity of the solutions, or through conditions in the Sobolev norm of solutions along the direction of the singular set.

math.AP↗
arXiv2019

Multiplicity results for fractional magnetic problems involving exponential growth

We study the following fractional elliptic equations of the type, \begin{equation*} (-Δ)^{\frac12}_A u = λu+f(|u|)u ,\;\textrm{in } \;(-1, 1),\; u=0\;\textrm{in } \;\mathbb R\setminus (-1, 1), \end{equation*} where $λ$ is a positive real parameter and $(-Δ)^{\frac12}_A$ is the fractional magnetic operator with $A:\mathbb R\to \mathbb R$ being a smooth magnetic field. Using a classical critical point theorems, we prove the existence of multiple solutions in the non-resonant case when the nonlinear term $f(t)$ has a critical exponential growth in the sense of Trudinger-Moser inequality.

math.AP↗