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José Francisco de Oliveira

Publications and source records attributed to José Francisco de Oliveira.

16 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).

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arXiv2026

Quantitative stability for the Trudinger-Moser inequality

We establich quantitative stability estimates for the Trudinger-Moser inequality on smooth, bounded domains in the Euclidean plane. More specifically, we prove that the deficit in the Trudinger-Moser inequality quadratically controls the distance to the set of optimizers if either (i) the exponential rate of growth is sufficiently small or (ii) the domain is a round disk. The latter estimate remains valid even in the critical case. Both proofs rely on a new spectral gap that we prove, which may be of independent interest. Additionally we show that the same stability estimate holds in the nondegenerate case, and that this occurs generically.

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arXiv2026

A Trudinger-Moser inequality under a refined constraint in fractional dimensions and extremal functions

We establish a Trudinger-Moser type inequality with a Tintarev-type constraint in fractional-dimensional spaces and prove the existence of maximizers in the critical regime. Our results provide a refinement of those in (Calc. Var. 52 (2015), 125-163) in the setting of fractional-dimensional spaces, as well as of those in (Ann. Global Anal. Geom. 54 (2018), 237-256) for classical Sobolev spaces.

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arXiv2025

On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation

We propose a natural gradient term for a class of second-order partial differential equations of the form \begin{equation}\nonumber M(x,Du,D^2u)+g(u)N(x,Du, D^2u)+f(x,u)=0 \;\;\mbox{in}\;\; Ω, \end{equation} where $Ω\subset\mathbb{R}^n$ is an open set, $f\in C(Ω\times \mathbb{R}, \mathbb{R})$, $M$ defines the partial differential operator, $N$ is a quadratic term driven by the gradient $Du$ and $M$ itself, and $g\in C(\mathbb{R},\mathbb{R})$. We establish conditions on the class of operators $M$ for the existence of a change of variables $v = Φ(u)$ that transforms the previous equation into another one of the form \begin{equation}\nonumber M(x,Dv, D^2v) + h(x,v)=0 \quad \text{in} \;\; Ω\end{equation} which does not depend on the quadratic term $N$. The results presented here unify previous findings for the Laplacian, $m$-Laplacian, and $k$-Hessian operators, which were derived separately by different authors and are restricted to $C^2$ solutions with fixed sign. Our work provides a more general framework, extending these findings to a broader class of nonlinear partial differential equations, including the infinity-Laplacian o\-pe\-ra\-tor. In addition, we also include both $C^2$ and viscosity solutions that may change sign. As an application, we also obtain an Aronsson-type result and investigate viscosity solutions for the Dirichlet problem associated with the infinity Laplace equation with its natural gradient term.

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arXiv2025

Adams-Trudinger-Moser inequalities of Adimurthi-Druet type regulated by the vanishing phenomenon and its extremals

Let $W^{m,\frac{n}{m}}(\mathbb{R}^n)$ with $1\le m < n$ be the standard higher order derivative Sobolev space in the critical exponential growth threshold. We investigate a new Adams-Adimurthi-Druet type inequality on the whole space $\mathbb{R}^n$ which is strongly influenced by the vanishing phenomenon. Specifically, we prove \begin{equation}\nonumber \sup_{\underset{\|\nabla^{m} u\|_{\frac{n}{m}}^{^{\frac{n}{m}}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}} \leq 1}{u\in W^{m,\frac{n}{m}}(\mathbb{R}^n)}} \int_{\mathbb{R}^n}Φ\left(β\left(\frac{1+α\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}{1-γα\|u\|_{\frac{n}{m}}^{\frac{n}{m}}}\right)^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\right) \mathrm{d}x<+\infty. \end{equation} where $0\le α<1$, $0<γ<\frac{1}α-1$ for $α>0$, $\nabla^{m} u$ is the $m$-th order gradient for $u$, $0\leβ\le β_0$, with $β_0$ being the Adams critical constant, and $Φ(t) = \operatorname{e}^{t}-\sum_{j=0}^{j_{m,n}-2}\frac{t^{j}}{j!}$ with $j_{m,n}=\min\{j\in\mathbb{N}\;:\: j\ge n/m\}$. In addition, we prove that the constant $β_0$ is sharp. In the subcritical case $β<β_0$, the existence and non-existence of extremal function are investigated for $n=2m$ and attainability is proven for $n=4$ and $m=2$ in the critical case $β=β_0$. Our method to analyze the extremal problem is based on blow-up analysis, a truncation argument recently introduced by DelaTorre-Mancini \cite{DelaTorre} and some ideas by Chen-Lu-Zhu \cite{luluzhu20}, who studied the critical Adams inequality in $\mathbb{R}^4$.

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arXiv2025

Trudinger-Moser type inequalities for the Hessian equation with logarithmic weights

We establish sharp Trudinger-Moser inequalities with logarithmic weights for the $k$-Hessian equation and investigate the existence of maximizers. Our analysis extends the classical results of Tian and Wang to $k$-admissible function spaces with logarithmic weights, providing a natural complement to the work of Calanchi and Ruf. Our approach relies on transforming the problem into a one-dimensional weighted Sobolev space, where we solve it using various techniques, including some radial lemmas and certain Hardy-type inequalities, which we establish in this paper, as well as a theorem due to Leckband.

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arXiv2024

On a quadratic gradient natural term for the Pucci extremal operators

We introduce a quadratic gradient type term for the Pucci extremal operators. Our analysis demonstrates that this proposed term extends the classical quadratic gradient term associated with the Laplace equation, and we investigate the impact of the Kazdan-Kramer transformation. As an application, we explore the existence, non-existence, uniqueness, Liouville-type results, and asymptotic behavior of solutions for the new class of Pucci equations under various conditions on both nonlinearity and domains.

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arXiv2024

Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schrödinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -Δu+λu=μ|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $λ\in\mathbb{R}$, $μ>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ and $2^*_{d}= 2(N-d)/(N-2)$ is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the equation with critical exponential growth \begin{equation}\nonumber \begin{aligned} &-Δu+λu=|x|^{-b}f(u) \;\;\mbox{in}\;\; \mathbb{R}^2 \end{aligned} \end{equation} where the nonlinearity $f(s)$ behaves like $\exp(s^2)$ as $s\to\infty$. We extend the existence results due to Alves-Ji-Miyagaki (Calc. Var. 61, 2022) from $b =d= 0$ to the case $0 < b,d < 2$.

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arXiv2024

On a supercritical Hardy-Sobolev type inequality with logarithmic term and related extremal problem

Our main goal is to investigate supercritical Hardy-Sobolev type inequalities with a logarithmic term and their corresponding variational problem. We prove the existence of extremal functions for the associated variational problem, despite the loss of compactness. As an application, we show the existence of weak solution to a general class of related elliptic partial differential equations with a logarithmic term.

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arXiv2023

On a Sobolev-type inequality and its minimizers

Critical Sobolev-type inequality for a class of weighted Sobolev spaces on the entire space is established. We also investigate the existence of extremal function for the associated variational problem. As an application, we prove the existence of a weak solution for a general class of critical semilinear elliptic equations related to the polyharmonic operator.

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arXiv2023

A gradient type term for the $k$-Hessian equation

In this paper, we propose a gradient type term for the $k$-Hessian equation that extends for $k>1$ the classical quadratic gradient term associated with the Laplace equation. We prove that such as gradient term is invariant by the Kazdan-Kramer change of variables. As applications, we ensure the existence of solutions for a new class of $k$-Hessian equation in the sublinear and superlinear cases for Sobolev type growth. The threshold for existence is obtained in some particular cases. In addition, for the Trudinger-Moser type growth regime, we also prove the existence of solutions under either subcritical or critical conditions.

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arXiv2022

On a sharp inequality of Adimurthi-Druet type and extremal functions

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi-Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger-Moser type inequality in (Calc.Var.Partial Differential Equations, \textbf{52} (2015) 125-163) for the entire space. In addition, we perform the two-step strategy of Carleson-Chang together blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under subcritical regime for some special cases.

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arXiv2022

Equivalence of critical and subcritical sharp Trudinger-Moser inequalities in fractional dimensions and extremal functions

We establish critical and subcritical sharp Trudinger-Moser inequalities for fractional dimensions on the whole space. Moreover, we obtain asymptotic lower and upper bounds for the fractional subcritical Trudinger-Moser supremum from which we can prove the equivalence between critical and subcritical inequalities. Using this equivalence, we prove the existence of maximizers for both the subcritical and critical associated extremal problems. As a by-product of this development, we can explicitly calculate the value of the critical supremum in some special situations.

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arXiv2020

Extremal functions for a supercritical k-Hessian inequality of Sobolev-type

Our main purpose in this paper is to investigate a supercritical Sobolev-type inequality for the $k$-Hessian operator acting on $Φ^{k}_{0,\mathrm{rad}}(B)$, the space of radially symmetric $k$-admissible functions on the unit ball $B\subset\mathbb{R}^{N}$. We also prove both the existence of admissible extremal functions for the associated variational problem and the solvability of a related $k$-Hessian equation with supercritical growth.

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