Searcharxiv⌕ Search

arXiv subjects

João Marcos do Ó

Publications and source records attributed to João Marcos do Ó.

At least 19 recordsLinked to original sources
arXiv2026

Minimax solutions for FitzHugh-Nagumo systems with critical exponential growth

We study a class of FitzHugh-Nagumo type systems with critical exponential growth in a smooth bounded domain of $\mathbb{R}^{2}$. The classical reduction method turns the strongly indefinite system into a scalar nonlocal problem whose quadratic part induces a norm equivalent to, but different from the standard Sobolev norm, so that the Trudinger-Moser threshold must be handled in this framework. We prove the existence of a ground state solution via a mountain pass argument combined with a Nehari level comparison. We also prove the existence, without imposing the monotonicity condition required in the ground state argument, of a nontrivial solution obtained through a saddle-point linking argument based on the spectral decomposition of $\operatorname{T}=-Δ+B$.

math.AP↗
arXiv2026

The $G$-equivariant Kazdan--Warner problem

We establish an equivariant analogue of the Kazdan--Warner trichotomy for admissible scalar curvature functions. Let $M$ be a closed connected manifold of dimension $n \ge 3$ equipped with an effective isometric action of a compact connected Lie group $G$ of cohomogeneity at least one and with no zero-dimensional orbits. All metrics and prescribed functions are taken to be $G$-invariant. We prove that, for such pairs $(M, G)$, the classical trichotomy does not extend verbatim. A distinct class emerges, consisting of \emph{totally $G$-positive} pairs, for which every $G$-invariant metric exhibits positive total scalar curvature. Each such pair admits a $G$-invariant metric of positive constant scalar curvature, but admits no metric of zero or negative constant scalar curvature. Every remaining pair falls into exactly one of three classes mirroring the original trichotomy.

math.DG↗
arXiv2026

Radial Sobolev embeddings on spherically symmetric Riemannian manifolds

We study Sobolev spaces of radial functions on spherically symmetric Riemannian manifolds. Using geodesic polar coordinates, we give a sharp one-dimensional reduction: a radial function belongs to the Sobolev space on the manifold if and only if its radial representation lies in an associated weighted Sobolev space on an interval, with weights determined explicitly by the metric. This characterization allows us to prove optimal Sobolev-type embeddings for radial functions into weighted Lebesgue spaces on both bounded and unbounded spherically symmetric manifolds. As further consequences, we establish new radial lemmas and decay estimates that capture the precise behaviour of radial Sobolev functions near the origin and at infinity. Our results unify and extend the classical radial embeddings in Euclidean and hyperbolic spaces.

math.AP↗
arXiv2025

Steady states of FitzHugh-Nagumo-type systems with sign-changing coefficients

We establish existence and multiplicity results for steady-state solutions of spatially heterogeneous FitzHugh-Nagumo-type systems, extending the existing theory from constant to variable coefficients that may change sign. Specifically, we study the system Specifically, we study the system \begin{align*} -Δu + a(x)v &= f(x,u) && \text{in } \mathbb{R}^N, \\ -Δv + b(x)v &= c(x)u && \text{in } \mathbb{R}^N. \end{align*} where $N \geqslant 3$, the coefficients $a,b,c : \mathbb{R}^N \to \mathbb{R}$ are $L^\infty_{\mathrm{loc}}$-functions bounded from below, and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function with subcritical growth. For assumptions permitting sign changes and non-coercivity of the coefficients, we prove the existence of a mountain pass solution. In the case where $a,b,c$ do not change sign, still allowing non-coercive behavior, we additionally establish the existence of componentwise positive and negative solutions.

math.AP↗
arXiv2025

Overdetermined elliptic problems on model Riemannian manifolds

We establish a rigidity theorem for annular sector-like domains in the setting of overdetermined elliptic problems on model Riemannian manifolds. Specifically, if such a domain admits a solution to the inhomogeneous Helmholtz equation satisfying both constant Dirichlet and constant Neumann boundary conditions, then the domain must be a spherical sector, and the solution must be radially symmetric. This result underscores the strong geometric constraints imposed by overdetermined boundary conditions, extending classical rigidity phenomena to this more general framework.

math.AP↗
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.

math.AP↗
arXiv2025

Finite energy solutions of quasilinear elliptic equations with Orlicz growth

We present a sufficient condition, expressed in terms of Wolff potentials, for the existence of a finite energy solution to the measure data $(p,q)$-Laplacian equation with a "sublinear growth" rate. Furthermore, we prove that such a solution is minimal. Additionally, a lower bound of a suitably generalized Wolff-type potential is necessary for the existence of a solution, even if the energy is not finite. Our main tools include integral inequalities closely associated with $(p,q)$-Laplacian equations with measure data and pointwise potential estimates, which are crucial for establishing the existence of solutions to this type of problem. This method also enables us to address other nonlinear elliptic problems involving a general class of quasilinear operators.

math.AP↗
arXiv2025

Sharp Sobolev and Adams-Trudinger-Moser inequalities for symmetric functions without boundary conditions on hyperbolic spaces

Embedding theorems for symmetric functions without zero boundary condition have been studied on flat Riemannian manifolds, such as the Euclidean space. However, these theorems have only been established on hyperbolic spaces for functions with zero boundary condition. In this work, we focus on sharp Sobolev and Adams-Trudinger-Moser embeddings for radial functions in hyperbolic spaces, considering both bounded and unbounded domains. One of the main features of our approach is that we do not assume boundary zero condition for symmetric functions on geodesic balls or the entire hyperbolic space. Our main results include Theorems 1.2, 1.3, and 1.4, which establish weighted Sobolev embedding theorems, and Theorems 1.5 together with 1.6, which present Adams-Trudinger-Moser type of embedding theorems. In particular, a key result is Theorem 1.1 which is a highly nontrivial comparison result between norms of the higher order covariant derivatives and higher order derivatives of the radial functions. Higher order asymptotic behavior of radial functions on hyperbolic spaces are established to prove our main theorems. This approach includes novel radial lemmata and decay properties of higher order radial Sobolev functions defined in hyperbolic space.

math.AP↗
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.

math.AP↗
arXiv2024

Hearing Exotic Smooth Structures

This paper explores the existence and properties of \emph{basic} eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie group. Our primary focus is on investigating the potential existence of homeomorphic yet not diffeomorphic smooth manifolds that can accommodate invariant metrics sharing common basic spectra. We establish the occurrence of such scenarios for specific homotopy spheres and connected sums. Moreover, the developed theory demonstrates that the ring of invariant admissible scalar curvature functions fails to recover the smooth structure in many examples. We show the existence of homotopy spheres with identical rings of invariant scalar curvature functions, irrespective of the underlying smooth structure.

math.DG↗
arXiv2024

Concentration-compactness via profile decomposition for systems of coupled Schrödinger equations of Hamiltonian type

We analyse Hamiltonian-type systems of second-order elliptic PDE invariant under a non-compact group and, consequently, involve a lack of compactness of the Sobolev embedding. We show that the loss of compactness can be compensated by using a concentration-compactness principle via weak profile decomposition for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the existence of ground states involves a reduction by the inversion method of the system to a fourth-order equation combined with a variational principle of a minimax nature. Among other results, including regularity and a Pohozaev-type identity, we also prove the non-existence of weak solutions for a class of Lane-Emden systems.

math.AP↗
arXiv2024

Quasilinear Lane-Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane-Emden-type systems involving measures in the "sublinear growth" rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff's potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

math.AP↗
arXiv2023

Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces

This paper introduces a novel higher order Adams inequality that incorporates an exact growth condition for a class of weighted Sobolev spaces. Our rigorous proof confirms the validity of this inequality and provides insights into the optimal nature of the critical constant and the exponent within the denominator. Furthermore, we apply this inequality to study a class of ordinary differential equations (ODEs), where we successfully derive both a concept of the weak solution and a comprehensive regularity theory.

math.AP↗
arXiv2023

Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

math.AP↗
arXiv2022

Touchdown solutions in general MEMS models

We study general equations modeling electrostatic MEMS devices \begin{equation} \begin{cases} \label{P} φ\big(r,- u'(r)\big)=λ\int_0^r\frac{f(s)}{g(u(s))}\,\mathrm{d}s, & r\in(0,1), \\ 0 < u(r) < 1, & r\in(0,1), \\ u(1) = 0, \tag{$P_λ$} \end{cases} \end{equation} where $φ$, $g$, $f$ are some functions on $[0,1]$ and $λ>0$ is a parameter. We obtain results on the existence and regularity of a touchdown solution to \eqref{P} and find upper and lower bounds on the respective pull-in voltage. In the particular case, when $φ(r,v) = r^α|v|^βv$, i.e., when the associated differential equation involves the operator $r^{-γ}(r^α|u'|^βu')'$, we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

math.AP↗
arXiv2022

Constant Q-curvature metrics with Delaunay ends: the nondegenerate case

We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.

math.DG↗
arXiv2022

Existence, multiplicity and classification results for solutions to $k$-Hessian equations with general weights

The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= λρ(|x|) (1-u)^q &\mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B. \end{cases} \end{equation} Here, $B$ denotes the unit ball in $\mathbb R^n\!$, $n>2k$, $λ$ is a positive parameter and $q>k$ with $k\in \mathbb N$. The function $rρ'(r)/ρ(r)$ satisfies very general conditions in the radial direction $r=|x|$. We show the existence, nonexistence, and multiplicity of solutions to Problem \eqref{Eq:Ma:0}. The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in \eqref{Eq:Ma:0}. Further, using the aforementioned non-autonomous system, we give a comprehensive characterization of $P_2$-, $P_3^+$-, $P_4^+$-solutions to the related problem \begin{equation*} \begin{cases} S_k(D^2 w)= ρ(|x|) (-w)^q, \\ w<0, \end{cases} \end{equation*} given on the entire space $\mathbb R^n\!$. In particular, we describe new classes of solutions: fast decay $P^+_3$-solutions and $P_4^+$-solutions.

math.AP↗