arXiv subjects

Mayra Soares

Publications and source records attributed to Mayra Soares.

15 recordsLinked to original sources

arXiv2026

Geometric approaches for improved regularity in fully nonlinear parabolic models

In this paper, we derive improved estimates for a class of fully nonlinear parabolic equations with continuous drift and admissible source terms of the form $$ \partial_{t}u - F(D^2u,x,t) + \langle B(x,t), Du\rangle = f(x, t, u^{+}, u^{-}) \quad \text{in}\quad Q_1. $$ Our analysis reveals two distinct regimes. In the first, $f=f(x,t)$ exhibits $\theta$-H\"{o}lder decay ($\theta\in(0,1)$), yielding improved gradient regularity at vanishing points via perturbative methods and geometric iteration, as well as nondegeneracy with explicit growth rates under a suitable structural condition. In the second, $f(\cdot,u^+,u^-)=(u^+)^\gamma-(u^-)^\gamma$ with $\gamma\in(0,1)$ (corresponding to an evolutionary semilinear two-phase model), we obtain enhanced regularity at branching points by combining a robust blow-up analysis with local derivative estimates for linear equations. Our results remain relevant even in linear settings with merely continuous data, linking to classical free boundary problems arising in mathematical physics and related areas.

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arXiv2026

Fully Nonlinear Elliptic Grad--Mercier Equations in Weighted Orlicz Spaces

In this article, we study the existence and global regularity results for the fully nonlinear elliptic Grad--Mercier type equations with oblique boundary conditions in the context of weighted Orlicz spaces. Our approach employs an asymptotic analysis in which global regularity is transferred from a limit profile, namely, the recession operator associated with the governing operator, using topological and stability methods. In addition to the main regularity result, we derive global weighted Orlicz estimates for the Hessian and establish global Morrey-type estimates for the problem. This article extends the results established by Caffarelli--Tomasetti (Comm. Pure Appl. Math. 76 (3): 604--615, 2023), Zhang et al. (Nonlinearity 39 (2): 025011, 2026), and Bessa (J. Funct. Anal. 286 (4): 110295, 2024).

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arXiv2025

On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity

We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $ Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} $ for a bounded smooth domain $ \Omega \subset \mathbb{R}^N $, $ N \ge 3 $, and $ 1 < p < 2^{*} $. This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight $ Q_{\Omega} $. We prove that, for $ p > 2 $ sufficiently close to $ 2 $, the problem admits a unique positive solution, which is nondegenerate. Our approach combines a detailed analysis of an associated eigenvalue problem involving $ Q_{\Omega} $ with variational methods and blow-up techniques in the asymptotically linear regime. We also provide a comprehensive study of the spectral properties of the corresponding linear problem, including the existence and qualitative behavior of eigenfunctions, sharp decay estimates, and symmetry results. In particular, we establish analogues of the Faber--Krahn and Hong--Krahn--Szeg{\"o} inequalities in this non-standard setting.

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arXiv2025

A New Approach to Inspect Weakly Coupled Logistic Systems and their Asymptotic Behavior

We consider the weakly coupled elliptic system of logistic type, \begin{equation}\label{LS} \begin{cases} -\Delta u &=\lambda_1 u- |u|^{p-2}u+ \beta |u|^{\frac{p}{2}-2}u |v|{^{\frac{p}{2}-1}}v\mbox{ in }\Omega, -\Delta v & =\lambda_2 v- |v|^{p-2}v+\beta |u|^{\frac{p}{2}-1}u|v|^{\frac{p}{2}-2}v \mbox{ in }\Omega, \ \ u,v &\in H_0^1(\Omega), \end{cases} \tag{$LS$} \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with $N\geq 2$, $2< p < 2^*$, and $\lambda_1(\Omega)< \lambda_1 \leq \lambda_2$. We say the system is competitive if $\beta<0$ and cooperative if $\beta>0$, for $\beta \in \mathbb{R}$. We prove the existence and multiplicity of solutions to the problem \eqref{LS} in alternative variational frameworks, depending on the range of the parameter $\beta.$ We do not rely on bifurcation or degree theory, which have been used in the literature for logistic-type problems. Instead, the novelty is to obtain min-max type solutions by exploiting the different geometry of the functional associated with the logistic problem. In case $N\geq 2$ and suitable values of $p$, we extend the existence results, for all $\beta$ in the whole line, and possibly for the classical case $N=3$ and $p=4$. Furthermore, we analyze the asymptotic behavior of such solutions as $\beta \to 0$ or $\beta \to \pm \infty.$} \bigskip \newline \textsc{Key words: Logistic System, Ground State Solution, Linking structure, seminodal Solution.}{\small}

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arXiv2025

On the existence of a positive solution to a nonlocal logistic system with nonlinear advection terms

In this paper, we study a nonlocal logistic system with nonlinear advection terms \begin{equation*} \left\{ \begin{array}{lcl} -\Delta u+\vec{\alpha}(x)\cdot \nabla (|u|^{p-1}u)&=&\left(a-\int_{\Omega}K_1(x,y)f(u,v)dy \right)u+bv\mbox{ in }\Omega,\\ -\Delta v+\vec{\beta}(x)\cdot \nabla (|v|^{q-1}v)&=&\left(d-\int_{\Omega}K_2(x,y)g(u,v)dy \right)v+cu\mbox{ in }\Omega,\\ \qquad \qquad \qquad \qquad u=v&=&0\mbox{ on }\partial\Omega, \end{array} \right. \end{equation*} where $\Omega\subset\mathbb{R}^N$, $N\geq1$, is a bounded domain with a smooth boundary, $\vec{\alpha}(x)=(\alpha_1(x),\cdots,\alpha_N(x))$ and $\vec{\beta}(x)=(\beta_1(x),\cdots,\beta_N(x))$ are flows satisfying suitable conditions, $p,q\geq1$, $a,b,c,d>0$ and $K_1,K_2:\Omega\times\Omega\rightarrow\mathbb{R}$ are nonnegative functions, with their specific conditions detailed below. The functions $f$ and $g$ satisfy some assumptions which allow us to use bifurcation theory to prove the existence of solution to problem $(P)$. It is important to highlight that the inclusion of the integral nonlocal term on the right-hand side makes the problem more representative of real-world situations.

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arXiv2025

Higher regularity estimates for solutions to $\infty$-Laplacian-type models

In this work, we tackle the higher regularity estimates of solutions to inhomogeneous $\infty-$Laplacian equations at interior critical points. Our estimates provide smoothness properties better than the corresponding available regularity for the model with bounded forcing terms. We explore several scenarios, thereby obtaining improved regularity estimates, which depend on the universal parameters of the model. Our findings connect with nowadays well-known estimates developed for obstacle and dead-core type problems.

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arXiv2024

Optimal pinwheel partitions and pinwheel solutions to a nonlinear Schrödinger system

We establish the existence of a solution to a nonlinear competitive Schrödinger system whose scalar potential tends to a positive constant at infinity with an appropriate rate. This solution has the property that all components are invariant under the action of a group of linear isometries and each component is obtained from the previous one by composing it with some fixed linear isometry. We call it a pinwheel solution. We describe the asymptotic behavior of the least energy pinwheel solutions when the competing parameter tends to zero and to minus infinity. In the latter case the components are segregated and give rise to an optimal pinwheel partition for the Schrödinger equation, that is, a partition formed by invariant sets that are mutually isometric through a fixed isometry.

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arXiv2022

Boundary weak Harnack estimates and regularity for elliptic PDE in divergence form

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies the Hopf-Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among the consequences is a boundary gradient estimate, due to Krylov and well-studied for non-divergence form equations, but completely novel in the divergence framework. Another consequence is a new more general version of the Hopf-Oleinik lemma.

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arXiv2022

Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations \begin{align*}\tag{$P_\lambda$} -\mathrm{div}(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x),\qquad u\in H_0^1(\Omega)\cap L^\infty(\Omega), \end{align*} where $\Omega\subset\mathbb{R}^n$, $n\geq 3$, is a bounded domain with a low-regularity boundary $\partial\Omega$. The coefficients $c, h \in L^p(\Omega)$ for some $p > n$, with $c^\pm \geq 0$ and $c_\lambda(x) := \lambda c^+(x) - c^-(x)$ for a real parameter $\lambda$. The matrix $A(x)$ is uniformly positive definite and bounded, while $M(x)$ is positive definite and bounded. Under suitable assumptions, we characterize the solution continuum of $(P_\lambda)$, including its bifurcation points. We establish existence and uniqueness results in the coercive case ($\lambda \leq 0$) and prove multiplicity results in the non-coercive case ($\lambda > 0$). \bigskip \textbf{Keywords}: Quasilinear elliptic equations, quadratic growth on the gradient, sub and super solutions.

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arXiv2022

Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations \begin{equation*} \begin{cases} -\varepsilon^2Δu_i+u_i=μ_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ellλ_{ij}β_{ij}|u_j|^{α_{ij}}|u_i|^{β_{ij} -2}u_i,\\ u_i \in H^1_0(Ω), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} in a bounded domain $Ω$ in $\mathbb{R}^N$, with $N\geq 4$, $\varepsilon>0$, $μ_i>0$, $λ_{ij}=λ_{ji}<0$, $α_{ij}, β_{ij}>1$, $α_{ij}=β_{ji}$, $α_{ij} + β_{ij} = p\in (2,2^*)$, and $2^{*}:=\frac{2N}{N-2}$. If $Ω$ is the unit ball we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as $\varepsilon\to 0$: solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system \begin{equation*} \begin{cases} -Δu_i+u_i=μ_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ellλ_{ij}β_{ij}|u_j|^{α_{ij}}|u_i|^{β_{ij} -2}u_i,\\ u_i \in H^1(\mathbb{R}^N), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} and solutions whose limit profile is a solution of the uncoupled system, i.e., after rescaling and translation, the limit profile of the $i$-th component is a positive or a nonradial sign-changing solution to the equation $$-Δu+u=μ_i|u|^{p-2}u,\qquad u \in H^1(\mathbb{R}^N), \qquad u\neq 0.$$

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arXiv2022

Energy estimates for seminodal solutions to an elliptic system with mixed couplings

We study the system of semilinear elliptic equations $$-Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix $(β_{ij})$ is symmetric and admits a block decomposition such that the entries within each block are positive or zero and all other entries are negative. We provide simple conditions on $(β_{ij})$, which guarantee the existence of fully nontrivial solutions, i.e., solutions all of whose components are nontrivial. We establish existence of fully nontrivial solutions to the system having a prescribed combination of positive and nonradial sign-changing components, and we give an upper bound for their energy when the system has at most two blocks. We derive the existence of solutions with positive and nonradial sign-changing components to the system of singularly perturbed elliptic equations $$-\varepsilon^2Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots,\ell,$$ in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\varepsilon\to 0$, and solutions whose components remain coupled all the way up to their limit.

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arXiv2019

An Indefinite Elliptic Problem on RN Autonomous at Infinity: The Crossing Effect of the Spectrum and the Nonlinearity

We present a new approach to solve a Schrödinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. In order to apply variational methods, we set up a suitable linking structure depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Our method allows us to circumvent the lack of compactness. The main novelty is that none monotonicity assumption is required on the nonlinearity, which may be sign-changing as well as the potential. Furthermore, depending on the nonlinearity, the limit of the potential at infinity may be non-positive, so that zero may be an interior point in the essential spectrum of the Schrödinger operator.

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arXiv2019

An abstract linking theorem applied to indefinite problems via spectral properties

An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite dimensional Hilbert Spaces. The main applications are given to Hamiltonian systems and Schrodinger equations. Here spectral properties of the operators are exploited and hypotheses of monotonicity on the nonlinearities are discarded.

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arXiv2018

Non-Cooperative Systems Modeling Repulsive Interaction of Bose-Einstein Condesates in RN

Inspired by so many possible applications of this class of problems, we seek solution for non-cooperative elliptic systems of two Schrodinger equations. General conditions are assumed under the potentials, which produces convenient spectral properties on the elliptic operator concerned, hence the non-cooperation characterizes it as a strongly indefinite problem. An Abstract Linking Theorem developed previously by the first two authors is the main tool, since variational approach is applied. Furthermore, super and asymptotically quadratic nonlinearities are considered.

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arXiv2018

Spectral Theory Approach for a Class of Radial Indefinite Variational Problems

Considering the radial nonlinear Schrodinger equation - Δu + V(x)u = g(x,u) in R^N, N \geq 3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring this problem is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating to the problem an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator A:= - Δ+ V(x) restricted to H^1_{rad}(R^N), the space of radially symmetric functions in H^1(R^N), for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.

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