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João Marcos do Ó

Publications and source records attributed to João Marcos do Ó.

At least 37 records · Page 2Linked to original sources
arXiv2022

Singular solutions to $k$-Hessian equations with fast-growing nonlinearities

We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular and analyze the intersection number depending on the parameter $k$ and the dimension $d$.

math.AP↗
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.

math.AP↗
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.

math.AP↗
arXiv2021

Qualitative properties for solutions to conformally invariant fourth order critical systems

We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and strongly positive. Furthermore, using a Pohozaev-type invariant, we prove the non-existence of semi-singular solutions, that is, all components equally blow-up in the neighborhood of origin. Namely, they are classified as multiples of the Emden--Fowler solution. Our results are natural generalizations of the famous classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] on the classical singular Yamabe equation.

math.AP↗
arXiv2021

Asymptotics for singular solutions to conformally invariant fourth order systems in the punctured ball

We study the asymptotic behavior for singular solutions to a critical fourth order system generalizing the constant $Q$-curvature equation. Our main result extends to the case of strongly coupled systems, the celebrated asymptotic classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] and [N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Invent. Math., (1999)]. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution. Besides, we obtain sharp a priori estimates for the decay rate of singular solutions near the origin. Consequently, we prove that sufficiently close to the isolated singularity solutions behave like the so-called Emden--Fowler solution. Our main theorem positively answers a question posed by [R. L. Frank and T. König, Anal. PDE (2019)] concerning the local behavior close to the isolated singularity for scalar solutions in the punctured ball.

math.AP↗
arXiv2021

Qualitative properties for solutions to subcritical fourth order systems

We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Gross--Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results of [J. Serrin, Acta Math. (1964)], [P.-L. Lions, J. Differential Equations (1980)], [P. Aviles, Comm. Math. Phys. (1987)], and [B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1981)]. On the technical level, we use the moving sphere method to classify the limit blow-up solutions to our system. Besides, applying asymptotic analysis techniques, we show that these solutions are indeed the local models near the isolated singularity. We also introduce a new fourth order nonautonomous Pohozaev functional, whose monotonicity properties yield improvement for the asymptotics results due to [R. Soranzo, Potential Anal. (1997)].

math.AP↗
arXiv2021

Compactness within the space of complete, constant Q-curvature metrics on the sphere with isolated singularities

In this paper we consider the moduli space of complete, conformally flat metrics on a sphere with k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize of each singular point. We prove that any set in the moduli space such that the distances between distinct punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a theorem of D. Pollack about singular Yamabe metrics. Along the way we define a radial Pohozaev invariant at each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest.

math.DG↗
arXiv2020

Singular solutions to Yamabe-type systems with prescribed asymptotics

Our primary purpose is to study a class of strongly coupled nonlinear elliptic systems with critical growth in a compact Riemannian manifold with constant scalar curvature. Using a gluing technique and perturbation arguments, we show the existence of singular solutions asymptotic to a Fowler-type solution near the isolated singularity.

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↗
arXiv2019

On Lane-Emden systems with singular nonlinearities and applications to MEMS

In this paper we analyse the Lane-Emden system \begin{equation} \left\{ \begin{alignedat}{3} -Δu = & \, \frac{λf(x)}{(1-v)^2} & \quad \text{in} & \quadΩ\\ -Δv = & \, \frac{μg(x)}{(1-u)^2} & \quad \text{in} & \quadΩ\\ 0\leq u &, v < 1 & \quad \text{in} & \quad Ω\\ u = v & = \, 0 & \text{on} & \quad \partialΩ\\ \end{alignedat} \right.\tag{$S_{λ, μ}$} \end{equation} where $λ$ and $μ$ are positive parameters and $Ω$ is a smooth bounded domain of $\mathbb{R}^N$ $( N \geq 1)$. Here we prove the existence of a critical curve $Γ$ which splits the positive quadrant of the $(λ,μ)\text{-plane}$ into two disjoint sets $\mathcal{O}_1$ and $\mathcal{O}_2$ such that the problem $(S_{λ, μ})$ has a smooth minimal stable solution $(u_λ,v_μ)$ in $\mathcal{O}_1$, while for $(λ,μ)\in\mathcal{O}_2$ there are no solutions of any kind. We also establish upper and lower estimates for the critical curve $Γ$ and regularity results on this curve if $N\leq 7$. Our proof is based on a delicate combination involving maximum principle and $L^p$ estimates for semi-stable solutions of $(S_{λ, μ}$).

math.AP↗
arXiv2019

Some elliptic problems with singular nonlinearity and advection for Riemannian manifolds

We are interested in regularity properties of semi-stable solutions for a class of singular semilinear elliptic problems with advection term defined on a smooth bounded domain of a complete Riemannian manifold with zero Dirichlet boundary condition. We prove uniform Lebesgue estimates and we determine the critical dimensions for these problems with nonlinearities of the type Gelfand, MEMS and power case. As an application, we show that extremal solutions are classical whenever the dimension of the manifold is below the critical dimension of the associated problem. Moreover, we analyze the branch of minimal solutions and we prove multiplicity results when the parameter is close to critical threshold and we obtain uniqueness on it. Furthermore, for the case of Riemannian models we study properties of radial symmetry and monotonicity for semi-stable solutions.

math.AP↗
arXiv2019

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions.

math.AP↗
arXiv2018

Ground states for a linearly coupled system of Schrödinger equations on $\mathbb{R}^{N}$

We study the following class of linearly coupled Schrödinger elliptic systems $$\left\{ \begin{array}{lr} -Δu+V_{1}(x)u=μ|u|^{p-2}u+λ(x)v, & \quad x\in\mathbb{R}^{N}, \\ -Δv+V_{2}(x)v=|v|^{q-2}v+λ(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $N\geq3$, $2 0$ such that the coupled system possesses positive ground state solution for all $μ\geqμ_{0}$. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case $p=q=2^{*}$, and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.

math.AP↗
arXiv2018

Existence of bound and ground states for fractional coupled systems in $\mathbb{R}^{N}$

In this work we consider the following class of nonlocal linearly coupled systems involving Schrödinger equations with fractional laplacian $$ \left\{ \begin{array}{lr} (-Δ)^{s_{1}} u+V_{1}(x)u=f_{1}(u)+λ(x)v, & x\in\mathbb{R}^{N}, (-Δ)^{s_{2}} v+V_{2}(x)v=f_{2}(v)+λ(x)u, & x\in\mathbb{R}^{N}, \end{array} \right. $$ where $(-Δ)^{s}$ denotes de fractional Laplacian, $s_{1},s_{2}\in(0,1)$ and $N\geq2$. The coupling function $λ:\mathbb{R}^{N} \rightarrow \mathbb{R}$ is related with the potentials by $|λ(x)|\leq δ\sqrt{V_{1}(x)V_{2}(x)}$, for some $δ\in(0,1)$. We deal with periodic and asymptotically periodic bounded potentials. On the nonlinear terms $f_{1}$ and $f_{2}$, we assume "superlinear" at infinity and at the origin. We use a variational approach to obtain the existence of bound and ground states without assuming the well known Ambrosetti-Rabinowitz condition at infinity. Moreover, we give a description of the ground states when the coupling function goes to zero.

math.AP↗
arXiv2018

Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas, and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [11] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.

math.AP↗
arXiv2018

Positive ground states for a class of superlinear $(p,q)$-Laplacian coupled systems involving Schrödinger equations

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -Δ_{p} u+a(x)|u|^{p-2}u=f(u)+ αλ(x)|u|^{α-2}u|v|^β, & x\in\mathbb{R}^{N}, -Δ_{q} v+b(x)|v|^{q-2}v=g(v)+ βλ(x)|v|^{β-2}v|u|^α, & x\in\mathbb{R}^{N}, \end{array} \right. \] where $N\geq3$ and $1\leq p\leq q<N$. Here the coefficient $λ(x)$ of the coupling term is related with the potentials by the condition $|λ(x)|\leqδa(x)^{α/p}b(x)^{β/q}$ where $δ\in(0,1)$ and $α/p+β/q=1$. We deal with periodic and asymptotically periodic potentials. The nonlinear terms $f(s), \; g(s)$ are "superlinear" at $0$ and at $\infty$ and are assumed without the well known Ambrosetti-Rabinowitz condition at infinity. Thus, we have established the existence of positive ground states solutions for a large class of nonlinear terms and potentials. Our approach is variational and based on minimization technique over the Nehari manifold.

math.AP↗
arXiv2017

Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr} (-Δ)^{1/2}u+V_{1}(x)u=f_{1}(u)+λ(x)v, & x\in\mathbb{R}, (-Δ)^{1/2}v+V_{2}(x)v=f_{2}(v)+λ(x)u, & x\in\mathbb{R}, \end{array} \right. $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $δ\in (0,1)$ such that $λ(x)\leqδ\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.

math.AP↗