arXiv subjects

Eduardo de Souza Böer

Publications and source records attributed to Eduardo de Souza Böer.

8 recordsLinked to original sources

arXiv2026

Symmetry and classification of positive standing waves of nonlinear Hartree type equations

This paper presents some qualitative properties of positive solutions to the strongly coupled system \[ \begin{cases} \displaystyle - Δu + τu = \frac{2 p}{p + q} \left( I_α\ast |v|^q \right) |u|^{p - 2} u &\text{in} ~ \mathbb{R}^N, \\ \\ \displaystyle - Δv + ηv = \frac{2 q}{p + q} \left( I_α\ast |u|^p \right) |v|^{q - 2} v &\text{in} ~ \mathbb{R}^N, \end{cases} \] with $τ, η> 0$, $N \in \mathbb{N}$, $0 < α< N$, \[ \max \left\{1, \frac{2 α}{N}\right\} < p, q < 2^* \quad \text{and} \quad \frac{2 (N + α)}{N} < p + q < 2_α^*, \] where $I_α$ denotes the Riesz potential, \[ 2^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 N}{N - 2}, &\text{if} ~ N \geq 3, \end{cases} \quad \text{and} \quad 2_α^* := \begin{cases} \infty, &\text{if} ~ N \in \{1, 2\}, \\ \frac{2 (N + α)}{N - 2}, &\text{if} ~ N \geq 3. \end{cases} \] More precisely, by means of the moving planes method, we prove that positive solutions to this system are radially symmetric and strictly radially decreasing when $p, q \geq 2$, and we obtain a classification result for positive ground states in the case $p = q$ and $τ= η$.

math.AP↗

arXiv2024

Standing waves for nonlinear Hartree type equations: existence and qualitative properties

We consider systems of the form \[ \left\{ \begin{array}{l} -Δu + u = \frac{2p}{p+q}(I_α\ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -Δv + v = \frac{2q}{p+q}(I_α\ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in } \mathbb{R}^N, \end{array} \right. \] for $α\in (0, N)$, $\max\left\{\frac{2α}{N}, 1\right\} < p, q < 2^*$ and $\frac{2(N+α)}{N} < p+ q < 2^{*}_α$, where $I_α$ denotes the Riesz potential, \[ 2^* = \left\{ \begin{array}{l}\frac{2N}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2, \end{array}\right. \quad \text{and} \quad 2^*_α = \left\{ \begin{array}{l}\frac{2(N+α)}{N-2} \ \ \text{for} \ \ N\geq 3,\\ +\infty \ \ \text{for} \ \ N =1,2. \end{array} \right. \] This type of systems arises in the study of standing wave solutions for a certain approximation of the Hartree theory for a two-component attractive interaction. We prove existence and some qualitative properties for ground state solutions, such as definite sign for each component, radial symmetry and sharp asymptotic decay at infinity, and a regularity/integrability result for the (weak) solutions. Moreover, we show that the straight lines $p+q=\frac{2(N+α)}{N}$ and $ p+ q = 2^{*}_α$ are critical for the existence of solutions.

math.AP↗

arXiv2022

Existence and multiplicity results for a class of Kirchhoff-Choquard equations with a generalized sign-changing potential

In the present work we are concerned with the following Kirchhoff-Choquard-type equation $$-M(||\nabla u||_{2}^{2})Δu +Q(x)u + μ(V(|\cdot|)\ast u^2)u = f(u) \mbox{ in } \mathbb{R}^2 , $$ for $M: \mathbb{R} \rightarrow \mathbb{R}$ given by $M(t)=a+bt$, $ μ>0 $, $ V $ a sign-changing and possible unbounded potential, $ Q $ a continuous external potential and a nonlinearity $f$ with exponential critical growth. We prove existence and multiplicity of solutions in the nondegenerate case and guarantee the existence of solutions in the degenerate case.

math.AP↗

arXiv2021

On the existence and multiplicity of solutions for the $ N $-Choquard logarithmic equation with exponential critical growth

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-Δ_N u + a(x)|u|^{N-2}u + λ(\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ in } \mathbb{R}^N $, where $ a:\mathbb{R}^N \rightarrow \mathbb{R} $, $ λ>0 $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ is continuous function that behaves like $ \exp(α|u|^{\frac{N}{N-1}}) $ at infinity, for $ α>0 $.

math.AP↗

arXiv2021

$ (p, N)-$Choquard logarithmic equation involving a nonlinearity with exponential critical growth: existence and multiplicity

The present work is concerned with the following version of Choquard Logarithmic equations $ -Δ_p u -Δ_N u + a|u|^{p-2}u + b|u|^{N-2}u + λ(\ln|\cdot|\ast G(u))g(u) = f(u) \textrm{ in } \mathbb{R}^N $ , where $ a, b, λ>0 $, $ \max\{\frac{N}{2}, 2 \} < p< N $, $f, g: \mathbb{R} \rightarrow \mathbb{R} $ are continuous functions that behave like $ \exp(α|u|^{\frac{N}{N-1}}) $ at infinity, for $ α>0 $, and that has polynomial growth, respectively, and $ G(s)=\int\limits_{0}^{s}g(τ)dτ$. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, using a version of the Symmetric Mountain-Pass Theorem, we get infinitely many solutions.

math.AP↗

arXiv2020

Existence and multiplicity of solutions for the fractional $p$-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth

In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-Δ)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ λ>0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical growth we prove the existence of infinitely many solutions, via genus theory.

math.AP↗

arXiv2020

The Choquard logarithmic equation involving fractional Laplacian operator and a nonlinearity with exponential critical growth

In the present work we investigate the existence and multiplicity of nontrivial solutions for the Choquard Logarithmic equation $(-Δ)^{\frac{1}{2}} u + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ in } \mathbb{R}$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under exponential critical and subcritical growth. Morever, when $ f $ has subcritical growth we guarantee the existence of infinitely many solutions, via genus theory.

math.AP↗

arXiv2020

The Choquard logarithmic equation involving a nonlinearity with exponential growth

In the present work we are concerned with the Choquard Logarithmic equation $-Δu + au + λ(\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ λ>0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

math.AP↗