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Gustavo de Paula Ramos

Publications and source records attributed to Gustavo de Paula Ramos.

15 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 $τ= η$.

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arXiv2026

Ground states of the defocusing nonlinear Schrödinger equation with a point interaction in dimensions 2 and 3

This paper is concerned with ground states of the defocusing nonlinear Schrödinger equation with a point interaction, \[ \mathrm{i} \partial_t ψ= -Δ_αψ+ ψ|ψ|^{p - 2} \quad \text{in} \quad \mathbb{R} \times \mathbb{R}^N, \] where $- Δ_α$ denotes the Laplacian of point interaction centered at the origin with inverse s-wave scattering length $- 2 (N - 1) πα$ and we suppose that either (i) $N = 2$, $α\in \mathbb{R}$ and $p > 2$ or (ii) $N = 3$, $α< 0$ and $2 < p < 3$. At sufficiently small masses, (i) we prove that this equation admits ground states, (ii) we obtain some qualitative properties of ground states and (iii) we obtain some results relating ground states with critical points of the associated action functional.

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arXiv2025

Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a, b} \leq 0$

We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - Δu + (V - ω) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\ \|u\|_{\mathscr{L}^2}^2 = μ, \end{cases} \] where \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0 \leq a, b \leq \infty$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $ω\in \mathbb{R}$, $u \colon \mathbb{R}^3 \to \mathbb{R}$. This problem is obtained by looking for standing waves of the Schrödinger equation coupled with the nonrelativistic gravitational potential prescribed by a family of fourth-order gravity theories. In this paper, (i) we obtain a complete picture of the existence/nonexistence of ground states of the associated autonomous problem for every possible geometry of $K_{a, b}$, (ii) we obtain conditions that ensure the existence of ground states of the nonautonomous problem when $K_{a, b} \leq 0$ and (iii) we prove that as \[ (a, b) \to (A, B) \in \left\{(0, 0), (\infty, \infty), (0, \infty)\right\}, \] ground states of this problem respectively converge to a ground state of (1) the Schrödinger equation, (2) the Choquard equation and (3) a rescaling of the Choquard equation.

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arXiv2025

Ground states of the defocusing NLSE with a point interaction

Suppose that either (i) $N = 2$, $α\in \mathbb{R}$ and $p > 2$ or (ii) $N = 3$, $α< 0$ and $2 < p < 3$. We prove that there exists an explicitly computable $μ_0 = μ_0 (N, α, p) > 0$ such that if $0 < μ< μ_0$, then the following normalized semilinear elliptic problem with a point interaction admits ground states: \[ \begin{cases} - Δ_αu + ωu + u |u|^{p - 2} = 0 &\text{in} ~ \mathbb{R}^N; \\ \|u\|_{\mathscr{L}^2}^2 = μ, \end{cases} \] where $- Δ_α$ denotes the Laplacian of point interaction (centered at the origin) with inverse scattering length $- 2 (N - 1) πα$ and we want to solve for $ω\in \mathbb{R}$, $u \colon \mathbb{R}^N \to \mathbb{R}$. We remark that this kind of solutions does not exist in the framework of the defocusing NLSE without a point interaction.

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arXiv2025

Ground states of the planar nonlinear Schrödinger--Newton system with a point interaction

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger--Newton system with a point interaction: \[ \begin{cases} - Δ_αu = w u + βu |u|^{p - 2} &\text{on} ~ \mathbb{R}^2; \\ - Δw = 2 π|u|^2 &\text{on} ~ \mathbb{R}^2; \\ \|u\|_{L^2}^2 = c, \end{cases} \] where $p > 2$; $α, β\in \mathbb{R}$ and $- Δ_α$ denotes the Laplacian of point interaction with scattering length $(- 2 πα)^{- 1}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem \[ \mathrm{i} ψ' (t) = - Δ_αψ(t) - (\log |\cdot| \ast |ψ(t)|^2) ψ(t) - βψ(t) |ψ(t)|^{p - 2}. \]

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arXiv2025

Sign-changing multi-peak standing waves of the NLSE with a point interaction

Consider the following semilinear problem with a point interaction in $\mathbb{R}^N$: \[- Δ_αu + ωu = u |u|^{p - 2},\] where $N \in \{2, 3\}$; $ω> 0$; $- Δ_α$ denotes the Hamiltonian of point interaction with inverse $s$-wave scattering length $- (4 πα)^{- 1}$ and we want to solve for $u \colon \mathbb{R}^N \to \mathbb{R}$. By means of Lyapunov--Schmidt reduction, we prove that this problem has sign-changing multi-peak solutions when either (1) $N = 2$, $α\in \mathbb{R}$, $p_* < p \leq 3$ and $ω$ is sufficiently large or (2) $N = 3$, $0 < α< \infty$, $p_* < p < 3$ and $ω$ is sufficiently small, where $2.45 < p_* := \frac{9 + \sqrt{113}}{8} < 2.46$.

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arXiv2025

Minimizers of mass-constrained functionals involving a nonattractive point interaction

We establish conditions to ensure the existence of minimizer for a class of mass-constrained functionals involving a nonattractive point interaction in three dimensions. The existence of minimizers follows from the compactness of minimizing sequences which holds when we can simultaneously rule out the possibilities of vanishing and dichotomy. The proposed method is derived from the strategy used to avoid vanishing in Adami, Boni, Carlone & Tentarelli (Calc. Var. 61, 195 (2022)) and the strategy used to avoid dichotomy in Bellazzini & Siciliano (J. Funct. Anal. 261, 9 (2011)). As applications, we prove the existence of ground states with sufficiently small mass for the following nonlinear problems with a point interaction: a Kirchhoff-type equation and the Schrödinger-Poisson system.

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arXiv2024

Asymptotic profile of least energy solutions to the nonlinear Schrödinger-Bopp-Podolsky system

Consider the following nonlinear Schrödinger--Bopp--Podolsky system in $\mathbb{R}^3$: \[ \begin{cases} - Δv + v + ϕv = v |v|^{p - 2}; \\ β^2 Δ^2 ϕ- Δϕ= 4 πv^2, \end{cases} \] where $β> 0$ and $3 < p < 6$, the unknowns being $v$, $ϕ\colon \mathbb{R}^3 \to \mathbb{R}$. We prove that, as $β\to 0$ and up to translations and subsequences, least energy solutions to this system converge to a least energy solution to the following nonlinear Schrödinger--Poisson system in $\mathbb{R}^3$: \[ \begin{cases} - Δv + v + ϕv = v |v|^{p - 2}; \\ - Δϕ= 4 πv^2. \end{cases} \]

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arXiv2024

Cluster semiclassical states of the nonlinear Schrödinger-Bopp-Podolsky system

Consider the following nonlinear Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$: $$ \begin{cases} -\varepsilon^2 Δu + (V + ϕ) u = u |u|^{p-1}; \\ a^2 Δ^2 ϕ- Δϕ= 4 πu^2, \end{cases} $$ where $a, \varepsilon > 0$; $1 < p < 5$; $V \colon \mathbb{R}^3 \to ]0, \infty[$ and we want to solve for $u, ϕ\colon \mathbb{R}^3 \to \mathbb{R}$. By means of Lyapunov-Schmidt reduction, we show that if $K \geq 2$, $z_0$ is a strict local minimum of $V$, $V$ is adequately flat in a neighborhood of $z_0$ and $\varepsilon$ is sufficiently small, then the system has a multipeak cluster solution with $K$ peaks placed at the vertices of a regular convex $K$-gon centered at $z_0$.

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arXiv2024

On the existence of ground states to Hartree-type equations in $\mathbb{R}^3$ with a delta potential

Consider the Hartree-type equation in $\mathbb{R}^3$ with a delta potential formally described by $$ i \partial_t ψ= - Δ_x ψ+ αδ_0 ψ- (I_β\ast |ψ|^p) |ψ|^{p - 2} ψ$$ where $α\in \mathbb{R}$; $0 < β< 3$ and we want to solve for $ψ\colon \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$. By means of a Pohožaev identity, we show that if $p = (3 + β) / 3$ and $α\geq 0$, then the problem has no ground state at any mass $μ> 0$. We also prove that if $$ \frac{3 + β}{3} < p < \min \left( \frac{5 + β}{3}, \frac{5 + 2 β}{4} \right), $$ which includes the physically-relevant case $p = β= 2$, then the problem admits a ground state at any mass $μ> 0$.

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arXiv2023

Existence and concentration of semiclassical bound states for a quasilinear Schrödinger-Poisson system

In the paper we consider the following quasilinear Schrödinger--Poisson system in the whole space $\mathbb R^{3}$ $$ \begin{cases} - \varepsilon^2 Δu + (V + ϕ) u = u |u|^{p - 1} \newline - Δϕ- βΔ_4 ϕ= u^2, \end{cases} $$ where $1 < p < 5, β> 0,V :\mathbb R^{3}\to ]0, \infty[$ and look for solutions $u,ϕ:\mathbb R^{3}\to \mathbb R$ in the semiclassical regime, namely when $\varepsilon\to 0.$ By means of the Lyapunov--Schmidt method we estimate the number of solutions by the cup-length of the critical manifold of the external potential $V$.

math.AP↗
arXiv2023

Concentrated solutions to the Schrödinger--Bopp--Podolsky system with a positive potential

Consider the Schrödinger--Bopp--Podolsky system \[ \begin{cases} -ε^2Δu+(V+Kϕ)u=u|u|^{p-1};\newline Δ^2ϕ-Δϕ=4πK u^2 \end{cases} ~\text{in}~\mathbb{R}^3 \] for sufficiently small $ε>0$, where $V,K\colon\mathbb{R}^3\to [0,\infty[$; $p\in ]1,5[$ are fixed and we want to solve for $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}$. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of $V$ and we establish the existence of solutions concentrated around critical points of $V$.

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arXiv2022

Existence and limit behavior of least energy solutions to constrained Schrödinger-Bopp-Podolsky systems in $\mathbb{R}^3$

Consider the following Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -Δu + ωu + ϕu = u|u|^{p-2},\newline -Δϕ+ a^2Δ^2ϕ=4πu^2,\newline \|u\|_{L^2}=ρ, \end{cases} \] where $a,ρ>0$ and our unknowns are $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}^3$ and $ω\in\mathbb{R}$. We prove that if $2 0$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $2 0$ is sufficiently small, then least energy solutions are radially symmetric up to translation and as $a\to 0$, they converge to a least energy solution of the Schrödinger-Poisson-Slater system under the same $L^2$-norm constraint.

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