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Claudianor O. Alves

Publications and source records attributed to Claudianor O. Alves.

At least 19 recordsLinked to original sources
arXiv2025

Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schrödinger equation $$ \left\{ \begin{array}{ll} -Δu = λu + f(u)\quad & \text{in } Ω, \mathcal{B}_{α,ζ,γ}u = 0 & \text{on } \partial Ω, \int_Ω |u|^2\,dx = μ, \end{array} \right. \leqno{(P)^μ_{α,ζ,γ}} $$ where $Ω\subset \mathbb{R}^N$ ($N \geq 1$) is a smooth bounded domain, $μ>0$ is prescribed, $λ\in \mathbb{R}$ is a part of the unknown which appears as a Lagrange multiplier, $f,g:\mathbb{R} \to \mathbb{R}$ are continuous functions satisfying some technical conditions. The boundary operator $\mathcal{B}_{α,ζ,γ}$ is defined by $$ \mathcal{B}_{α,ζ,γ}u=αu+ζ\frac{\partial u}{\partial η}-γg(u), $$ where $α,ζ,γ\in \{0,1\}$ and $η$ denotes the outward unit normal on $\partialΩ$. Moreover, we highlight several further applications of our approach, including the nonlinear Schrödinger equations with critical exponential growth in $\mathbb{R}^{2}$, the nonlinear Schrödinger equations with magnetic fields, the biharmonic equations, and the Choquard equations, among others.

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arXiv2025

Existence and multiplicity of normalized solutions for the generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^2$

In this paper, we study the existence and {multiplicity} of nontrivial solitary waves for the generalized Kadomtsev-Petviashvili equation with prescribed {$L^2$-norm} \begin{equation*}\label{Equation1} \left\{\begin{array}{l} \left(-u_{x x}+D_x^{-2} u_{y y}+λu-f(u)\right)_x=0,{\quad x \in \mathbb{R}^2, } \\[10pt] \displaystyle \int_{\mathbb{R}^2}u^2 d x=a^2, \end{array}\right.%\tag{$\mathscr E_λ$} \end{equation*} where $a>0$ and $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. For the case $f(t)=|t|^{q-2}t$, with $2 0$, we prove the existence of normalized ground state solutions which corresponds to a local minimum of the associated energy functional. In this case, we further show that there exists a sequence $(a_n) \subset (0,a_0)$ with $a_n \to 0$ as $n \to+\infty$, such that for each $a=a_n$, the problem admits a second solution with positive energy. To the best of our knowledge, this is the first work that studies the existence of solutions for the generalized Kadomtsev-Petviashvili equations under the $L^2$-constraint, which we refer to them as the normalized solutions.

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arXiv2024

Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $\mathbb{R}^2$

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(εx,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2, $$ where $ε>0$ and $V$ is a double-well potential with minima at $t=α$ and $t=β$ with $α<β$. Here, we consider some class of functions $A(x,y)$ that are oscillatory in the variable $y$ and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity.

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arXiv2024

Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems

In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen-Cahn type given by $$ -\left(ϕ(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R}, $$ where $V$ is a double-well potential with minima at $t=\pmα$ and $ϕ:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $ϕ(t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $ϕ(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator.

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arXiv2023

Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains

In this paper we will prove the existence of a positive solution for a class of Schrödinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -Δu &+ u =Q(x)u\log u^2,\;\;\mbox{in}\;\;Ω,\nonumber &\mathcal{B}u=0 \,\,\, \mbox{on} \,\,\, \partial Ω, \end{aligned} \right. \end{equation} where $Ω\subset \mathbb{R}^N$, $N \geq 3$, is an \textit{exterior domain}, i.e., $Ω^c=\mathbb{R}^N \setminus Ω$ is a bounded smooth domain where $\mathcal{B}u=u$ or $\mathcal{B}u=\frac{\partial u}{\partial ν}$. We have used new approach that allows us to apply the usual $C^1$-variational methods to get a nontrivial solutions for these classes of problems.

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arXiv2023

[Existence of multiple solutions for a Schröndiger logarithmic equation

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2Δu + V(x)u & =u\log u^2,\;\;\mbox{in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.

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arXiv2023

Multiple normalized solutions to a logarithmic Schrödinger equation via Lusternik-Schnirelmann category

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by \begin{align*} \left\{ \begin{aligned} &-ε^2 Δu+V( x)u=λu+u \log u^2, \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}ε^N, \end{aligned} \right. \end{align*} where $a, ε>0, λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and $V: \mathbb{R}^N \rightarrow[-1, \infty)$ is a continuous function. Our analysis demonstrates that the number of normalized solutions of the equation is associated with the topology of the set where the potential function $V$ attains its minimum value. To prove the main result, we employ minimization techniques and use the Lusternik-Schnirelmann category. Additionally, we introduce a new function space where the energy functional associated with the problem is of class $C^1$.

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arXiv2023

On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^N$ via penalization method

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems \begin{align*} \left\{ \begin{aligned} &-ε^2Δu+V(x)u=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N, &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}ε^N, \end{aligned} \right. \end{align*} where $a,ε>0$, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V:\mathbb{R}^N \to [0,\infty)$ is a continuous function, and $f$ is a continuous function with $L^2$-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential $V$ attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer.

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arXiv2022

A nonsmooth variational approach to semipositone quasilinear problems in $\mathbb{R}^N$

This paper concerns the existence of a solution for the following class of semipositone quasilinear problems \begin{equation*} \left \{ \begin{array}{rclcl} -Δ_p u = h(x)(f(u)-a),\ & u > 0 & \mbox{in} & \mathbb{R}^N, \end{array} \right. \end{equation*} where $1 0$, $ f:[0,+\infty) \to [0,+\infty)$ is a function with subcritical growth and $f(0)=0$, while $h:\mathbb{R}^N \to (0,+\infty)$ is a continuous function that satisfies some technical conditions. We prove via nonsmooth critical points theory and comparison principle, that a solution exists for $a$ small enough. We also provide a version of Hopf's Lemma and a Liouville-type result for the $p$-Laplacian in the whole $\mathbb{R}^N$.

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arXiv2022

The method of the energy function and applications

In this work, we establish a new method to find critical points of differentiable functionals defined in Banach spaces which belong to a suitable class ($\mathcal{J}$) of functionals. Once given a functional $J$ in the class ($\mathcal{J}$), the central idea of the referred method consists in defining a real function $ζ$ of a real variable, called {\it energy function}, which is naturally associated to $J$ in the sense that the existence of real critical points for $ζ$ guarantees the existence of critical points for the functional $J$. As a consequence, we are able to solve some variational elliptic problems, whose associated energy functional belongs to ($\mathcal{J}$) and provide a version of the mountain pass theorem for functionals in the class ($\mathcal{J}$) that allows us to obtain mountain pass solutions without the so-called Ambrosetti-Rabinowitz condition.

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arXiv2021

Existence of solution for a class of heat equation involving the 1-Laplacian operator

This paper concerns the existence of global solutions for the following class of heat equation involving the 1-Laplacian operator of the Dirichlet problem $$ \left\{ \begin{array}{llc} u_{t}-Δ_1 u=f(u) & \text{in}\ & Ω\times (0, +\infty) , u =0 & \text{in} & \partialΩ\times (0, +\infty), u(x,0)=u_{0}(x)& \text{in} &Ω, \end{array}\right. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N \geq 1$ and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function satisfying some technical conditions, and $Δ_1 u=\text{div}\left(\frac{Du}{|Du|}\right)$ denotes the 1-Laplacian operator. The existence of global solution is done by using an approximation technique that consists in working with a class of $p$-Laplacian problem associated with $(P)$ and then taking the limit when $ p \to 1^+$ to get our results.

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arXiv2021

A Berestycki-Lions type result for a class of degenerate elliptic problems involving the Grushin Operator

In this work we study the existence of nontrivial solution for the following class of semilinear degenerate elliptic equations $$ -Δ_γ u + a(z)u = f(u) ~~ \mbox{in} ~~ \mathbb{R}^{N}, $$ where $Δ_γ$ is known as the {\it Grushin operator}, $z:=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{k}$ and $m+k=N\geq 3$, $f$ and $a$ are continuous function satisfying some technical conditions. In order to overcome some difficulties involving this type of operator, we have proved some compactness results that are crucial in the proof of our main results. For the case $a=1$, we have showed a Berestycki-Lions type result.

math.AP↗
arXiv2021

Existence of normalized solutions for the planar Schrödinger-Poisson system with exponential critical nonlinearity

In the present work we are concerned with the existence of normalized solutions to the following Schrödinger-Poisson System $$ \left\{ \begin{array}{ll} -Δu + λu + μ(\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2 , \\ \intR |u(x)|^2 dx = c,\ c> 0 , \end{array} \right. $$ for $μ\in \R $ and a nonlinearity $f$ with exponential critical growth. Here $ λ\in \R$ stands as a Lagrange multiplier and it is part of the unknown. Our main results extend and/or complement some results found in \cite{Ji} and \cite{[cjj]}.

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arXiv2021

Multiple solutions for a class of quasilinear problems with double criticality

We establish multiplicity results for the following class of quasilinear problems $$ \left\{ \begin{array}{l} -Δ_Φu=f(x,u) \quad \mbox{in} \quad Ω, \\ u=0 \quad \mbox{on} \quad \partial Ω, \end{array} \right. \leqno{(P)} $$ where $Δ_Φu=\text{div}(φ(x,|\nabla u|)\nabla u)$ for a generalized N-function $Φ(x,t)=\int_{0}^{|t|}φ(x,s)s\,ds$. We consider $Ω\subset\mathbb{R}^N$ to be a smooth bounded domain that contains two disjoint open regions $Ω_N$ and $Ω_p$ such that $\overline{Ω_N}\cap\overline{Ω_p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-Δ_Φ$ behaves like $-Δ_N$ on $Ω_N$ and $-Δ_p$ on $Ω_p$. We assume the nonlinearity $f:Ω\times\mathbb{R}\to\mathbb{R}$ of two different types, but both behaves like $e^{α|t|^\frac{N}{N-1}}$ on $Ω_N$ and $|t|^{p^*-2}t$ on $Ω_p$ as $|t|$ is large enough, for some $α>0$ and $p^*=\frac{Np}{N-p}$ being the critical Sobolev exponent for $1<p<N$. In this context, for one type of nonlinearity $f$, we provide multiplicity of solutions in a general smooth bounded domain and for another type of nonlinearity $f$, in an annular domain $Ω$, we establish existence of multiple solutions for the problem $(P)$ that are nonradial and rotationally nonequivalent.

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arXiv2021

Multiplicity of solutions for a class of quasilinear problems involving the $1$-Laplacian operator with critical growth

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ \left\{ \begin{array}{l} - Δ_1 u +ξ\frac{u}{|u|} =λ|u|^{q-2}u+|u|^{1^*-2}u, \quad\text{in }Ω, u=0, \quad\text{on } \partialΩ. \end{array} \right. $$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N \geq 2$ and $ξ\in\{0,1\}$. Moreover, $λ> 0$, $q \in (1,1^*)$ and $1^*=\frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $ξ=1$, $Ω= \{x \in \mathbb{R}^N\,:\,r < |x| < r+1\}$, $N\geq 2$, $N \not = 3$ and $r > 0$. In the second one, $Ω$ is a smooth bounded domain, $ξ=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.

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arXiv2021

Normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^{N}$

In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a>0$, $λ\in \mathbb{R}$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=μ|u|^{q-2}u+|u|^{2^*-2}u$ with $q \in (2+\frac{4}{N},2^*)$, $μ>0$ and $2^*=\frac{2N}{N-2}$ when $N \geq 3$. Our main results complement some recent results for $N \geq 3$ and it is totally new for $N=2$.

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arXiv2021

Multiplicity of normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^{N}$

In this paper we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-Δu=λu+μ|u|^{q-2}u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \end{aligned} \right. \end{align*} where $a,μ>0$, $λ\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q \in (2,2+\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=|u|^{2^*-2}u$ when $N \geq 3$ and $2^{*}=\frac{2N}{N-2}$.

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