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Claudianor O. Alves

Publications and source records attributed to Claudianor O. Alves.

At least 55 records · Page 3Linked to original sources
arXiv2019

Existence, multiplicity and concentration for a class of fractional $p\&q$ Laplacian problems in $\mathbb{R}^{N}$

In this work we consider the following class of fractional $p\&q$ Laplacian problems \begin{equation*} (-Δ)_{p}^{s}u+ (-Δ)_{q}^{s}u + V(\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $1< p<q<\frac{N}{s}$, $(-Δ)^{s}_{t}$, with $t\in \{p,q\}$, is the fractional $t$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{C}^{1}$-function with subcritical growth. Applying minimax theorems and the Ljusternik-Schnirelmann theory, we investigate the existence, multiplicity and concentration of nontrivial solutions provided that $\varepsilon$ is sufficiently small.

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arXiv2018

Existence of solutions for a class of fractional elliptic problems on exterior domains

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-Δ)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $Ω$},\\ u \geq 0 \quad \mbox{in} \quad Ω\quad \mbox{and} \quad u \not\equiv 0, \\ u=0 \quad \mathbb{R}^N \setminus Ω, \end{array} \right. \end{equation*} involving the fractional Laplacian operator $(-Δ)^{s}$, where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is an exterior domain with (non-empty) smooth boundary $\partial Ω$ and $p\in (2, 2_{s}^{*})$. The main technical approach is based on variational and topological methods. The variational analysis that we perform in this paper dealing with exterior domains is quite general and may be suitable for other goals too.

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arXiv2018

A minimum problem with free boundary and subcritical growth in Orlicz spaces

The aim of this paper is to study the heterogeneous optimization problem \begin{align*} \mathcal {J}(u)=\int_Ω(G(|\nabla u|)+qF(u^+)+hu+λ_{+}χ_{\{u>0\}} )\text{d}x\rightarrow\text{min}, \end{align*} in the class of functions $ W^{1,G}(Ω)$ with $ u-φ\in W^{1,G}_{0}(Ω)$, for a given function $φ$, where $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|)\text{d}x<\infty$. The functions $G$ and $F$ satisfy structural conditions of Lieberman's type that allow for a different behavior at $0$ and at $\infty$. {}{Moreover, $F$ allows for a subcritical growth.} Given functions $q,h$ and constant $λ_+\geq 0$, we address several regularity results for minimizers of $\mathcal {J}(u)$, including local $C^{1,α}-$, and local Log-Lipschitz continuities for minimizers of $\mathcal {J}(u)$ with $λ_+=0$, and {}{$λ_+\geq 0$} respectively. We also establish growth rate near the free boundary for each non-negative minimizer of $\mathcal {J}(u)$ with $λ_+=0$, and $λ_+>0$ respectively. Furthermore, under additional assumption that $F\in C^1([0,+\infty); [0,+\infty))$, local Lipschitz regularity is carried out for non-negative minimizers of $\mathcal {J}(u)$ with $λ_{+}>0$.

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arXiv2018

Existence of solution for a nonlocal dispersal model with nonlocal term via bifurcation theory

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =u \left(λ- \int_ΩQ(x,y) |u(y)|^p dy \right) , \ \mbox{in} \ Ω, \] where $Ω\subset \mathbb{R}^{N}$, $N\geq 1$, is a bounded connected open, $p>0$, $λ$ is a real parameter, $Q:Ω\times Ω\to \mathbb{R}$ is a nonnegative function, and $L_0 : C(\overlineΩ) \to (\overlineΩ)$ is a nonlocal dispersal operator. The existence of solution is obtained via bifurcation theory.

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arXiv2018

Existence and profile of ground-state solutions to a $1-$Laplacian problem in $\mathbb{R}^N$

In this work we prove the existence of ground state solutions for the following class of problems \begin{equation*} \left\{ \begin{array}{ll} \displaystyle - Δ_1 u + (1 + λV(x))\frac{u}{|u|} & = f(u), \quad x \in \mathbb{R}^N, \\ u \in BV(\mathbb{R}^N), & \end{array} \right. \label{Pintro} \end{equation*} \end{abstract} where $λ> 0$, $Δ_1$ denotes the $1-$Laplacian operator which is formally defined by $Δ_1 u = \mbox{div}(\nabla u/|\nabla u|)$, $V:\mathbb{R}^N \to \mathbb{R}$ is a potential satisfying some conditions and $f:\mathbb{R} \to \mathbb{R}$ is a subcritical and superlinear nonlinearity. We prove that for $λ> 0$ large enough there exists ground-state solutions and, as $λ\to +\infty$, such solutions converges to a ground-state solution of the limit problem in $Ω= \mbox{int}( V^{-1}(\{0\}))$.

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arXiv2018

Existence of heteroclinic solution for a double well potential equation in an infinite cylinder of $\mathbb{R}^N$

This paper concernes with the existence of heteroclinic solutions for the following class of elliptic equations $$ -Δ{u}+A(εx, y)V'(u)=0, \quad \mbox{in} \quad Ω, $$ where $ε>0$, $Ω=\R \times \D$ is an infinite cylinder of $\mathbb{R}^N$ with $N \geq 2$. Here, we have considered a large class of potential $V$ that includes the Ginzburg-Landau potential $V(t)=(t^{2}-1)^{2}$ and two geometric conditions on the function $A$. In the first condition we assume that $A$ is asymptotic at infinity to a periodic function, while in the second one $A$ satisfies $$ 0<A_0=A(0,y)=\inf_{(x,y) \in Ω}A(x,y) < \liminf_{|(x,y)| \to +\infty}A(x,y)=A_\infty<\infty, \quad \forall y \in \D. $$

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arXiv2018

Existence and concentration phenomena for a class of indefinite variational problems with critical growth

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -Δu+V(x)u=A(εx)f(u), \quad x \in \R^{N}, \eqno{(P)_ε} $$ where $N \geq 2$, $ε>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that satisfies $$ 0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)} $$ $f:\R\rightarrow\R$ is a continuous function having critical growth, $V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with $0\notinσ(Δ+V)$. By using variational methods, we prove the existence of solution for $ε$ small enough. After that, we show that the maximum points of the solutions concentrate around of a maximum point of $A$.

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arXiv2017

A Berestycki-Lions type result and applications

In this paper we show an abstract theorem involving the existence of critical points for a functional $I$, which permit us to prove the existence of solutions for a large class of Berestycki-Lions type problems. In the proof of the abstract result we apply the deformation lemma on a special set associated with $I$, which we call of Pohozaev set.

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arXiv2017

Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -Δu=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.

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arXiv2017

Existence of solution for a class of quasilinear problem in Orlicz-Sobolev space without $Δ_2$-condition

\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -Δ_Φ{u}=f(u), \quad \mbox{in} \quad Ωu=0, \quad \mbox{on} \quad \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a smooth bounded domain, $f:\mathbb{R} \to \mathbb{R}$ is a continuous function verifying some conditions, and $Φ:\mathbb{R} \to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known $Δ_2$-condition, then the Orlicz-Sobolev space $W^{1,Φ}_0(Ω)$ can be non reflexive. As main model we have the function $Φ(t)=(e^{t^{2}}-1)/2$. Here, we study some situations where it is possible to work with global minimization, local minimization and mountain pass theorem, however some estimates are not standard for this type of problem.

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arXiv2017

Minimization of quotients with variable exponents

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, $p\in C^{1}(\overlineΩ),$ $q\in C(\overlineΩ)$ and $l,j\in\mathbb{N}.$ We describe the asymptotic behavior of the minimizers of the Rayleigh quotient $\frac{\Vert\nabla u\Vert_{lp(x)}}{\Vert u\Vert_{jq(x)}}$, first when $j\rightarrow\infty$ and after when $l\rightarrow\infty.$

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arXiv2017

Singularly perturbed critical Choquard equations

In this paper we study the semiclassical limit for the singularly perturbed Choquard equation $$ -\vr^2Δu +V(x)u =\vr^{μ-3}\Big(\int_{\R^3} \frac{Q(y)G(u(y))}{|x-y|^μ}dy\Big)Q(x)g(u) \quad \mbox{in $\R^3$}, $$ where $0<μ<3$, $\vr$ is a positive parameter, $V,Q$ are two continuous real function on $\R^3$ and $G$ is the primitive of $g$ which is of critical growth due to the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on the nonlinearity $g$, we first establish the existence of ground states for the critical Choquard equation with constant coefficients in $\R^3$. Next we establish existence and multiplicity of semi-classical solutions and characterize the concentration behavior by variational methods.

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arXiv2017

Ground state solution for a class of indefinite variational problems with critical growth

In this paper we study the existence of ground state solution for an indefinite variational problem of the type $$ \left\{\begin{array}{l} -Δu+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)} $$ where $N \geq 2$, $V,W:\mathbb{R}^N \to \mathbb{R}$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ are continuous functions verifying some technical conditions and $f$ possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, $V$ is $\mathbb{Z}^N$-periodic, $W$ goes to 0 at infinity and $f$ is asymptotically periodic.

math.AP↗
arXiv2017

An elliptic system with logarithmic nonlinearity

In the present paper we study the existence of solutions for some classes of singular systems involving the p(x) and q(x) Laplacian operators. The approach is based on bifurcation theory and subsupersolution method for systems of quasilinear equations involving singular terms.

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arXiv2016

Existence and concentration of solution for a non-local regional Schrödinger equation with competing potentials

In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation $$ \left\{ \begin{array}{l} ε^{2α}(-Δ)_ρ^α u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^α(\mathbb{R}^n) \end{array} \right. $$ where $ε$ is a positive parameter, $0< α< 1$, $1 2α$; $(-Δ)_ρ^α$ is a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius $ρ(x)>0$, $ρ, Q, K$ are competing functions. We study the existence of ground state and we analyze the behavior of semi-classical solutions as $ε\to 0$.

math.AP↗
arXiv2016

Existence and multiplicity of solutions for a nonlinear Schrödinger equation with non-local regional diffusion

In this article we are interested in the following non-linear Schrödinger equation with non-local regional diffusion $$ (-Δ)_{ρ_ε}^αu + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^α(\mathbb{R}^n), \qquad\qquad(P_ε) $$ where $ε>0$, $0< α< 1$, $(-Δ)_{ρ_ε}^α$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $ρ_ε(x)=ρ(εx)>0$, where $ρ$ is a continuous function. We give general conditions on $ρ$ and $f$ which assure the existence and multiplicity of solution for $(P_ε)$.

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