arXiv subjects
Publications and source records attributed to Claudianor O. Alves.
In this paper, we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by Szulkin combined with some properties of the weak$^*$ topology.
In this paper we prove a Lions type result for a large class of Orlicz-Sobolev space that can be nonreflexive and use this result to show the existence of solution for a large class of quasilinear problem on a nonreflexive Orlicz-Sobolev space.
This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -Δu+ λV(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in H^1(\mathbb{R}^{N}), \\ \end{array} \right. $$ where $N \geq 1$, $λ>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ has a potential well $Ω: =\text{int}\, V^{-1}(0)$ which possesses $k$ disjoint bounded components $Ω=\bigcup_{j=1}^{k}Ω_{j}$. Using the variational methods, we prove that if the parameter $λ>0$ is large enough, then the equation has at least $2^{k}-1$ multi-bump positive solutions.
This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a discontinuous function and asymptotically linear at infinity, $λ=0$ is in a spectral gap of $-Δ+V$, and $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$. Here, by employing Variational Methods for Locally Lipschitz Functionals, we establish the existence of solution when $f$ is periodic and non periodic
This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a $\mathbb{Z}^{N}$-periodic Caratheodory function and $λ=0$ does not belong to the spectrum of $-Δ+V$. Here, $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$.
In this paper we study the existence of solution for a class of variational inequality in whole $\mathbb{R}^N$ where the nonlinearity has a critical growth for $N \geq 2$. By combining a penalization scheme found in del Pino and Felmer [18] with a penalization method due to Bensoussan and Lions [9], we improve a recent result by Alves, Barros and Torres [1].
IIn this paper we consider the problem $$ \left\{ \begin{array}{rcl} -Δu+V_λ(x)u=(I_μ*|u|^{2^{*}_μ})|u|^{2^{*}_μ-2}u \ \ \mbox{in} \ \ \mathbb{R}^{N},\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\leqno{(P_λ)} $$ where $V_λ=λ+V_{0}$ with $λ\geq 0$, $V_0\in L^{N/2}(\R^N)$, $I_μ=\frac{1}{|x|^μ}$ is the Riesz potential with $0<μ<\min\{N,4\}$ and $2^{*}_μ=\frac{2N-μ}{N-2}$ with $N\geq 3$. Under some smallness assumption on $V_0$ and $λ$ we prove the existence of two positive solutions of $(P_λ)$. In order to prove the main result, we used variational methods combined with degree theory.
In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$ \left\{ \begin{array}{lcl} -Δu=P(x)\Big( g(u)+f(x)\Big) \mbox{ in } \mathbb{R}^N,\\ u \in D^{1,2}(\R^N),\ \lim_{|x|\to +\infty}u(x)=0, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $P\in C(\R^N,\R^+)$, $f\in C(\R^N)\cap L^{\infty}(\R^N)$ and $g\in C^1(\R)$. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.
In this paper we use the dynamical methods to establish the existence of nontrivial solution for a class of nonlocal problem of the type $$ \left\{\begin{array}{l} -a\left(x,\int_Ωg(u)\,dx \right)Δu =f(u), \quad x \in Ω\\ u=0, \hspace{2 cm} x \in \partial Ω, \end{array}\right. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N \, ( N \geq 2)$ is a smooth bounded domain and $a:\overlineΩ \times \mathbb{R} \to \mathbb{R}$ and $g,f: \mathbb{R} \to \mathbb{R}$ are $C^1$-functions that satisfy some technical conditions.
We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality.
In this article we are concerned with the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential. Under a local assumption on the potential $V$, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.
This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0$, $N \geq 1$ and $V$ is a continuous function with a global minimum. Using variational method, we prove that for small enough $ε>0$, the "shape" of the graph of the function $V$ affects the number of nontrivial solutions.
In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $Ω\subset \R^N$ is a bounded set with smooth boundary, $(-Δ)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus Ω} \frac{u(x) - u(y)}{|x-y|^{N+2s}}dy,\;\;x\in Ω. $$
In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -Δu+V(x)u=A(εx)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)_ε} $$ where $N \geq 1$, $ε$ is a positive parameter, $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with subcritical growth and $V,A: \mathbb{R}^{N} \to \mathbb{R}$ are continuous functions verifying some technical conditions. Here $V$ is a $\mathbb{Z}^N$-periodic function, $0 \not\in σ(-Δ+ V)$, the spectrum of $-Δ+V$, and $$ 0 < \inf_{x \in \R^{N}}A(x)\leq \displaystyle\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x \in \R^{N}}A(x). $$
In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-Δ)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R \setminus (a,b)\\ \mathcal{N}_{1/2}u(x) &= 0\;\;\mbox{in}\;\;(a,b), \end{aligned} \right. \end{equation} where $a,b\in \R$ with $a<b$, $(-Δ)^{\frac{1}{2}}$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_{1/2}u(x) = \frac{1}π \int_{\R\setminus (a,b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy,\;\;x\in [a,b]. $$
In this paper we study the existence of solution for a class of elliptic problem in whole $\mathbb{R}^N$ without the well known Ambrosetti-Rabinowitz condition. Here, we do not assume any monotonicity condition on $f(s)/s$ for $s>0$.
In this article we use the variational method developed by Szulkin \cite{szulkin} to prove the existence of a positive solution for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $ε>0, N \geq 1$ and $V$ is a saddle-like potential.
In this paper we use variational methods to establish a Berestycki-Lions type result for a class of generalized Kadomtsev-Petviashvili equation in $\mathbb{R}^N$. The positive and zero mass cases are considered.