arXiv subjects
Publications and source records attributed to Giovany M. Figueiredo.
In this work we study the existence of ground-state solutions of Dirac equations with potentials which are allowed to vanish at infinity. The approach is based on minimization of the energy functional over a generalized Nehari set. Some conditions on the potentials are given in order to overcome the lack of compactness.
We show the existence of a nodal solution with two nodal domains for a generalized Kirchhoff equation of the type $$ -M\left(\displaystyle\int_ΩΦ(|\nabla u|)dx\right)Δ_Φu = f(u) \ \ \mbox{in} \ \ Ω, \ \ u=0 \ \ \mbox{on} \ \ \partialΩ, $$ where $Ω$ is a bounded domain in $\mathbf{R}^N$, $M$ is a general $C^{1}$ class function, $f$ is a superlinear $C^{1}$ class function with subcritical growth, $Φ$ is defined for $t\in \mathbf{R}$ by setting $ Φ(t)=\int_0^{|t|}ϕ(s)sds$, $Δ_Φ$ is the operator $Δ_Φu:=div(ϕ(|\nabla u|)\nabla u)$. The proof is based on a minimization argument and a quantitative deformation lemma.
We study the following class of nonlinear Choquard equation, $$ -Δu +V(x)u =\Big( \frac{1}{|x|^μ}\ast F(u)\Big)f(u) \quad \mbox{in} \quad \R^N, $$ where $0<μ<N$, $N \geq 3$, $V$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions on the potential $V$, which include the case $V(\infty)=0$, that is, $V(x)\to 0$ as $|x|\to +\infty$, we prove existence of a nontrivial solution for the above equation by penalization method.
In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.
In this paper, we are going to study the existence of solution for the following Kirchhoff problem $$ \left\{ \begin{array}{l} M\biggl(\displaystyle\int_{\mathbb{R}^{3}}|\nabla u|^{2} dx +\displaystyle\int_{\mathbb{R}^{3}} λa(x)+1)u^{2} dx\biggl) \biggl(- Δu + (λa(x)+1)u\biggl) = f(u) \mbox{ in } \,\,\, \mathbb{R}^{3}, \\ \mbox{}\\ u \in H^{1}(\mathbb{R}^{3}). \end{array} \right. $$ Assuming that the nonnegative function $a(x)$ has a potential well with $int (a^{-1}(\{0\}))$ consisting of $k$ disjoint components $Ω_1, Ω_2, ....., Ω_k$ and the nonlinearity $f(t)$ has a subcritical growth, we are able to establish the existence of positive multi-bump solutions by variational methods.
In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k - dimensional sphere of RN, where k is between 1 and N-1, as a parameter goes to 0. The radius of such sphere is related with the local minimum of a function which takes into account the potential. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.
In this work we study the following class of problems in $\mathbb R^{N}, N>2s$ $$ \varepsilon^{2s} (-Δ)^{s}u + V(z)u=f(u), \,\,\, u(z) > 0 $$ where $0<s<1$, $(-Δ)^{s}$ is the fractional Laplacian, $\varepsilon$ is a positive parameter, the potential $V:\mathbb{R}^N \to\mathbb{R}$ %is a continuous functions and the nonlinearity $f:\mathbb R \to \mathbb R$ satisfy suitable assumptions; in particular it is assumed that $V$ achieves its positive minimum on some set $M.$ By using variational methods we prove existence, multiplicity and concentration of maxima of positive solutions when $\varepsilon\to 0^{+}$. In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the "topological complexity" of the set $M$.
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem involving N-functions and theory of locally Lispchitz continuous functionals.
This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz-Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lispchitz continuous functionals.
The main goal this work is to prove two results like Strauss and Lions for Orlicz-Sobolev spaces. After, we use these results for study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.
We establish the existence of a positive ground state solution for a Kirchhoff problem in $\mathbb{R}^2$ involving critical exponential growth, that is, the nonlinearity behaves like $\exp(α_{0}s^{2})$ as $|s| \to \infty$, for some $α_{0}>0$. In order to obtain our existence result we used minimax techniques combined with the Trudinger-Moser inequality.
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{\varepsilon^3} \int_{\mathbb{R}^3} V(x) u^{2}\right) \left[-\varepsilon^2Δu + V(x)u \right],$$ $M:\mathbb{R}_+\to \mathbb{R}_+$ and $V:\mathbb{R}^3 \to \mathbb{R}$ are continuous functions which verify some hypotheses.
The multiplicity of positive weak solutions for a quasilinear Schrödinger equations $-L_p u +(λA(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq ε^{p}Δ_p u +ε^{p}Δ_p (u^2)u$, $A$ is a nonnegative continuous function and nonlinear term $h$ has a subcritical growth. We achieved our results by using minimax methods and Lusternik-Schnirelman theory of critical points.
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(μx))^{2}u= μ|u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ Ω, \ \ \ \ u \in H^{1}_{0}(Ω,\mathbb{C}), $$ where $Ω\subset \mathbb{R}^{N} (N \geq 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $Ω$.
We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Schödinger equation. The main result is established by using minimax methods and Lusternik-Schnirelman theory of critical points.
In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = λ|u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $Ω$} u = 0 & \mbox{on $\partial Ω$}, \end{array} \right. $$ where $Ω$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ and $1 < q<2$. In order to employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions of the modified problem is necessary to recover solutions of the original one.