arXiv subjects

Edir Junior Ferreira Leite

Publications and source records attributed to Edir Junior Ferreira Leite.

6 recordsLinked to original sources

arXiv2024

Some applications of the Nehari manifold method to functionals in $C^1(X \setminus \{0\})$

Given a real Banach space $X$, we show that the Nehari manifold method can be applied to functionals which are $C^1$ in $X \setminus \{0\}$. In particular we deal with functionals that can be unbounded near $0$, and prove the existence of a ground state and infinitely many critical points for such functionals. These results are then applied to three classes of problems: the {\it prescribed energy problem} for a family of functionals depending on a parameter, problems involving the {\it affine} $p$-Laplacian operator, and degenerate Kirchhoff type problems.

math.AP↗

arXiv2023

Least energy solutions for affine $p$-Laplace equations involving subcritical and critical nonlinearities

The paper is concerned with Lane-Emden and Brezis-Nirenberg problems involving the affine $p$-laplace nonlocal operator $Δ_p^{\cal A}$, which has been introduced in \cite{HJM5} driven by the affine $L^p$ energy ${\cal E}_{p,Ω}$ from convex geometry due to Lutwak, Yang and Zhang \cite{LYZ2}. We are particularly interested in the existence and nonexistence of positive $C^1$ solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ${\cal E}_{p,Ω}$ and by the comparison ${\cal E}_{p,Ω}(u) \leq \Vert u \Vert_{W^{1,p}_0(Ω)}$ generally strict.

math.AP↗

arXiv2023

Fixed Point Theorem: Variants, Affine Context and Some Consequences

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are defined through the affine $L^{p}$ functional $\mathcal{E}_{p,Ω}^p$ introduced by Lutwak, Yang and Zhang in the work $\textit{Sharp affine $L_p$ Sobolev inequalities}$, J. Differential Geom. 62 (2002), 17-38 for $p > 1$ that is non convex and does not represent a norm in $\mathbb{R}^m$. Moreover, we address results for discontinuous functional at a point. As an application, we study critical points of the sequence of affine functionals $Φ_m$ on a subspace $W_m$ of dimension $m$ given by \[ Φ_m(u)=\frac{1}{p}\mathcal{E}_{p, Ω}^{p}(u) - \frac{1}α\|u\|^α_{L^α(Ω)}- \int_Ωf(x)u dx, \] where $1<α<p$, $[W_m]_{m \in \mathbb{N}}$ is dense in $W^{1,p}_0(Ω)$ and $f\in L^{p'}(Ω)$, with $\frac{1}{p}+\frac{1}{p'}=1$.

math.FA↗

arXiv2023

Principal curves to fractional $m$-Laplacian systems and related maximum and comparison principles

In this paper we develop a comprehensive study on principal eigenvalues and both the (weak and strong) maximum and comparison principles related to an important class of nonlinear systems involving fractional $m$-Laplacian operators. Explicit lower bounds for principal eigenvalues of this system in terms of the diameter of $Ω$ are also proved. As application, given $λ,μ\geq 0$ we measure explicitly how small has to be $\text{diam}(Ω)$ so that weak and strong maximum principles associated to this problem hold in $Ω$.

math.AP↗

arXiv2021

Minimization to the Zhang's energy on $BV(Ω)$ and sharp affine Poincaré-Sobolev inequalities

We prove the existence of minimizers for some constrained variational problems on $BV(Ω)$, under subcritical and critical restrictions, involving the affine energy introduced by Zhang in \cite{Z}. Related functionals have non-coercive geometry and properties like semicontinuity and affine compactness are deeper in the weak* topology. As a by-product of the theory, extremal functions are shown to exist for various affine Poincaré-Sobolev type inequalities.

math.FA↗

arXiv2016

A priori bounds and positive solutions for non-variational fractional elliptic systems

In this paper we study strongly coupled elliptic systems in non-variational form involving fractional Laplace operators. We prove Liouville type theorems and, by mean of the blow-up method, we establish a priori bounds of positive solutions for subcritical and superlinear nonlinearities in a coupled sense. By using those latter, we then derive the existence of positive solutions through topological methods.

math.AP↗