arXiv subjects

Maxwell L. Silva

Publications and source records attributed to Maxwell L. Silva.

3 recordsLinked to original sources

arXiv2026

On singular problems in nonreflexive fractional Orlicz-Sobolev spaces

In this work, we deal with existence and uniqueness of positive solution $u_s$ for the singular quasilinear problem $(-Δ_Φ)^su=u^{-γ}$ in the nonreflexive fractional Orlicz-Sobolev $ W^{s}_0L^Φ(Ω)$ for $0<s<1$. Furthermore, we show that $u_s$ converges in $L^Φ(Ω)$ to the unique positive solution $u\in W^{1}_0L^Φ(Ω)$ of the problem $-Δ_Ψu=u^{-γ}$ as $s \uparrow 1$, where $Ψ$ is an appropriate $N$-function equivalent to the $N$-function $Φ$. The main difficulties to obtain existence of weak solutions for both singular quasilinear problems are that their associate energy functionals may not be well-defined on their whole natural workspaces due to the lack of the reflexivity and the presence of the singular term. To overcome these difficulties, we will use the minimization method and present a new approach to building appropriate test functions to prove that the problems have positive minimizers that we showed to be weak solutions of them, respectively.

math.AP↗

arXiv2025

Singular nonlocal elliptic systems via nonlinear Rayleigh quotient

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-Δ)^su +V_1(x)u = λ\frac{a(x)}{u^p} + \fracα{α+β}θ|u|^{α- 2}u|v|^β, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-Δ)^sv +V_2(x)v= λ\frac{b(x)}{v^q}+ \fracβ{α+β}θ|u|^α|v|^{β-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$ where $ 0 0, λ> 0, N > 2s$, and $s \in (0,1)$. The potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ are continuous functions which are bounded from below. Under our assumptions, we prove that there exists the largest positive number $λ^* > 0$ such that our main problem admits at least two positive solutions for each $λ\in (0, λ^*)$. Here we apply the nonlinear Rayleigh quotient together with the Nehari method. The main feature is to minimize the energy functional in Nehari set which allows us to prove our results without any restriction on the size of parameter $θ> 0$. Moreover, we shall consider the multiplicity of solutions for the case $λ= λ^*$ where degenerated points are allowed.

math.AP↗

arXiv2022

On prescribed energy saddle-point solutions to indefinite problems

A minimax variational principle for saddle-point solutions with prescribed energy levels is introduced. The approach is based on the development of the linking theorem to the energy level nonlinear generalized Rayleigh quotients. An application to indefinite elliptic Dirichlet problems is presented. Among the consequences, the existence of solutions with zero-energy levels is obtained.

math.AP↗