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Marcos L. M. Carvalho

Publications and source records attributed to Marcos L. M. Carvalho.

8 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↗
arXiv2024

Quasilinear elliptic problems via nonlinear Rayleigh quotient

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -Δ_Φu = λa(x) |u|^{q-2}u + |u|^{p-2}u, & x\inΩ, u = 0, & x \in \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N, N \geq 2,$ is a smooth bounded domain, $1 < q < \ell \leq m < p < \ell^*$ and $Φ: \mathbb{R} \to \mathbb{R}$ is suitable $N$-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number $λ^* > 0$ in such way that our main problem admits at least two distinct solutions for each $λ\in (0, λ^*)$. Furthermore, using some fine estimates and some extra assumptions on $Φ$, we prove the existence of at least two positive solutions for $λ= λ^*$ and $λ\in (λ^*, \overlineλ)$ where $\overlineλ > λ^*$.

math.AP↗
arXiv2021

Choquard equations via nonlinear Rayleigh quotient for concave-convex nonlinearities

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -Δu +V(x) u &=& (I_α* |u|^p)|u|^{p-2}u+ λ|u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ where $λ> 0, N \geq 3, α\in (0, N)$. The potential $V$ is a continuous function and $I_α$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_α < p < 2^*_α$ where $2_α=(N+α)/N$, $2_α=(N+α)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $λ> 0$ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $λ_n > 0$ such that our main problem admits at least two positive solutions for each $λ\in (0, λ_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $λ_n > 0$ is optimal in some sense which allow us to apply the Nehari method.

math.AP↗
arXiv2020

Compact embedding theorems and a Lions' type Lemma for fractional Orlicz-Sobolev spaces

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrödinger equations, taking into account unbounded or bounded potentials.

math.AP↗
arXiv2019

Separating of critical points on the Nehari manifold via the nonlinear generalized Rayleigh quotients

In this paper, we deal with equations of variational form which Nahari manifolds can contain more than two different types of critical points. We introduce a method of separating critical points on the Nahari manifold, based on the use of nonlinear generalized Rayleigh quotients. The method is illustrated by establishing the existence of positive solutions, ground states and multiplicity results for a two-parameter nonlinear elliptic boundary problem with polynomial nonlinearities.

math.AP↗
arXiv2019

Equivalent conditions for existence of three solutions for a problem with discontinuous and strongly-singular terms

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz-Sobolev space. The presence of both strongly-singular and non-continuous terms bring up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $W_0^{1,Φ}(Ω)$. To overcome this obstacle, we established an optimal condition for the existence of $W_0^{1,Φ}(Ω)$-solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $W_0^{1,Φ}(Ω)$ to apply techniques of convex analysis and generalized gradient in Clarke sense.

math.AP↗
arXiv2017

Existence and regularity of positive solutions of quasilinear elliptic problems with singular semilinear term

This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Φ$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{\infty}(Ω)$-regularity for positive solutions

math.AP↗