arXiv subjects

M. L. M. Carvalho

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

11 recordsLinked to original sources

arXiv2026

Elliptic problems involving a diffusion operator of vectorial $p\&q$-Laplacian type

We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator \[ \mathcal L_Au=-\sum_{i=1}^N\partial_i\!\left( a_i(|\partial_i u|^{p_i})|\partial_i u|^{p_i-2} \partial_i u\right), \] where the functions $a_i$ satisfy structural growth and monotonicity conditions of $(p_i,q_i)$-type. This framework contains, among other examples, the anisotropic $\vec p$-Laplacian, the orthotropic $p\&q$-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, $r=p_1<p_N$, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave--convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz--Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration--compactness.

math.AP↗

arXiv2026

Extremal parameters, multiplicity and nonexistence for a singular elliptic problem

We consider the following class of singular elliptic problems \[ \begin{cases} -Δu = -\dfrac{u}{|u|^{β+1}}χ_{\{|u|>0\}} +λ|u|^{p-1}u, & \text{in }Ω,\\[1mm] u=0, & \text{on }\partialΩ, \end{cases} \] where $Ω\subset\mathbb R^N$ is a bounded smooth domain, $N\geq3$, $0<β,p<1$, and $λ>0$. The energy functional is not of class $C^1$ in $H_0^1(Ω)$, and the standard critical point theory does not apply. We prove that a nontrivial nonnegative weak solution exists if and only if $λ$ exceeds a critical value $\barλ$, which we identify and locate strictly between two Rayleigh-type parameters $0<λ_*<\barλ<λ^*<+\infty$. This gives a sharp existence threshold. We also prove that for every $λ$ in a neighborhood on the left of $λ^*$, there exist at least two distinct nontrivial nonnegative weak solutions. The first is a minimizer on the Nehari manifold, with an energy changing sign at $λ^*$. The second has a strictly positive energy. For $λ<λ^*$ sufficiently close to $λ^*$, both solutions have positive energy. This contrasts with previous multiplicity results for the same problem, in which the two solutions obtained have energies of opposite signs.

math.AP↗

arXiv2025

Superlinear fractional $Φ$-Laplacian type problems via the nonlinear Rayleigh quotient with two parameters

In this work, we establish the existence and multiplicity of weak solutions for nonlocal elliptic problems driven by the fractional $Φ$-Laplacian operator, in the presence of a sign-indefinite nonlinearity. More specifically, we investigate the following nonlocal elliptic problem: \begin{equation*} \left\{\begin{array}{rcl} (-Δ_Φ)^s u +V(x)u & = & μa(x)|u|^{q-2}u-λ|u|^{p-2}u \mbox{ in }\, \mathbb{R}^N, \\ u\in W^{s,Φ}(\mathbb{R}^N),&& \end{array} \right. \end{equation*} where $s \in (0,1), N \geq 2$ and $μ, λ>0$. Here, the potentials $V, a : \mathbb{R}^N \to \mathbb{R}$ satisfy some suitable hypotheses. Our main objective is to determine sharp values for the parameters $λ> 0$ and $μ> 0$ where the Nehari method can be effectively applied. To achieve this, we utilize the nonlinear Rayleigh quotient along with a detailed analysis of the fibering maps associated with the energy functional. Additionally, we study the asymptotic behavior of the weak solutions to the main problem as $λ\to 0$ or $μ\to +\infty$.

math.AP↗

arXiv2023

Asymptotic behavior of Musielak-Orlicz-Sobolev modulars

In this article we study the asymptotic behavior of anisotropic nonlocal nonstandard growth seminorms and modulars as the fractional parameter goes to 1. This gives a so-called Bourgain-Brezis-Mironescu type formula for a very general family of functionals. In the particu\-lar case of fractional Sobolev spaces with variable exponent, we point out that our proof asks for a weaker regularity of the exponent than the considered in previous articles.

math.AP↗

arXiv2023

On Fractional Musielak-Sobolev spaces and applications to nonlocal problems

In this work, we establish some abstract results on the perspective of the fractional Musielak-Sobolev spaces, such as: uniform convexity, Radon-Riesz property with respect to the modular function, $(S_{+})$-property, Brezis-Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems \begin{equation*} \left\{ \begin{array}{ll} (-Δ)_{Φ_{x,y}}^s u = f(x,u),& \mbox{in }Ω, u=0,& \mbox{on }\mathbb{R}^N\setminus Ω, \end{array} \right. \end{equation*} where $N\geq 2$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial Ω$ and $f:Ω\times \mathbb{R} \rightarrow \mathbb{R}$ is a Carathéodory function not necessarily satisfying the Ambrosetti-Rabinowitz condition. Such class of problems enables the presence of many particular operators, for instance, the fractional operator with variable exponent, double-phase and double-phase with variable exponent operators, anisotropic fractional $p$-Laplacian, among others.

math.AP↗

arXiv2018

Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (ϕ1, ϕ2)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.

math.AP↗

arXiv2017

Multiplicity of Solutions for Quasilinear Elliptic Problems

It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $Φ$-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In order to prove our main results we employ variational methods, regularity results and truncation techniques.

math.AP↗

arXiv2017

Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $Φ$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.

math.AP↗

arXiv2016

Concave-convex effects for critical quasilinear elliptic problems

It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $Φ$-Laplacian operator. One of these solutions is obtained as ground state solution by applying the well known Nehari method. The semilinear term in the quasilinear equation is a concave-convex function which presents a critical behavior at infinity. The concentration compactness principle is used in order to recover the compactness required in variational methods.

math.AP↗

arXiv2016

Sign changing solutions for quasilinear superlinear elliptic problems

Results on existence and multiplicity of solutions for a nonlinear elliptic problem driven by the $Φ$-Laplace operator are established. We employ minimization arguments on suitable Nehari manifolds to build a negative and a positive ground state solutions. In order to find a nodal solution we employ additionally the well known Deformation Lemma and Topological Degree Theory.

math.AP↗

arXiv2013

Nonlinear Boundary Value Problems via Minimization on Orlicz-Sobolev Spaces

We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (ϕ(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} Ω$ under Dirichlet boundary conditions, where $Ω\subset {\bf R}^{N}$ is a bounded smooth domain, $ϕ: (0,\infty)\longrightarrow (0,\infty)$ is a suitable continuous function and $f: Ω\times {\bf R} \to {\bf R}$ satisfies the Carathéodory conditions, while $h$ is a measure.

math.AP↗