# Marcos Leandro Mendes CarvalhoUniversidade Federal de Goiás | UFG · Departamento de Matemática

Marcos Leandro Mendes Carvalho

PhD

## About

53

Publications

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222

Citations

Citations since 2017

Introduction

Additional affiliations

January 2006 - November 2015

## Publications

Publications (53)

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 o...

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 existe...

It is established L p estimates for the fractional Φ-Laplacian operator defined in bounded domains, where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with Moser's iteration, we prove that any weak solution for fractional Φ-Laplacian operator defined in bounded domains belongs to L...

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form −Δpu=f(u), with p>1. We deal with relatively unexplored cases when f is non-Lipschitz at 0, f(0)=0 and f(u)<0, u∈(0,r), for some r<+∞. Using the nonlinear generalized Rayleigh quotients method we find a ran...

We consider a fractional double phase Robin problem involving variable order and variable exponents. The nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz-type condition. By using a variational methods, we investigate the multiplicity of solutions.

In this paper we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by Szulkin [34] combined with some properties of the weak * topology.

We discuss the existence of multiple positive solutions leading to the occurrence of an S-shaped bifurcation curve to the equations of the form $$ -\Delta_p u= f(\mu,\lambda, u)~ \mbox{in} ~\Omega \subset \mathbb{R}^N $$ where $\Delta_p$ is a $p$-Laplacian, $p>1$, $N\geq 1$, $\mu, \lambda \in \mathbb{R}$. We deal with relatively unexplored cases wh...

It is established $L^{p}$ estimates for the fractional $\Phi$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $\Phi$-Laplacian operator defined in bounded dom...

This paper deals with nonlinear elliptic boundary value problems with complicated geometry of nonlinearities. A new method for obtaining multiple solutions based on a recursive use of the non-linear generalized Rayleigh quotients to split Nehari manifold into subsets without the degeneracies is introduced. The method is applied to prove the multipl...

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,...

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 brings up difficulties in associating a differentiable functional to the problem...

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form
\begin{document}$ \begin{equation*} \begin{cases} -\Delta u +V(x) u = (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u \, {\rm{\;in\;}}\, \mathbb{R}^N, \\ \ u\in H^1( \mathbb{R}^N) \end{cases} \end{equ...

It is established existence of minimal W 1,Φ loc (Ω)-solutions on some appropriated set for the quasilinear elliptic problem −∆ Φ u = λf (x, u) + µh(x, u, ∇u) in Ω, u > 0 in Ω, u = 0 on ∂Ω, where f may have indefinite sign and behave in a strongly singular way at u = 0, h has sublinear growth, λ > 0 and µ ≥ 0 are real parameters. The main results i...

It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space R N. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the pr...

It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ \begin{array}{rcl} -\Delta u +V(x) u &=& (I_\alpha* |u|^p)|u|^{p-2}u+ \lambda |u|^{q-2}u, \, u \in H^1(\mathbb{R}^{N}), \end{array} $$ where $\lambda > 0, N \geq 3, \alpha \in (0, N)$. The potentia...

In this paper we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by Szulkin [32] combined with some properties of the weak * topology.

In this paper we prove a Lions type result for a large class of Orlicz-Sobolev space that can be nonreflexive and use this result to show the existence of solution for a large class of quasilinear problem on a nonreflexive Orlicz-Sobolev space.

We consider a fractional double phase Robin problem involving variable order and variable exponents. The nonlinearity $f$ is a Carath\'{e}odory function satisfying some hypotheses which do not include the Ambrosetti-Rabinowitz type condition. By using Variational methods, we investigate the multiplicity of solutions.

It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space \begin{document}$ \mathbb{R}^N $\end{document}. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general...

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,...

In this paper we prove a Lieb type result in an Orlicz-Sobolev space that can be non-reflexive and use this result to show the existence of solution for a large class of quasilinear problem on a non-reflexive Orlicz-Sobolev space.

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (Φ 1 , Φ 2)-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...

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 s...

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by the Φ-Laplacian operator. One of the 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 space framework....

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 w...

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 w...

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known Φ -Laplacian operator given by {-ΔΦu=g(x,u),inΩ,u=0,on∂Ω,where Δ Φ u: = div (ϕ(| ∇ u|) ∇ u) and Ω ⊂ R N , N≥ 2 , is a bounded domain with smooth boundary ∂Ω. Our work concerns on nonlinearities g which can be homogeneous or non-homogeneou...

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $\Phi$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-\Delta_\Phi u= g(x,u), & \mbox{in}~\Omega, u=0, & \mbox{on}~\partial \Omega, \end{array} \right. \end{equation*} where $\Delta_{\Phi}u :=\mbox{d...

It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by (\phi 1, \phi 2)-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 co...

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by the Φ-Laplacian operator. One of the 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 space framework....

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...

It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by the Φ-Laplacian operator. One of the 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 space framework....

In this work we consider existence and uniqueness of solutions for a quasilinear elliptic problem, which may be singular at the origin. Furthermore, we consider a comparison principle for subsolutions and supersolutions just in $W_{loc}^{1,\Phi}(\Omega)$ to the problem
$$
\left\{\
\begin{array}{l}
\displaystyle-\Delta_\Phi u= f(x,u) ~\mbox{in}~\Ome...

This paper deals with existence of positive solutions for a class of quasilinear elliptic systems involving the $\Phi$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbartion techniques and comparison arguments in the setting of Orlicz-Sobolev spaces

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 T...

It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $\Phi$-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.

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 [4] and a comparison princ...

It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $\Phi$-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 p...

Results on existence and multiplicity of solutions for a nonlinear elliptic problem driven by the $\Phi$-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...

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the (Formula presented.)-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the (Formula presented.)-Laplacian operator is not homogen...

In this work, we employ minimization arguments and topological degree theory for mappings of type

We study existence of multiple positive solutions for the nonlinear eigenvalue problem -div(θ(¯δu¯)δu) = δf (u) in ω, u = 0 on ω, where ω RN is a bounded domain with smooth boundary ω, θ : (0,∞) → (0,∞) is a suitable C1-function, δ < 0 is a parameter and f : [0,∞) → R is a sign-changing continuous function. We show existence of a finite number of s...

Using variational arguments, we establish the existence of multiple solutions for quasilinear elliptic problems driven by the Φ-Laplacian operator. A major point is that we ensure compactness without the well-known Ambrosetti–Rabinowitz superlinearity condition.

We study the existence and regularity of the solution to the multivalued equation - ΔΦu ∈ ∂j(u) + λh in Ω, where Ω ⊂ RN is a bounded smooth domain, Φ is an N-function, ΔΦ is the corresponding Φ-Laplacian, λ > 0 is a parameter, h is a measurable function, and j is a continuous function with critical growth where ∂j(u) denotes its subdifferential. We...

We exploit minimization of locally Lipschitz functionals defined on Orlicz-Sobolev spaces along with convexity techniques, to investigate existence of solution of the multivalued equation\ \ $-\Delta_{\Phi} u \in \partial j(.,u) + h$\ \ in $\Omega$, where $\Omega \subset {\bf R}^N$ is a bounded smooth domain, $\Phi: {\bf R} \to [0,\infty)$ is an N-...

In 2009 Loc and Schmitt established a result on sufficient conditions for
multiplicity of solutions of a class of nonlinear eignvalue problems for the
p-Laplace operator under Dirichlet boundary conditions, extending an earlier
result of 1981 by Peter Hess for the Laplacian. Results on necessary conditions
for existence were also established. In th...

We develop arguments on convexity and minimization of energy functionals on
Orlicz-Sobolev spaces to investigate existence of solution to the equation
$\displaystyle -\mbox{div} (\phi(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in}
\Omega$ under Dirichlet boundary conditions, where $\Omega \subset {\bf R}^{N}$
is a bounded smooth domain, $\phi : (0,\i...

We develop arguments on the critical point theory for locally Lipschitz
functionals on Orlicz-Sobolev spaces, along with convexity and compactness
techniques to investigate existence of solution of the multivalued equation
$\displaystyle - \Delta_{\Phi} u \in \partial j(.,u) + \lambda h \mbox{in}
\Omega$, where $\Omega \subset {\bf R}^{N}$ is a bou...

In this work we develop arguments on the critical point theory for locally Lipschitz functionals
on Orlicz-Sobolev spaces, along with convexity, minimization and compactness
techniques to investigate existence of solution of the multivalued equation
−∆Φu ∈ ∂ j(.,u) +λh in Ω,
where Ω ⊂ RN is a bounded domain with boundary smooth ∂Ω, Φ : R → [0,∞) is...

## Projects

Projects (4)

The objective is to study the existence and multiplicity of solutions for a class of nonlinear elliptical problems

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Multiplicity of Solutions