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Marcos T. O. Pimenta

Publications and source records attributed to Marcos T. O. Pimenta.

13 recordsLinked to original sources
arXiv2026

Existence of solutions for elliptic problems involving the $(1,q)$-Laplacian operator and a discontinuous superlinear nonlinearity

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(Ω)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $β\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

math.AP↗
arXiv2025

Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

In this paper we consider degenerate Kirchhoff-type equations of the form \[-ϕ(Ξ(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } Ω,\] \[\phantom{aaiaaaaaaaaa}ϕ(Ξ(u)) \mathcal{B}(u) \cdot ν= g(x,u) \quad \text{on } \partialΩ,\] where $Ω\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partialΩ$, $\mathcal{A}$ denotes the double phase operator given by \begin{align*} \mathcal{A}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u \right)\quad \text{for }u\in W^{1,\mathcal{H}}(Ω), \end{align*} $ν(x)$ is the outer unit normal of $Ω$ at $x \in \partialΩ$, \[\mathcal{B}(u)=|\nabla u|^{p-2}\nabla u + μ(x) |\nabla u|^{q-2}\nabla u,\] \[\phantom{aaaiaaaa}Ξ(u)= \int_Ω\left(\frac{|\nabla u|^p+|u|^p}{p}+μ(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x,\] $1 0$ and $ζ\geq 1$, and $f\colonΩ\times\mathbb{R}\to\mathbb{R}$, $g\colon\partialΩ\times\mathbb{R}\to\mathbb{R}$ are Carathéodory functions that grow superlinearly and subcritically. We prove the existence of a nodal ground state solution to the problem above, based on variational methods and minimization of the associated energy functional $\mathcal{E}\colon W^{1,\mathcal{H}}(Ω) \to\mathbb{R}$ over the constraint set \[\mathcal{C}=\Big\{u \in W^{1,\mathcal{H}}(Ω)\colon u^{\pm}\neq 0,\, \left\langle \mathcal{E}'(u),u^+ \right\rangle= \left\langle \mathcal{E}'(u),-u^- \right\rangle=0 \Big\},\] whereby $\mathcal{C}$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.

math.AP↗
arXiv2022

On a quasilinear elliptic problem involving the 1-laplacian operator and a discontinuous nonlinearity

In this work, we study a quasilinear elliptic problem involving the 1-laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - β)$. Our approach is based on an analysis of the associated p-laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $β\to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

math.AP↗
arXiv2021

Multiplicity of solutions for a class of quasilinear problems involving the $1$-Laplacian operator with critical growth

The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ \left\{ \begin{array}{l} - Δ_1 u +ξ\frac{u}{|u|} =λ|u|^{q-2}u+|u|^{1^*-2}u, \quad\text{in }Ω, u=0, \quad\text{on } \partialΩ. \end{array} \right. $$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$, $N \geq 2$ and $ξ\in\{0,1\}$. Moreover, $λ> 0$, $q \in (1,1^*)$ and $1^*=\frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $ξ=1$, $Ω= \{x \in \mathbb{R}^N\,:\,r < |x| < r+1\}$, $N\geq 2$, $N \not = 3$ and $r > 0$. In the second one, $Ω$ is a smooth bounded domain, $ξ=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.

math.AP↗
arXiv2018

Multiplicity of solutions for resonant and non-resonant asymptotically linear elliptic problems

Results about existence of a signed ground state solution and multiple solutions (if $f$ is odd with respect to the second variable) are proven for a class of asymptotically linear elliptic problems involving a Carathéodory type nonlinearity satisfying assumptions weaker than (fF) in \cite{GLZ} or $(F_{2})_{+}$ in \cite{CM}. A close relation between the behaviour of $\lim_{|t|\to \infty}f(x, t)/|t|$ and the number of solutions is stablished.

math.AP↗
arXiv2018

Existence and profile of ground-state solutions to a $1-$Laplacian problem in $\mathbb{R}^N$

In this work we prove the existence of ground state solutions for the following class of problems \begin{equation*} \left\{ \begin{array}{ll} \displaystyle - Δ_1 u + (1 + λV(x))\frac{u}{|u|} & = f(u), \quad x \in \mathbb{R}^N, \\ u \in BV(\mathbb{R}^N), & \end{array} \right. \label{Pintro} \end{equation*} \end{abstract} where $λ> 0$, $Δ_1$ denotes the $1-$Laplacian operator which is formally defined by $Δ_1 u = \mbox{div}(\nabla u/|\nabla u|)$, $V:\mathbb{R}^N \to \mathbb{R}$ is a potential satisfying some conditions and $f:\mathbb{R} \to \mathbb{R}$ is a subcritical and superlinear nonlinearity. We prove that for $λ> 0$ large enough there exists ground-state solutions and, as $λ\to +\infty$, such solutions converges to a ground-state solution of the limit problem in $Ω= \mbox{int}( V^{-1}(\{0\}))$.

math.AP↗
arXiv2017

Existence of solution for a class of quasilinear problem in Orlicz-Sobolev space without $Δ_2$-condition

\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -Δ_Φ{u}=f(u), \quad \mbox{in} \quad Ωu=0, \quad \mbox{on} \quad \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a smooth bounded domain, $f:\mathbb{R} \to \mathbb{R}$ is a continuous function verifying some conditions, and $Φ:\mathbb{R} \to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known $Δ_2$-condition, then the Orlicz-Sobolev space $W^{1,Φ}_0(Ω)$ can be non reflexive. As main model we have the function $Φ(t)=(e^{t^{2}}-1)/2$. Here, we study some situations where it is possible to work with global minimization, local minimization and mountain pass theorem, however some estimates are not standard for this type of problem.

math.AP↗
arXiv2017

Radial sign-changing solutions to biharmonic nonlinear Schrödinger equations

In this work we obtain three radial solutions of a biharmonic stationary Schrödinger equation, being one positive, one negative and one that changes sign. The Dual Decompostion Method is used to split the natural second order Sobolev space considered in order to apply the appropriate variational approach.

math.AP↗
arXiv2015

Nodal solutions of a NLS equation concentrating on lower dimensional spheres

In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and concentration of nodal solutions which concentrate around a k - dimensional sphere of RN, where k is between 1 and N-1, as a parameter goes to 0. The radius of such sphere is related with the local minimum of a function which takes into account the potential. Variational methods are used together with the penalization technique in order to overcome the lack of compactness.

math.AP↗
arXiv2013

Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth

In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = λ|u|^{q-2}u+ |u|^{2^*-2}u & \mbox{in $Ω$} u = 0 & \mbox{on $\partial Ω$}, \end{array} \right. $$ where $Ω$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ and $1 < q<2$. In order to employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions of the modified problem is necessary to recover solutions of the original one.

math.AP↗