arXiv subjects

M. T. O. Pimenta

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

4 recordsLinked to original sources

arXiv2017

Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions

In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.

math.AP↗

arXiv2017

On existence and concentration of solutions to a class of quasilinear problems involving the $1-$Laplace operator

In this work we use variational methods to prove results on existence and concentration of solutions to a problem in $\mathbb{R}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV(\mathbb{R}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.

math.AP↗

arXiv2016

Existence of bounded variation solutions for a $1-$Laplacian problem with vanishing potentials

In this work it is studied a quasilinear elliptic problem in the whole space $\mathbb{R}^N$ involving the $1-$Laplacian operator, with potentials which can vanish at infinity. The Euler-Lagrange functional is defined in a space whose definition resembles $BV(\mathbb{R}^N)$ and, in order to avoid working with extensions of it to some Lebesgue space, we state and prove a version of the Mountain Pass Theorem without the Palais-Smale condition to Lipschitz continuous functionals.

math.AP↗

arXiv2016

Strauss' and Lions' type results in $BV(\mathbb{R}^N)$ with an application to $1-$Laplacian problem

In this work we state and prove versions of some classical results, in the framework of functionals defined in the space of functions of bounded variation in $\mathbb{R}^N$. More precisely, we present versions of the Radial Lemma of Strauss, the compactness of the embeddings of the space of radially symmetric functions of $BV(\mathbb{R}^N)$ in some Lebesgue spaces and also a version of the Lions Lemma, proved in his celebrated paper of 1984. As an application, we state and prove a version of the Mountain Pass Theorem without the Palais-Smale condition in order to get existence of a ground-state bounded variation solution of a quasilinear elliptic problem involving the $1-$Laplacian operator in $\mathbb{R}^N$. This seems to be the very first work dealing with stationary problems involving this operator in the whole space.

math.AP↗