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

2016/10/24 by Giovany M. Figueiredo, Figueiredo, G. M., Marcos T. O. Pimenta +1
Computer Science · Mathematics · #35J62 #35J93 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1610.07369

openalex publication_date 2016/10/24 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

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 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(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 ℝ^N. This seems to be the very first work dealing with stationary problems involving this operator in the whole space.

Citations

• Existence of solitary waves in higher dimensions

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