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

Abstract

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

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

“…We would like point out that there is in the literature few papers involving the 1-Laplacian operator in the whole R N . In fact the authors know only the papers due to Alves and Pimenta [1] and Figueiredo and Pimenta [13,14]. In [1], Alves and Pimenta have studied the existence and concentration of solution for the following class of problem…”

Existence of solution for a class of heat equation involving the p(x) Laplacian with triple regime

“…Indeed, the energy functional is not C 1 and we find some hindrances to show that functionals defined in this space satisfy compactness conditions like the Palais-Smale. Meanwhile, a lot of attention has been paid recently to that space, for example see [3,[8][9][10]12,13,16,22,29,30] and references therein, since it is the natural environment in which minimizers of many problems can be found, especially in problems involving interesting physical situations, in capillarity theory and existence of minimal surfaces and as application of variational approach to image restoration.…”

Strauss’ and Lions’ Type Results in $${BV (\mathbb{R}^N}$$ B V ( R N ) with an Application to an 1-La

“…In [14] it is studied a problem related to (1), but in the presence of vanishing potentials, where it is proved the existence of a nontrivial ground-state solution. In [1] the authors show the existence and concentration of a sequence of solutions of a singularly perturbed version of (1), with a potential satisfying geometrical conditions like in the celebrated paper [20].…”