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

Abstract

In this work we prove the existence of ground state solutions for the following classwhere λ > 0, ∆ 1 denotes the 1−Laplacian operator which is formally defined by ∆ 1 u = div(∇u/|∇u|), V : R N → R is a potential satisfying some conditions and f : R → R is a subcritical and superlinear nonlinearity. We prove that for λ > 0 large enough there exists ground-state solutions and, as λ → +∞, such solutions converges to a ground-state solution of the limit problem in Ω = int(V −1 ({0})).

Ground state solution for a non-autonomous 1-Laplacian problem involving periodic potential in ℝ N

“…For the case of V (x) = constant and f (x, s) = K (x)g (s), in [6], the existence of ground state bounded variation solution is obtained when V is vanishing potential and suitable conditions under of K , g . Moreover, Alves, Figueiredo and Pimenta in [2] got the similar results when V is steep potential and autonomous f (x, s) = f (s).…”

“…Recently, more and more attention has been paid to p-Laplace operator ∆ p u = div(|∇u| p−2 ∇u), p ∈ [1, +∞). For more details about the applications, see [2]- [7] and the references therein. However, there are few results about the 1-Laplacian problem, partially because of the compactness of sequences as p → 1 occurs in weak norms, and partially because the associated energy functional is no longer smooth and strictly convex.…”

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

“…Before concluding this introduction, for those readers interested in problems involving the 1-Laplacian operator, we would like to cite Alves [1,2], Alves and Pimenta [4], Alves, Figueiredo and Pimenta [5], Bellettini, Caselles and Novaga [14], Chang [20], Demengel [24], Figueiredo and Pimenta [29,30], Mercaldo, Rossi, Segura de León and Trombetti [37], Mercaldo, Segura de León and Trombetti [38], Molino Salas and Segura de León [40], Ortiz Chata and Pimenta [41].…”