Existence and profile of ground-state solutions to a 1-Laplacian problem in ℝ^N

2018/04/18 by Alves, Claudianor O., Figueiredo, Giovany M., Pimenta, Marcos T. O.
#35J62 #35J93 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1804.07618

Abstract

In this work we prove the existence of ground state solutions for the following class of problems \ - Δ₁ u + (1 + λV(x))(u)/(|u|) · amp; = f(u), x ∈ ℝ^N,
u ∈ BV(ℝ^N), · amp; . \endabstract where λ> 0, Δ₁ denotes the 1-Laplacian operator which is formally defined by Δ₁ 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\)).

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