A remark on least energy solutions in 𝐑^{𝐍}

Abstract

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R N \mathbf {R}^N : − Δ u = g ( u ) , u ∈ H 1 ( R N ) , \begin{equation*} -\Delta u = g(u),\, u \in H^1(\mathbf {R}^N), \end{equation*} where N ≥ 2 N\geq 2 . Without the assumption of the monotonicity of t ↦ g ( t ) t t\mapsto \frac {g(t)}{t} , we show that the mountain pass value gives the least energy level.