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

Abstract

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