Nonlinear scalar field equations in $\mathbb R^{N}$: mountain pass and symmetric mountain pass approaches

Abstract

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb{R}^N$: \begin{gather*} -\Delta u=g(u) \quad \text{in }\mathbb{R}^N,\\ u\in H^1(\mathbb R^N). \end{gather*} We give an extension of the existence results due to H. Berestycki, T. Gallouët and O. Kavian [Equations de Champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Ser. I Math. 297, 307–310].

We take a mountain pass approach in $H^1(\mathbb{R}^N)$ and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity.