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

Jun Hirata, Norihisa Ikoma, Kazunaga Tanaka

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 [< i> Equations de Champs scalaires euclidiens
non linéaires dans le plan< /i> , C. R. Acad. Sci. Paris Ser. I Math. < b> 297< /b> , 307–310].< /p>

< p> 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.

Keywords


Nonlinear scalar field equations; radially symmetric solutions; minimax methods

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