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Nonlinear scalar field equations in \(\mathbb{R}^N\): Mountain pass and symmetric mountain pass approaches. (English) Zbl 1203.35106

Summary: We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in \(\mathbb R^N\):
\[ -\Delta u= g(u) \quad\text{in }\mathbb R^N, \qquad u\in H^1(\mathbb R^N). \]
We give an extension of the existence results due to H. Berestycki, T. Gallouet and O. Kavian [C. R. Acad. Sci., Paris, Sér. I 297, 307–310 (1983; Zbl 0544.35042)].
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.

MSC:

35J61 Semilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 0544.35042
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