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Nonlinear scalar field equations with Berestycki–Lions’ nonlinearity on large domains

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Abstract

We prove the existence of solutions for the following semilinear elliptic equation:

$$\begin{aligned} - \Delta u = g(u) \quad \text {in } \Omega , \quad u \in H^1_0(\Omega ). \end{aligned}$$

Here \(\Omega \) is a suitable large domain and g satisfies the completely same conditions as Berestycki–Lions’ conditions. Those conditions of g are known as “almost sufficient and necessary conditions” to the existence of nontrivial solutions of the equations defined in \(\mathbb {R}^N\). The main difficulty to prove the existence of solutions of the equation is that we can not obtain bounded Palais–Smale sequences. To overcome this difficulty, we modify the corresponding functional, which is a new idea introduced in our previous paper.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers JP15K17567, JP18K03356, JP20K03691.

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Authors and Affiliations

  1. Department of Mathematics, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama, 338-8570, Japan

    Yohei Sato

  2. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo, 152-8551, Japan

    Masataka Shibata

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  1. Yohei Sato
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  2. Masataka Shibata
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Correspondence to Yohei Sato.

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Sato, Y., Shibata, M. Nonlinear scalar field equations with Berestycki–Lions’ nonlinearity on large domains. J Elliptic Parabol Equ 6, 711–731 (2020). https://doi.org/10.1007/s41808-020-00079-5

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  • DOI: https://doi.org/10.1007/s41808-020-00079-5

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