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Chapter https://doi.org/10.1007/978-3-031-85757-7_14 is devoted to the proof of Theorem 1.9 . Namely, we prove that the semilinear Robin problem 1.2 \(\displaystyle \begin {cases} Au = \lambda \left (1 + \varepsilon u\right )^{m}\,\exp \left [\frac {u}{1+\varepsilon u}\right ] &\mbox{in}\ D, \\ B{\boldsymbol \gamma }u = a(x^{\prime }) \frac {\partial u}{\partial {\boldsymbol \nu }} + b(x^{\prime })u = 0 &\mbox{on}\ \partial D \end {cases} \) is uniquely solvable for \(\lambda \) sufficiently small if \(0 < \varepsilon < (1/(1+\sqrt {1-m}))^{2}\) under the conditions (H.1) and (H.2) (see Fig. 1.7 ).

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Proof of Theorem 1.9 (Unique Solvability for \(\lambda \) Sufficiently Small)

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  • 发表日期：

    2025年

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  • Kazuaki Taira

摘要

Chapter https://doi.org/10.1007/978-3-031-85757-7_14 is devoted to the proof of Theorem 1.9 . Namely, we prove that the semilinear Robin problem 1.2 \(\displaystyle \begin {cases} Au = \lambda \left (1 + \varepsilon u\right )^{m}\,\exp \left [\frac {u}{1+\varepsilon u}\right ] &\mbox{in}\ D, \\ B{\boldsymbol \gamma }u = a(x^{\prime }) \frac {\partial u}{\partial {\boldsymbol \nu }} + b(x^{\prime })u = 0 &\mbox{on}\ \partial D \end {cases} \) is uniquely solvable for \(\lambda \) sufficiently small if \(0 < \varepsilon < (1/(1+\sqrt {1-m}))^{2}\) under the conditions (H.1) and (H.2) (see Fig. 1.7 ).