TY - JOUR AU - Shibata, Tetsutaro PY - 2022/07/30 Y2 - 2024/03/29 TI - Asymptotic behavior of bifurcation curve of nonlinear eigenvalue problem with logarithmic nonlinearity JF - Topological Methods in Nonlinear Analysis JA - TMNA VL - 60 IS - 1 SE - DO - 10.12775/TMNA.2021.040 UR - https://apcz.umk.pl/TMNA/article/view/39287 SP - 99 - 110 AB - We study the following nonlinear eigenvalue problem$$-u''(t) = \lambda u(t)^p\log(1+u(t)), \quad u(t) > 0,\quad t \in I := (-1,1), \quad u(\pm 1) = 0,$$%where $p \ge 0$ is a given constant and $\lambda > 0$ is a parameter.It is known that, for any given $\alpha > 0$, there exists a unique classical solution pair$(\lambda(\alpha), u_\alpha)$ with $\alpha = \Vert u_\alpha\Vert_\infty$.We establish the asymptotic formulas for the bifurcation curves $\lambda(\alpha)$ and the shapeof solution $u_\alpha$ as $\alpha \to \infty$ and $\alpha \to 0$. ER -