Asymptotic behavior of bifurcation curve of nonlinear eigenvalue problem with logarithmic nonlinearity

Abstract

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 shape of solution $u_\alpha$ as $\alpha \to \infty$ and $\alpha \to 0$.