We consider the nonlinear eigenvalue problem $$ [D(u)u']' + \lambda f(u) = 0, \ \ u(t) > 0, \ \ t \in I := (0,1), \ \ u(0) = u(1) = 0, $$ where $D(u) = u^p$, $f(u) = u^{q} + \sin (u^n)$ and $\lambda > 0$ is a bifurcation parameter. Here, $p \ge 0$, $n > 0$ and $q > 0$ are given constants and $k:=(p + q + 1)/2 \in \mathbb{N}$. This equation is motivated by the mathematical model of animal dispersal and invasion and $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(\alpha)$. We establish new precise asymptotic expansion formulas for $\lambda(\alpha)$ as $\alpha \to \infty$. In particular, we obtain the precise asymptotic behavior of $\lambda(\alpha)$ which tends to $0$ with oscillation as $\alpha \to \infty$.

中文翻译：

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具有非线性扩散的ODEs振动分岔曲线的渐近展开。

我们考虑非线性特征值问题$$ [D（u）u']'+ \ lambda f（u）= 0，\ \ u（t）> 0，\ \ t \ in I：=（0,1）， \ \ u（0）= u（1）= 0，$$其中$ D（u）= u ^ p $，$ f（u）= u ^ {q} + \ sin（u ^ n）$和$ \ lambda> 0 $是分叉参数。在这里，给定$ p \ ge 0 $，$ n> 0 $和$ q> 0 $常数，并且$ k：=（p + q + 1）/ 2 \ in \ mathbb {N} $。该方程由动物扩散和入侵的数学模型驱动，而$ \ lambda $由与$ \相关联的解$ u_ \ lambda $的最大范本$ \ alpha = \ Vert u_ \ lambda \ Vert_ \ infty $参数化lambda $并写为$ \ lambda = \ lambda（\ alpha）$。我们为$ \ lambda（\ alpha）$建立新的精确渐近展开公式，公式为$ \ alpha \至\ infty $。尤其是，