In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $\lambda$ and on $\kappa$. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $\lambda\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{\kappa+1}-1$ solutions for sufficiently negative $\lambda$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $\lambda$ and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.

翻译：本文研究了一个推广Moore与Nehari在文献[43]中所研究问题的超线性一维椭圆边值问题。具体而言，我们处理非线性项前具有任意数量κ ≥ 1个消失区域的逐段常权函数。我们从分析和数值角度研究了正解的数量（该数量依赖于参数λ的值和κ）。我们的主要结果具有双重性。一方面，我们解析研究了在权函数消失区域中当λ↓-∞时解的行为。该结果引导我们猜想：当λ足够负时，存在2^{κ+1}-1个解。另一方面，我们通过数值模拟结果支持这一猜想，这些模拟还揭示了λ全局分支图的结构以及正解的轮廓特征。最后，我们补充了其他数值结果，表明相同的高重数性对更大类别的权函数依然成立——这类权函数可任意接近存在正解唯一性的情形。