This work discusses the existence of the limit as p goes to 1 of the nontrivial solutions to the one-dimensional problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( |u_x|^{p-2} u_x\right) _x = \lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u}&{} \quad 0< x < 1\\ u(0)=u(1)=0, &{} \end{array}\right. } \end{aligned}$$

where \(\lambda \) is a positive parameter and the exponents p, q satisfy \(1< p < q\). We prove that solutions do converge to a limit function, which solves in a proper sense a Dirichlet problem involving the 1-Laplacian operator.

中文翻译：

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一维逻辑反应和p-拉普拉斯扩散，随着p趋于一

这项工作讨论了当p变为一维问题的非平凡解中的1时极限的存在：

$$ \ begin {aligned} {\ left \ {\ begin {array} {ll}-\ left（| u_x | ^ {p-2} u_x \ right）_x = \ lambda | {u} | ^ {{p } -2} {u}-| {u} | ^ {{q} -2} {u}＆{} \ quad 0 <x <1 \\ u（0）= u（1）= 0，＆{ } \ end {array} \ right。} \ end {aligned} $$

其中\（\ lambda \）是一个正参数，指数p，  q满足\（1 <p <q \）。我们证明了解确实收敛到极限函数，该极限函数在适当的意义上解决了涉及1-Laplacian算子的Dirichlet问题。