Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-01-18 , DOI: 10.1007/s00033-020-01466-7 Vincenzo Ambrosio , Dušan Repovš
In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem:
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap W^{1, q}({\mathbb {R}}^{N}), \quad u>0 \text{ in } {\mathbb {R}}^{N}, \end{array} \right. \end{aligned}$$where \(\varepsilon >0\) is a small parameter, \(1< p<q<N\), \(\Delta _{r}u={{\,\mathrm{div}\,}}(|\nabla u|^{r-2}\nabla u)\), with \(r\in \{p, q\}\), is the r-Laplacian operator, \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a continuous function satisfying the global Rabinowitz condition, and \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small \(\varepsilon \).
中文翻译：
$$ {\ mathbb {R}} ^ {N} $$ RN中（p，q）-Laplacian问题的多重性和集中性结果
在本文中，我们研究以下（p， q）-Laplacian问题的正解的多重性和集中性：
$$ \ begin {aligned} \ left \ {\ begin {array} {ll}-\ Delta _ {p} u-\ Delta _ {q} u + V（\ varepsilon x）\ left（| u | ^ { p-2} u + | u | ^ {q-2} u \ right）= f（u）＆{} \ text {in} {\ mathbb {R}} ^ {N}，\\ u \ in W ^ {1，p}（{\ mathbb {R}} ^ {N}）\ cap W ^ {1，q}（{\ mathbb {R}} ^ {N}），\ quad u> 0 \ text {在} {\ mathbb {R}} ^ {N}中，\ end {array} \ right。\ end {aligned} $$其中\（\ varepsilon> 0 \）是一个小参数，\（1 <p <q <N \），\（\ Delta _ {r} u = {{\，\ mathrm {div} \，}}（ | \ nabla u | ^ {r-2} \ nabla u）\）与\（r \ in \ {p，q \} \）在一起，是r-拉普拉斯算子\（V：{\ mathbb {R }} ^ {N} \ rightarrow {\ mathbb {R}} \）是一个满足全局Rabinowitz条件的连续函数，并且\（f：{\ mathbb {R}} \ rightarrow {\ mathbb {R}} \）是具有亚临界增长的连续功能。使用适当的变分参数和Ljusternik–Schnirelmann类别理论，我们研究了正解的数量与该集合的拓扑之间的关系，其中V在小情况下达到最小值\（\ varepsilon \） 。
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