In this paper, we study the following nonlocal problem: $$ \textstyle\begin{cases} - (a-b \int _{\Omega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= \lambda \vert u \vert ^{q-2}u, & x\in \Omega , \\ u=0, & x\in \partial \Omega , \end{cases} $$ where Ω is a smooth bounded domain in $\mathbb{R}^{N}$ with $N\ge 3$ , $a,b>0$ , $1< q<2$ and $\lambda >0$ is a parameter. By virtue of the variational method and Nehari manifold, we prove the existence of multiple positive solutions for the nonlocal problem. As a co-product of our arguments, we also obtain the blow-up and the asymptotic behavior of these solutions as $b\searrow 0$ .

中文翻译：

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一类非局部问题的正解的新多重性

在本文中，我们研究以下非局部问题：$$ \ textstyle \ begin {cases}-（ab \ int _ {\ Omega} \ vert \ nabla u \ vert ^ {2} \，dx）\ Delta u = \ lambda \ vert u \ vert ^ {q-2} u，＆x \ in \ Omega，\\ u = 0，＆x \ in \ partial \ Omega，\ end {cases} $$其中Ω是光滑有界域在$ \ mathbb {R} ^ {N} $中，其中$ N \ ge 3 $，$ a，b> 0 $，$ 1 <q <2 $和$ \ lambda> 0 $是一个参数。借助于变分方法和Nehari流形，我们证明了非局部问题存在多个正解。作为我们争论的共同产物，我们还获得了这些解的爆炸和渐近行为，如$ b \ searrow 0 $。