In this paper, we study a Kirchhoff type elliptic problem, \[ \begin{cases} \displaystyle - \Big ( 1+\alpha \int_{\Omega}|\nabla u|^2dx \Big ) \Delta u =\lambda u^q+u^3,\ u > 0 \text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \] where $\Omega\subset \mathbb{R}^4$ is a bounded domain with smooth boundary $\partial \Omega$ and we assume $\alpha,\lambda > 0$ and $1\le q < 3$. For $q=1$, we prove the existence of possibly multiple solutions for $\alpha > 0$ and $\lambda\ge \lambda_1$ in suitable intervals, where $\lambda_1 > 0$ is the first eigenvalue of $-\Delta$ on $\Omega$. On the other hand for $q\in (1,3)$, we show the existence of a solution for all small $\alpha > 0$ and all $\lambda > 0$. We establish the results by the method of the Nehari manifold and the concentration compactness analysis for Palais-Smale sequences.

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第四维Kirchhoff型椭圆型问题的正解的存在性和多重性

在本文中，我们研究了Kirchhoff型椭圆问题，\ [\ begin {cases} \ displaystyle-\ Big（1+ \ alpha \ int _ {\ Omega} | \ nabla u | ^ 2dx \ Big）\ Delta u = \ lambda u ^ q + u ^ 3，\ u> 0 \ text {in} \ Omega，\\ u = 0 \ text {on} \ partial \ Omega，\ end {cases} \]其中$ \ Omega \ subset \ mathbb {R} ^ 4 $是一个具有光滑边界$ \ partial \ Omega $的有界域，我们假定$ \ alpha，\ lambda> 0 $和$ 1 \ le q <3 $。对于$ q = 1 $，我们证明在适当的时间间隔内存在$ \ alpha> 0 $和$ \ lambda \ ge \ lambda_1 $可能存在多个解，其中$ \ lambda_1> 0 $是$-\的第一个特征值$ \ Omega $上的Delta $。另一方面，对于$ q \ in（1,3）$，我们显示了所有小的$ \ alpha> 0 $和所有$ \ lambda> 0 $的解的存在。