Abstract In this paper, we study following Kirchhoff type equation: −a+b∫Ω|∇u|2dxΔu=f(u)+h in Ω,u=0 on ∂Ω. $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis.

中文翻译：

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Kirchhoff 型方程的存在性、多重性和不存在性结果

摘要 在本文中，我们研究了以下 Kirchhoff 型方程：−a+b∫Ω|∇u|2dxΔu=f(u)+h in Ω,u=0 on ∂Ω。$$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d} x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}。\end{array} \right。\end{array}$$ 我们首先考虑Ω ⊂ ℝ3 是有界域的情况。上述方程至少存在一或两个正解是通过使用单调性技巧获得的。不存在标准也通过相应的 Pohožaev 身份建立。特别是，我们展示了 3-sublinear 情况以及临界情况的不存在属性。在非线性的一般假设下，