We study the following Kirchhoff type equation: −a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN, $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N =3, a,b>0 $ a,b \gt 0 $ , 1<q<2<p<min{4,2∗} $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2 N /( N − 2), k ∈ C (ℝ N ) is bounded and m ∈ L p /( p − q ) (ℝ N ). By imposing some suitable conditions on functions k ( x ) and m ( x ), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H1(RN)↪Lr(RN)(2≤r<2∗) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

中文翻译：

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一类具有不定非线性的基尔霍夫型方程的多重正解

我们研究以下基尔霍夫型方程：-a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN, $$\begin {等式*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\对)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R }^{N}, \end{array} \end{equation*}$$ 其中 N =3, a,b>0 $ a,b \gt 0 $ , 1<q<2<p<min{4, 2∗} $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2 N /( N − 2), k ∈ C (ℝ N )是有界的，并且 m ∈ L p /( p − q ) (ℝ N )。通过对函数 k ( x ) 和 m ( x ) 施加一些合适的条件，我们首先介绍了一些新技术来恢复 Sobolev 嵌入 H1(RN)↪Lr(RN)(2≤r<2∗) $ H 的紧致性^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ;