Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

Abstract

AbstractWe study the following Kirchhoff type equation:−a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN,$$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$whereN=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^{*}\} $, 2≤=2N/(N − 2),k ∈ C(ℝN) is bounded andm ∈ Lp/(p−q)(ℝN). By imposing some suitable conditions on functionsk(x) andm(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embeddingH1(RN)↪Lr(RN)(2≤r<2∗)$ 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.