| Summary: | Abstract In this paper, we consider the following p-Kirchhoff equation: P [M(∥u∥p)]p−1(−Δpu+V(x)|u|p−2u)=f(x,u),x∈RN, $$\begin{aligned} \bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \bigl(-\Delta_{p} u+V(x) \vert u \vert ^{p-2}u \bigr)=f(x,u), \quad x\in{\mathbb {R}}^{N}, \end{aligned}$$ where f(x,u)=λg(x)|u|q−2u+h(x)|u|r−2u,1<q<p<r<p∗ $f(x,u)=\lambda g(x)|u|^{q-2}u+h(x)|u|^{r-2}u,1< q< p< r< p^{*}$ ( p∗=NpN−p $p^{*}=\frac{Np}{N-p}$ if N≥p,p∗=∞ $N\ge p,p^{*}=\infty$ if N≤p $N\le p$). Using variational methods, we prove that, under proper assumptions, there exist λ0,λ1>0 $\lambda_{0},\lambda_{1}>0$ such that problem (P) has a solution for all λ∈[0,λ0) $\lambda\in[0,\lambda_{0})$ and has a sequence of solutions for all λ∈[0,λ1) $\lambda\in[0,\lambda_{1})$.
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