Multi-bump solutions for a class of Kirchhoff type problems with critical growth in $\mathbb{R}^N$

Abstract

Using variational methods, we establish existence of multi-bump solutions for a class of Kirchhoff type problems $$ -\bigg(1+b\int_{\mathbb{R}^N}|\nabla u|^pdx\bigg)\Delta_pu + (\lambda V(x) + Z(x))u^{p-1} = \alpha f(u) + u^{p^\ast-1}, $$ where $f$ is a continuous function, $V, Z\colon \mathbb{R}^N \rightarrow\mathbb{R}$ are continuous functions verifying some hypotheses. We show that if the zero set of $V$ has several isolated connected components $\Omega_1,\ldots,\Omega_k$ such that the interior of $\Omega_i$ is not empty and $\partial\Omega_i$ is smooth, then for $\lambda > 0$ large enough there exists, for any non-empty subset $\Gamma \subset \{1,\ldots,k\}$, a bump solution trapped in a neighbourhood of $\bigcup\limits_{j\in \Gamma}\Omega_j$. The results are also new for the case $p=2$.