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

DOI: http://dx.doi.org/10.12775/TMNA.2016.040

#### 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 $p=2$.

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 $p=2$.

#### Keywords

Kirchhoff type problems; potential well; multi-bump solutions; variational methods

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