## Topological Methods in Nonlinear Analysis

### 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 $p=2$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 1 (2016), 71-101.

Dates
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.tmna/1475266372

Digital Object Identifier
doi:10.12775/TMNA.2016.040

Mathematical Reviews number (MathSciNet)
MR3561423

Zentralblatt MATH identifier
1373.35125

#### Citation

Liang, Sihua; Zhang, Jihui. Multi-bump solutions for a class of Kirchhoff type problems with critical growth in $\mathbb{R}^N$. Topol. Methods Nonlinear Anal. 48 (2016), no. 1, 71--101. doi:10.12775/TMNA.2016.040. https://projecteuclid.org/euclid.tmna/1475266372

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