Multi-bump solutions for a class of Kirchhoff type problems with critical growth in $\mathbb{R}^N$
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Kirchhoff type problems, potential well, multi-bump solutions, variational methodsAbstrakt
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 $p=2$.Pobrania
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2016-05-29
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LIANG, Sihua & ZHANG, Jihui. Multi-bump solutions for a class of Kirchhoff type problems with critical growth in $\mathbb{R}^N$. Topological Methods in Nonlinear Analysis [online]. 29 maj 2016, T. 48, nr 1, s. 71–101. [udostępniono 22.7.2024].
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