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Using variational methods and depending on a parameter
$\unicode[STIX]{x1D706}$
we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$
:
for the subcritical case (
$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$
$\unicode[STIX]{x1D6FE}=0$
) and also for the critical case (
$\unicode[STIX]{x1D6FE}=1$
).
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