We consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$:(*)

\begin{align}\left\{\begin{array}{ll}-\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\\\u=0, & x\in\partial \Omega,\\\end{array}\right.\end{align}

$b\geq0$

$\partial\Omega\neq\emptyset$

$\mathbb{R}^{3}\backslash\Omega$

$u\in H_{0}^{1}(\Omega)$

$f\in C^1(\mathbb{R},\mathbb{R})$

\begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}

$H^1(\mathbb R^3)$

$\mathbb R^3\setminus \Omega$

We consider the Null Mass nonlinear field equation (𝒫)

$$\left\{ {\matrix{ {-\Delta u = f(u){\rm in}\;\;\Omega } \hfill \hfill \hfill \hfill \cr {u > 0} \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \hfill \cr {u \vert_{\partial \Omega } = 0} \cr } } \right.$$

${\open R}^N \setminus \Omega $

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 $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

$\unicode[STIX]{x1D6FE}=0$

$\unicode[STIX]{x1D6FE}=1$

We establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as under quite general assumptions upon f and g.

In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying ${\mathbb R}^2$. We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.