Solutions to a nonlinear Neumann problem in three-dimensional exterior domains (Q1671202)

The authors prove the existence of multipeak solutions to a nonlinear elliptic Neumann problem involving nearly critical Sobolev exponent, in three-dimensional exterior domains. More precisely, consider the problem: \[\begin{cases} -\Delta u=u^{2^\star-1-\varepsilon},\; u>0 \quad&\text{ in } \mathbb{R}^N\setminus\Omega, \\ \frac{\partial u}{\partial \nu}=0 &\text{ on }\partial\Omega,\end{cases} \] where \(\Omega\) is a smooth and bounded domain in \( \mathbb{R}^N\), \(N\geq 3\), such that \( \mathbb{R}^N\setminus\Omega\) is connected, \(2^\ast=2N/(N-2)\) is the critical Sobolev exponent and \(\varepsilon\) is a strictly positive number, assumed to be small. This problem could be considered asymptotically critical. In fact the techniques used in its study as \(\varepsilon\) goes to zero are the same as those to be used in the case of critical nonlinearities. The existence of multipeak solution was studied by \textit{S. Yan} [Adv. Differ. Equ. 7, No. 8, 919--950 (2002; Zbl 1208.35064)] in dimensions \(N\geq 4\). In this paper the authors complete these results considering the case of dimension \(N=3\). Specifically, by considering the case of positive local maximum of the mean curvature of \(\partial \Omega\) (resp. local minimum) they obtain the existence of a multipeak solution of the previous problem for small \(\varepsilon>0\). An asymptotic behaviour of this solutions as \(\varepsilon\) goes to zero is also obtained.