Multispike Solutions for a slightly subcritical elliptic problem with non-power nonlinearity
Mohamed Ben Ayed, Habib Fourti, Rabeh Ghoudi · Discrete and Continuous Dynamical Systems · 2022
In this paper, we are concerned with the following elliptic equation$ \left\{\begin{array}{rrl}-\Delta u& = & |u|^{4/(n-2)}u/[\ln (e+|u|)]^\varepsilon \hbox{ in } \Omega,\\ u& = &0 \hbox{ on }\partial \Omega, \end{array} \right. $where $ \Omega $ is a smooth bounded open domain in $ \mathbb{R}^n, \ n\geq 4 $ and $ \varepsilon>0 $. By using a Ljapunov-Schmidt reduction method, Clapp et al. in Journal of Diff. Eq. (Vol 275) proved that there exists a single-peak positive solution for small $ \varepsilon $. This solution blows up at a non-degenerate critical point of the Robin function as $ \varepsilon $ goes to $ 0 $.Here we construct positive as well as changing sign solutions concentrated at several points inside the domain $ \Omega $ at the same time. More precisely, we build solutions which blow up (positively or negatively) at distinct points which form a non-degenerate critical point of a function defined explicitly in terms of the Green function and its regular part. Our proof follows the finite reduction method introduced by Bahri, Li and Rey in Calc. Var. and PDE (Vol 3).