Solutions with many mixed positive and negative interior spikes for a semilinear Neumann problem

We study the existence of sign-changing multiple interior spike solutions for the following Neumann problem $$\varepsilon^2\Delta v-v+f(v) = 0 \,\, {\rm in} \,\, \Omega, \quad \frac{\partial v}{\partial \nu} = 0 \,\, {\rm on} \,\, \partial \Omega,$$ where ?? is a smooth bounded domain of ${\mathbb {R}^N}$ , ?? is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. No symmetry on ?? is assumed. To our knowledge, only positive interior peak solutions have been obtained for this problem and it remains a question whether or not multiple interior peak solutions with mixed positive and negative peaks exist. In this paper we assume that ?? is a two-dimensional strictly convex domain and, provided that k is sufficiently large, we construct a (k?+?1)-peak solutions with k positive interior peaks aligned on a closed curve near ??? and 1 negative interior peak located in a more centered part of ??.