Multiple interior peak solutions for some singularly perturbed Neumann problems.

*(English)* Zbl 1061.35502
Summary: We consider the problem ${\u03f5}^{2}{\Delta}u-u+f\left(u\right)=0,$ $u>0$, in ${\Omega},\partial u/\partial \nu =0$ on $\partial $, where ${\Omega}$ is a bounded smooth domain in ${\mathbb{R}}^{N},\u03f5>0$ is a small parameter, and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions that concentrate, as $\u03f5$ approaches zero, at a critical point of the mean curvature function $H\left(P\right),\phantom{\rule{4pt}{0ex}}P\in \partial {\Omega}$. It is also proved that this equation has single interior spike solutions at a local maximum point of the distance function $d(P,\partial {\Omega}),P\in {\Omega}$. In this paper, we prove the existence of interior $K$-peak $(K\ge 2)$ solutions at the local maximum points of the following function: $\varphi ({P}_{1},{P}_{2},\cdots ,{P}_{K})={min}_{i,k,l=1,\cdots ,K;k\ne l}(d({P}_{i},\partial {\Omega}),\frac{1}{2}|{P}_{k}-{P}_{l}\left|\right)$. We first use the Lyapunov-Schmidt reduction method to reduce the problem to a finite-dimensional problem. Then we use a maximizing procedure to obtain multiple interior spikes. The function $\varphi ({P}_{1},\cdots ,{P}_{K})$ appears naturally in the asymptotic expansion of the energy functional.

##### MSC:

35J25 | Second order elliptic equations, boundary value problems |

35B25 | Singular perturbations (PDE) |

47J30 | Variational methods (nonlinear operator equations) |

47N20 | Applications of operator theory to differential and integral equations |