The paper deals with the study of the problem
where is a smooth bounded domain in and is a subcritical exponent. This problem may be viewed as a prototype of pattern formation in biology and it is also related to the steady state problem for a chemotactic aggregation model with logarithmic sensitivity. The above problem has been intensively studied by several authors (Lin, Ni, Takagi, Gui, Wang and others) who obtained strong results on the existence of a least energy solution, the study of asymptotics as etc. The author applies a method which is originally due to Li and Nirenberg in order to construct solutions with single peak and multi-peak on . Another result establishes the existence of multi-peak solutions with a prescribed number of local maximum points in .
The paper is interesting and the powerful methods developed by the authors allow the study of wide classes of boundary value problems.