On the multiplicity and concentration of positive solutions to a Kirchhoff-type problem with competing potentials

Abstract

We study the multiplicity and concentration of positive solutions for a Kirchhoff-type problem involving competing weight potentials and the nonlinearity K(x)|u|p−2u(2 < p < 4) in R3. Such a problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because (PS) sequence may not be bounded. In this paper, by the decomposition of the Nehari manifold and Ljusternik–Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. This is perhaps the first work of studying such problems via the Ljusternik–Schnirelmann category. An interesting point is that unlike some methods that work only for the case of R3 {see the work of G. Li and H. Ye [J. Differ. Equations 257, 566–600 (2014)] and Liu et al. [J. Differ. Equations 266, 5912–5941 (2019)]}, our method is also applicable for the bounded domain case.