Multiple positive solutions for a logarithmic Kirchhoff type problem in [formula omitted

Abstract

By using the variational methods and the maximum principle, we investigate the following logarithmic Kirchhoff type problem (P)a+b∫R3|∇u|2+V(x)u2dx[−Δu+V(x)u]=|u|p−1ulog|u|,where a>0, b>0, 1<p<3. Under some assumptions on the potential function V(x), we prove that (P) has only trivial solution for large b>0; and (P) owns two positive solutions for small b>0. Moreover, we obtain that the value of parameter b determines the number of positive solutions under the effects of both local term and nonlocal term.