On the effect of space dimension and potential on the multiplicity of positive and nodal solutions for Kirchhoff equations

Abstract

The paper is concerned with the multiplicity of positive and nodal solutions for the following Kirchhoff equation arising in the study of string or membrane vibrations−(a+b∫RN|∇u|2dx)Δu+λV(x)u=|u|p−2u,inRN, where N≥3, a,b and λ are positive parameters, the potential V(x)∈C(RN,R+), and 2<p<min⁡{4,2⁎=2NN−2}. Because the complicated competition between nonlocal Kirchhoff term and sub-cubic nonlinearity, the standard nodal Nehari manifold method does not work. As we shall see, under some suitable assumptions on potential V(x), we prove that the above problem admits at least two positive solutions and two nodal solutions, where a novel constraint manifold and some analytical tricks are introduced to overcome this obstacle. An interesting finding is that the space dimension and potential have a strong influence on the number of positive or nodal solutions. Moreover, we also give the asymptotic behavior of solutions as λ→+∞ and b→0+. These results improve and generalize the previous results in the literature.